problem stringlengths 11 5.47k | reasoning stringlengths 708 124k | deepseek_solution stringlengths 0 10.3k | ground_truth_solution stringlengths 97 9.16k |
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Two players alternately replace the stars in the expression
\[*x^{2000}+*x^{1999}+...+*x+1 \]
by real numbers. The player who makes the last move loses if the resulting polynomial has a real root $t$ with $|t| < 1$ , and wins otherwise. Give a winning strategy for one of the players. | Okay, so there's this game between two players who take turns replacing the stars in the polynomial \[*x^{2000} + *x^{1999} + \dots + *x + 1\] with real numbers. The last player to make a move loses if the resulting polynomial has a real root \( t \) with \( |t| < 1 \); otherwise, they win. I need to figure out a winni... | The problem involves two players alternately replacing the stars in the polynomial \(*x^{2000} + *x^{1999} + \dots + *x + 1\) with real numbers. The player who makes the last move loses if the resulting polynomial has a real root \(t\) with \(|t| < 1\). The goal is to determine a winning strategy for one of the players... | 1. **Identify the players and their roles**: Let \( A \) be the first player and \( B \) be the second player. The players alternately replace the stars in the polynomial \( *x^{2000} + *x^{1999} + \ldots + *x + 1 \) with real numbers. The player who makes the last move loses if the resulting polynomial has a real root... |
A *mirrored polynomial* is a polynomial $f$ of degree $100$ with real coefficients such that the $x^{50}$ coefficient of $f$ is $1$ , and $f(x) = x^{100} f(1/x)$ holds for all real nonzero $x$ . Find the smallest real constant $C$ such that any mirrored polynomial $f$ satisfying $f(1) \ge C$ has a co... | Okay, let's try to tackle this problem about mirrored polynomials. So, the question is asking for the smallest real constant C such that any mirrored polynomial f with f(1) ≥ C must have a complex root z with |z| = 1.
First, let me make sure I understand what a mirrored polynomial is. The problem states that it's a p... | To solve the problem, we analyze mirrored polynomials of degree 100 with real coefficients and the coefficient of \( x^{50} \) equal to 1. These polynomials satisfy \( f(x) = x^{100}f(1/x) \), implying roots come in reciprocal pairs. The goal is to find the smallest real constant \( C \) such that if \( f(1) \ge C \), ... | 1. **Understanding the Problem:**
We are given a mirrored polynomial \( f \) of degree 100 with real coefficients such that the coefficient of \( x^{50} \) is 1, and \( f(x) = x^{100} f(1/x) \) for all nonzero real \( x \). We need to find the smallest real constant \( C \) such that if \( f(1) \geq C \), then \( f ... |
For positive real numbers $a, b, c, d,$ satisfying the following conditions: $a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$ , prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$ | Okay, let's try to tackle this problem. So, we have positive real numbers a, b, c, d with two conditions: a(c² - 1) = b(b² + c²), and d ≤ 1. We need to prove that d(a√(1 - d²) + b²√(1 + d²)) ≤ (a + b)c / 2. Hmm, that seems a bit complex, but let's break it down step by step.
First, let's make sure I understand all the... | Given positive real numbers \(a, b, c, d\) satisfying the conditions \(a(c^2 - 1) = b(b^2 + c^2)\) and \(d \leq 1\), we need to prove that:
\[
d(a \sqrt{1 - d^2} + b^2 \sqrt{1 + d^2}) \leq \frac{(a + b)c}{2}
\]
### Key Steps and Analysis:
1. **Given Condition**:
The equation \(a(c^2 - 1) = b(b^2 + c^2)\) can be r... | 1. Given the conditions \(a(c^2 - 1) = b(b^2 + c^2)\) and \(d \leq 1\), we need to prove that:
\[
d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}
\]
2. From the given equation \(a(c^2 - 1) = b(b^2 + c^2)\), we can rewrite it as:
\[
a(c^2 - 1) = b(b^2 + c^2) \implies a c^2 - a = b^3 + b c^2
... |
Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, ... | Okay, let's see. So, the problem is about Jamie, Linda, and Don buying bundles of roses from a flower shop and then selling them at a fair. They each paid the same price per bundle at the flower shop, and then sold them at a higher fixed price at the fair. The ones they didn't sell, they donated to the fair organizers.... | **Solution:**
Let's denote:
- \( c \) = cost price per bundle at the flower shop (in dollars)
- \( s \) = selling price per bundle at the fair (in dollars)
**Given:**
1. Jamie bought 20 bundles, sold 15, and made a \$60 profit:
\[
15s - 20c = 60 \quad \text{(Equation 1)}
\]
2. Linda bought 34 bundles, sold 2... | 1. Let \( b \) be the price of one bundle when bought at the flower shop, and let \( s \) be the price of one bundle when sold at the fair.
2. Jamie bought 20 bundles and sold 15 of them. Her profit is given by:
\[
15s - 20b = 60
\]
3. Linda bought 34 bundles and sold 24 of them. Her profit is given by:
\[
... |
Given an integer $n \ge 2$ , find all sustems of $n$ functions $ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0$ $f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0$ $...$ $f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0$ | Alright, let me try to tackle this problem step by step. The problem is to find all systems of n functions f₁, f₂, ..., fₙ: R → R such that for all real numbers x and y, a set of equations hold. Each equation seems to follow a pattern where the first function in the system, when evaluated at x raised to some power, min... | The only solutions are systems where each function is a constant function, either all identically zero or all identically two.
**Step-by-Step Explanation:**
1. **Analyze the First Functional Equation:**
The first equation is \( f_1(x) + f_1(y) = f_2(x)f_2(y) \). By setting \( y = 0 \), we find that \( f_1 \) and ... | 1. **Insert \( x = y \) in the first equation:**
\[
f_1(x) - f_2(x)f_2(x) + f_1(x) = 0 \implies 2f_1(x) = f_2(x)^2
\]
This implies:
\[
f_2(x)^2 = 2f_1(x)
\]
2. **Consider the expression \((f_1(x) + f_1(y))^2\):**
\[
(f_1(x) + f_1(y))^2 = (f_2(x)f_2(y))^2 = 4f_1(x)f_1(y)
\]
This implies... |
What is the sum of the squares of the roots of the equation $x^2 -7 \lfloor x\rfloor +5=0$ ? | Okay, let's tackle this problem: find the sum of the squares of the roots of the equation \( x^2 -7 \lfloor x \rfloor +5=0 \). Hmm, this looks a bit tricky because it involves the floor function. The floor function, denoted by \( \lfloor x \rfloor \), gives the greatest integer less than or equal to \( x \). So, for ex... | To find the sum of the squares of the roots of the equation \( x^2 - 7 \lfloor x \rfloor + 5 = 0 \), we need to consider the intervals where \( \lfloor x \rfloor = n \) is an integer. For each integer \( n \), the equation becomes \( x^2 - 7n + 5 = 0 \). The roots of this quadratic equation are \( x = \pm \sqrt{7n - 5}... | 1. Given the equation \(x^2 - 7 \lfloor x \rfloor + 5 = 0\), we need to find the sum of the squares of the roots of this equation.
2. By definition of the floor function, we have:
\[
x - 1 < \lfloor x \rfloor \leq x
\]
Multiplying by 7:
\[
7x - 7 < 7 \lfloor x \rfloor \leq 7x
\]
3. Substituting \... |
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called to be the power of the triple. What is the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$ ? $
\textbf{(A)}\ 9000
\qquad\textbf{(B)}\ 8460
\qquad\textbf{(C)}\ 7290
\qquad\textbf{(D)}\ 6150
\qquad... | Okay, so I need to solve this problem where I have to find the sum of powers of all triples (x, y, z) where each of x, y, z is a positive integer less than or equal to 9. The "power" of a triple is defined as the sum of the largest number and the smallest number in the triple. The answer choices are given, so I need to... | To solve the problem of finding the sum of powers of all triples \((x, y, z)\) where \(x, y, z \leq 9\), we need to consider the power of each triple, defined as the sum of the largest and smallest numbers in the triple.
First, we recognize that the power of each triple \((x, y, z)\) is \( \text{max}(x, y, z) + \text... | 1. **Define the power of a triple**: The power of a triple \((x, y, z)\) is defined as the sum of the largest and smallest numbers in the triple. Without loss of generality, assume \(x \geq y \geq z\). Thus, the power of the triple \((x, y, z)\) is \(x + z\).
2. **Pairing triples**: We pair each triple \((x, y, z)\) w... |
The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$ , in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the larg... | Okay, so I need to find the largest integer M such that, no matter how you label the 2011x2011 grid with numbers 1 to 2011², there will always be at least two neighboring cells whose labels differ by at least M. The grid is turned into a torus by identifying the left and right edges, and the top and bottom edges. That ... | To determine the largest positive integer \( M \) such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least \( M \), we consider the following steps:
1. **Understanding the Problem**: We need to find the maximum \( M \) such that any labelling of a ... | To solve this problem, we need to determine the largest positive integer \( M \) such that, no matter how we label the cells of a \( 2011 \times 2011 \) array with the integers \( 1, 2, \ldots, 2011^2 \), there exist two neighboring cells with the difference of their labels at least \( M \).
1. **Understanding the Pro... |
Solve the system of equations for positive real numbers: $$ \frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1 $$ | Okay, so I need to solve this system of equations for positive real numbers x, y, and z. The equations are:
1. \( \frac{1}{xy} = \frac{x}{z} + 1 \)
2. \( \frac{1}{yz} = \frac{y}{x} + 1 \)
3. \( \frac{1}{zx} = \frac{z}{y} + 1 \)
Hmm, let's start by trying to understand the structure of these equations. Each equation i... | To solve the system of equations for positive real numbers:
1. \( \frac{1}{xy} = \frac{x}{z} + 1 \)
2. \( \frac{1}{yz} = \frac{y}{x} + 1 \)
3. \( \frac{1}{zx} = \frac{z}{y} + 1 \)
we start by analyzing the structure of each equation. Each equation is reciprocal in nature and involves products of two variables in the ... | To solve the system of equations for positive real numbers, we start with the given equations:
\[
\frac{1}{xy} = \frac{x}{z} + 1, \quad \frac{1}{yz} = \frac{y}{x} + 1, \quad \frac{1}{zx} = \frac{z}{y} + 1
\]
1. **Rewrite the equations:**
Let's rewrite each equation in a more convenient form:
\[
\frac{1}{xy} ... |
Let $f:[0,1]\to\mathbb R$ be a function satisfying $$ |f(x)-f(y)|\le|x-y| $$ for every $x,y\in[0,1]$ . Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$ , $a_i\le b_i$ in the plane such that $$ \{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\t... | Okay, so I have this problem here where I need to show that for a Lipschitz function f with constant 1 on the interval [0,1], and for any ε > 0, there exists a countable family of rectangles covering the graph of f. Each rectangle R_i has dimensions a_i × b_i with a_i ≤ b_i, and the sum of all a_i is less than ε. Also,... | To solve the problem, we utilize the properties of Lipschitz functions and measure theory to construct the required countable cover of rectangles.
**Step-by-Step Explanation:**
1. **Lipschitz Function Properties:**
- The function \( f: [0,1] \to \mathbb{R} \) is 1-Lipschitz, meaning \( |f(x) - f(y)| \leq |x - y| \... | 1. **Understanding the Problem:**
We are given a function \( f: [0,1] \to \mathbb{R} \) that satisfies the Lipschitz condition:
\[
|f(x) - f(y)| \leq |x - y| \quad \text{for all } x, y \in [0,1].
\]
We need to show that for every \(\varepsilon > 0\), there exists a countable family of rectangles \( (R_i)... |
The cards in a stack of $2n$ cards are numbered consecutively from $1$ through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A$ . The remaining cards form pile $B$ . The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A$ , respecti... | Okay, let me try to tackle this problem step by step. First, let me make sure I understand the problem correctly.
We have a stack of 2n cards numbered from 1 to 2n, with 1 being the top card and 2n the bottom. The stack is divided into two equal piles: pile A consists of the top n cards (1 through n), and pile B cons... | To solve the problem, we need to determine the number of cards in a magical stack where card number 131 retains its original position after restacking.
1. **Understanding the Restacking Process**:
- A stack of \(2n\) cards is divided into two piles: pile A (cards 1 to \(n\)) and pile B (cards \(n+1\) to \(2n\)).
... | 1. **Understanding the Problem:**
- We have a stack of \(2n\) cards numbered from 1 to \(2n\).
- The top \(n\) cards are removed to form pile \(A\), and the remaining \(n\) cards form pile \(B\).
- The cards are then restacked by taking cards alternately from the tops of pile \(B\) and \(A\).
- We need to f... |
Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number | Okay, so I need to find integers x such that 2^x + x² + 25 is a cube of a prime number. Let's start by breaking down the problem.
First, let's denote the prime number as p. Then, the equation becomes:
2^x + x² + 25 = p³
Our goal is to find integers x where this equation holds true, and p is a prime.
Since both 2^x... | To determine the integers \( x \) such that \( 2^x + x^2 + 25 \) is the cube of a prime number, we start by denoting the prime number as \( p \). This gives us the equation:
\[ 2^x + x^2 + 25 = p^3 \]
We need to find non-negative integers \( x \) (since \( 2^x \) must be an integer) and prime numbers \( p \) such tha... | 1. Let \( 2^x + x^2 + 25 = p^3 \) where \( p \) is a prime number. Since \( p^3 > 25 \) and \( p^3 \) must be greater than \( 2^x \) and \( x^2 \), we know \( p \) must be greater than 2. Therefore, \( p \) is an odd prime number.
2. If \( x < 0 \), then \( 2^x \not\in \mathbb{Z} \) (since \( 2^x \) would be a fractio... |
Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$ . Find all integer values that $P$ can take. | Okay, let's tackle this problem step by step. The question is: Given positive integers x < y, find all possible integer values that P can take, where P is defined as (x³ - y)/(1 + xy).
First, let me make sure I understand the problem correctly. We need to find all integer values P such that there exist positive integ... | Given positive integers \( x < y \), we need to find all possible integer values that \( P \) can take, where \( P \) is defined as \( P = \frac{x^3 - y}{1 + xy} \).
First, we rewrite the equation \( P = \frac{x^3 - y}{1 + xy} \) as:
\[
x^3 - y = P(1 + xy)
\]
Rearranging to solve for \( y \), we get:
\[
y = \frac{x^3 ... | 1. We start with the given expression \( P = \frac{x^3 - y}{1 + xy} \) and need to find all integer values that \( P \) can take, given \( x < y \) are positive integers.
2. Consider the case where \( x = p^3 \) and \( y = p^5 - p \). Substituting these into the expression for \( P \):
\[
P = \frac{(p^3)^3 - (p... |
There are $15$ points on the plane, coloured blue, red and green. It is known that the sum of all pairwise distances between red and blue points equals $51$ , between red and green points equals $39$ , between blue and green points equals $1$ . What are the possible amounts of red, blue and green points? | Okay, let's try to tackle this problem step by step. So, we have 15 points in total, colored blue, red, and green. The problem gives us the sums of all pairwise distances between different color groups: red-blue distances sum to 51, red-green sum to 39, and blue-green sum to 1. We need to find the possible numbers of r... | Given 15 points on a plane colored blue, red, and green, we know the sums of all pairwise distances between different color groups: red-blue distances sum to 51, red-green distances sum to 39, and blue-green distances sum to 1. We need to find the possible numbers of red (R), blue (B), and green (G) points.
1. **Key E... | Given the problem, we need to find the possible amounts of red, blue, and green points such that the sum of all pairwise distances between red and blue points equals $51$, between red and green points equals $39$, and between blue and green points equals $1$. Let $r$, $b$, and $g$ be the number of red, blue, and green ... |
A frog is placed on each cell of a $n \times n$ square inside an infinite chessboard (so initially there are a total of $n \times n$ frogs). Each move consists of a frog $A$ jumping over a frog $B$ adjacent to it with $A$ landing in the next cell and $B$ disappearing (adjacent means two cells sharing a side... | Okay, so I have this problem about frogs on an n x n chessboard. The goal is to prove that at least [n²/3] moves are needed to reach a configuration where no more moves can be made. Let me try to wrap my head around this.
First, let me understand the setup. We have an n x n square on an infinite chessboard, with a fro... | To prove that at least \( \left\lfloor \frac{n^2}{3} \right\rfloor \) moves are needed to reach a terminal configuration where no more moves are possible, we use a three-coloring argument with weighted potential functions.
**Step-by-Step Explanation:**
1. **Three-Coloring the Chessboard:**
- Assign each cell \((i,... | 1. **Define Efficient and Inefficient Moves:**
- Call a move **efficient** if the frog lands outside of the initial \( n \times n \) square.
- Call a move **inefficient** if the frog lands inside the initial \( n \times n \) square.
2. **Initial and Final Configuration:**
- Initially, there are \( n^2 \) frog... |
The polynomial $P(x) = x^3 + \sqrt{6} x^2 - \sqrt{2} x - \sqrt{3}$ has three distinct real roots. Compute the sum of all $0 \le \theta < 360$ such that $P(\tan \theta^\circ) = 0$ .
*Proposed by Lewis Chen* | Alright, let's try to tackle this problem step by step. The polynomial given is P(x) = x³ + √6 x² - √2 x - √3. We need to find the sum of all angles θ between 0 and 360 degrees such that tan θ° is a root of P(x). So, essentially, we're looking for all θ where tan θ satisfies the equation P(x) = 0, and then summing thos... | The polynomial \( P(x) = x^3 + \sqrt{6} x^2 - \sqrt{2} x - \sqrt{3} \) has three distinct real roots. We need to compute the sum of all \( 0 \le \theta < 360 \) such that \( P(\tan \theta^\circ) = 0 \).
1. **Identify the roots and corresponding angles**:
- Each root \( r_i = \tan \theta_i \) corresponds to two angl... | 1. Given the polynomial \( P(x) = x^3 + \sqrt{6} x^2 - \sqrt{2} x - \sqrt{3} \), we know it has three distinct real roots. Let these roots be \( \tan A \), \( \tan B \), and \( \tan C \) where \( A, B, C \in [0^\circ, 180^\circ) \).
2. By Vieta's formulas, the sum of the roots of the polynomial \( P(x) = 0 \) is given... |
There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly $ k$ direct routes to $ k$ other towns;
2) if two towns are not connected by a direct route... | Okay, let's try to figure out this problem step by step. So, we have 25 towns, and we need to set up two-directional flight routes between them. The conditions are:
1) Each town has exactly k direct routes to k other towns. So, each town has degree k in the graph terminology.
2) If two towns are not directly connecte... | To find the smallest \( k \) for which we can set up two-direction flight routes connecting 25 towns such that each town has exactly \( k \) direct routes and any two non-connected towns have a common neighbor, we need to consider the following steps:
1. **Graph Theory Interpretation**: The problem translates to findi... | 1. **Define the problem in terms of graph theory:**
- We have 25 towns, which we can represent as vertices in a graph.
- We need to find the smallest \( k \) such that each vertex has exactly \( k \) edges (degree \( k \)).
- Additionally, for any two vertices that are not directly connected, there must be a v... |
**p1.** What is the units digit of the product of the first seven primes?**p2.**In triangle $ABC$ , $\angle BAC$ is a right angle and $\angle ACB$ measures $34$ degrees. Let $D$ be a point on segment $ BC$ for which $AC = CD$ , and let the angle bisector of $\angle CBA$ intersect line $AD$ at $E$ . Wha... | Okay, let's tackle problem p1 first. The question is asking for the units digit of the product of the first seven primes. Alright, so the first seven primes are 2, 3, 5, 7, 11, 13, 17. Let me list them out to make sure: 2, 3, 5, 7, 11, 13, 17. Yep, that's seven primes.
Now, the units digit of their product. The units ... | To determine the units digit of the product of the first seven primes, we list the first seven primes: 2, 3, 5, 7, 11, 13, and 17. The units digit of a product depends only on the units digits of the factors.
The units digits of these primes are 2, 3, 5, 7, 1, 3, and 7. Multiplying these units digits step by step:
1... | 4. The quadratic equation \(kx^2 + (k + 5)x + 5 = 0\) has only one real solution if its discriminant is zero. The discriminant \(\Delta\) of the quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[
\Delta = b^2 - 4ac
\]
For our equation, \(a = k\), \(b = k + 5\), and \(c = 5\):
\[
\Delta = (k + 5)... |
The function $f(x) = x^2 + (2a + 3)x + (a^2 + 1)$ only has real zeroes. Suppose the smallest possible value of $a$ can be written in the form $p/q$ , where $p, q$ are relatively prime integers. Find $|p| + |q|$ . | Okay, let's see. I need to find the smallest possible value of 'a' such that the quadratic function f(x) = x² + (2a + 3)x + (a² + 1) only has real zeros. Then, express that value as a fraction p/q where p and q are coprime, and find |p| + |q|. Hmm, right.
First, I remember that for a quadratic equation ax² + bx + c =... | To determine the smallest possible value of \( a \) such that the quadratic function \( f(x) = x^2 + (2a + 3)x + (a^2 + 1) \) only has real zeros, we need to ensure that the discriminant of the quadratic equation is non-negative.
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c \) is given by \( D = ... | 1. To determine the conditions under which the quadratic function \( f(x) = x^2 + (2a + 3)x + (a^2 + 1) \) has real zeros, we need to ensure that its discriminant is non-negative. The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[
\Delta = b^2 - 4ac
\]
For the given fu... |
In $\triangle ABC$ , $X, Y$ , are points on BC such that $BX=XY=YC$ , $M , N$ are points on $AC$ such that $AM=MN=NC$ . $BM$ and $BN$ intersect $AY$ at $S$ and $R$ and respectively. If the area of $\triangle ABC$ is $1$ , find the area of $SMNR$ . | Okay, let me try to solve this geometry problem. It's about triangle ABC with points divided into thirds on sides BC and AC. Then, lines BM and BN intersect AY at S and R respectively. The area of triangle ABC is 1, and we need to find the area of quadrilateral SMNR. Hmm, this seems a bit complex, but let's break it do... | To solve the problem, we start by assigning coordinates to the vertices of triangle \( \triangle ABC \). We place point \( A \) at \( (0, 0) \), point \( C \) at \( (3, 0) \), and point \( B \) at \( (0, \frac{2}{3}) \) to ensure the area of \( \triangle ABC \) is 1.
Points \( X \) and \( Y \) on \( BC \) are such th... | 1. **Determine the area of $\triangle BMN$:**
Since $AM = MN = NC$, $M$ and $N$ divide $AC$ into three equal segments. Similarly, $BX = XY = YC$ divides $BC$ into three equal segments. Therefore, $\triangle BMN$ is a smaller triangle within $\triangle ABC$ with vertices on the midpoints of $AC$ and $BC$.
The are... |
How many positive integers are there such that $\left \lfloor \frac m{11} \right \rfloor = \left \lfloor \frac m{10} \right \rfloor$ ? ( $\left \lfloor x \right \rfloor$ denotes the greatest integer not exceeding $x$ .) $
\textbf{(A)}\ 44
\qquad\textbf{(B)}\ 48
\qquad\textbf{(C)}\ 52
\qquad\textbf{(D)}\ 54
\qquad\... | Okay, let's try to figure out how to solve this problem. The question is asking for the number of positive integers m such that the floor of m/11 is equal to the floor of m/10. The floor function, as a reminder, gives the greatest integer less than or equal to the given number. The answer choices are given from A to E,... | To determine the number of positive integers \( m \) such that \( \left\lfloor \frac{m}{11} \right\rfloor = \left\lfloor \frac{m}{10} \right\rfloor \), we start by denoting this common value as \( k \). This implies the inequalities:
\[
k \leq \frac{m}{11} < k + 1 \quad \text{and} \quad k \leq \frac{m}{10} < k + 1.
\]
... | To solve the problem, we need to find the number of positive integers \( m \) such that \(\left \lfloor \frac{m}{11} \right \rfloor = \left \lfloor \frac{m}{10} \right \rfloor\).
1. **Define the floor functions:**
Let \( k = \left \lfloor \frac{m}{11} \right \rfloor = \left \lfloor \frac{m}{10} \right \rfloor \). T... |
The function $f(x) = ax^2 + bx + c$ satisfies the following conditions:
$f(\sqrt2)=3$ and $ |f(x)| \le 1$ for all $x \in [-1, 1]$ .
Evaluate the value of $f(\sqrt{2013})$ | Okay, so I need to figure out the value of f(√2013) for the quadratic function f(x) = ax² + bx + c. The given conditions are that f(√2) = 3 and |f(x)| ≤ 1 for all x in the interval [-1, 1]. Hmm, let's start breaking this down step by step.
First, since f is a quadratic function, it has the general form ax² + bx + c. T... | Given the quadratic function \( f(x) = ax^2 + bx + c \) that satisfies the conditions \( f(\sqrt{2}) = 3 \) and \( |f(x)| \le 1 \) for all \( x \in [-1, 1] \), we need to evaluate the value of \( f(\sqrt{2013}) \).
1. **Understanding the Constraints**:
- The function \( f(x) \) must be bounded by 1 in absolute valu... | 1. Given the function \( f(x) = ax^2 + bx + c \) and the conditions \( f(\sqrt{2}) = 3 \) and \( |f(x)| \le 1 \) for all \( x \in [-1, 1] \), we need to evaluate \( f(\sqrt{2013}) \).
2. First, we use the condition \( f(\sqrt{2}) = 3 \):
\[
a(\sqrt{2})^2 + b(\sqrt{2}) + c = 3 \implies 2a + b\sqrt{2} + c = 3
\... |
Let $p$ be an odd prime number less than $10^5$ . Granite and Pomegranate play a game. First, Granite picks a integer $c \in \{2,3,\dots,p-1\}$ .
Pomegranate then picks two integers $d$ and $x$ , defines $f(t) = ct + d$ , and writes $x$ on a sheet of paper.
Next, Granite writes $f(x)$ on the paper, Pomegran... | Okay, so I need to solve this problem where Granite and Pomegranate play a game with certain rules involving an odd prime number p less than 10^5. The goal is to find the sum of all such primes p where Pomegranate has a winning strategy. Let me try to break down the problem step by step.
First, let me make sure I unde... | To solve the problem, we need to determine the primes \( p \) less than \( 10^5 \) for which Pomegranate has a winning strategy. The key insight is that Pomegranate can force the game to end on her turn if the multiplicative order of \( c \) modulo \( p \) is even for all \( c \) chosen by Granite. This happens if and ... | 1. **Define the game and the function:**
- Granite picks an integer \( c \in \{2, 3, \dots, p-1\} \).
- Pomegranate picks two integers \( d \) and \( x \), and defines the function \( f(t) = ct + d \).
- The sequence starts with \( x \), and each player alternately writes \( f \) applied to the last number wri... |
Find all pairs $(x,y)$ , $x,y\in \mathbb{N}$ for which $gcd(n(x!-xy-x-y+2)+2,n(x!-xy-x-y+3)+3)>1$ for $\forall$ $n\in \mathbb{N}$ . | Alright, so I need to find all natural number pairs (x, y) such that for every natural number n, the greatest common divisor (gcd) of two expressions is greater than 1. The expressions are n times (x! - xy - x - y + 2) plus 2, and n times (x! - xy - x - y + 3) plus 3. Hmm, okay. Let me try to parse this step by step.
... | To find all pairs \((x, y)\) where \(x, y \in \mathbb{N}\) such that \(\gcd(n(x! - xy - x - y + 2) + 2, n(x! - xy - x - y + 3) + 3) > 1\) for all \(n \in \mathbb{N}\), we start by defining the expressions inside the gcd:
Let \(A = x! - xy - x - y + 2\) and \(B = x! - xy - x - y + 3\). We need \(\gcd(nA + 2, nB + 3) > ... | 1. We start with the given condition:
\[
\gcd(n(x! - xy - x - y + 2) + 2, n(x! - xy - x - y + 3) + 3) > 1 \quad \forall \, n \in \mathbb{N}
\]
2. Let us denote:
\[
a = n(x! - xy - x - y + 2) + 2
\]
\[
b = n(x! - xy - x - y + 3) + 3
\]
3. We need to find the greatest common divisor (gcd) of ... |
Find the area of the region in the coordinate plane satisfying the three conditions $\star$ $x \le 2y$ $\star$ $y \le 2x$ $\star$ $x + y \le 60.$ | Alright, let's see. I need to find the area of the region in the coordinate plane that satisfies three conditions: x ≤ 2y, y ≤ 2x, and x + y ≤ 60. Hmm, okay. So, these are linear inequalities, and the region they define should be a polygon, probably a triangle or a quadrilateral. Let me break it down step by step.
Fir... | To find the area of the region in the coordinate plane satisfying the conditions \( x \leq 2y \), \( y \leq 2x \), and \( x + y \leq 60 \), we first identify the intersection points of the lines defining these inequalities.
1. **Intersection of \( x \leq 2y \) and \( x + y \leq 60 \)**:
- Substitute \( y = \frac{x}... | 1. **Identify the inequalities and their corresponding lines:**
- The inequality \( x \leq 2y \) can be rewritten as \( y \geq \frac{x}{2} \). This represents the region above the line \( y = \frac{x}{2} \).
- The inequality \( y \leq 2x \) represents the region below the line \( y = 2x \).
- The inequality \(... |
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$ \deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i) $$ and find when the equality... | Okay, so I need to prove that for n ≥ 2, given n real coefficient polynomials P₁(x), P₂(x), ..., Pₙ(x) that are not all the same and have positive leading coefficients, the degree of the polynomial S(x) = P₁ⁿ + P₂ⁿ + ... + Pₙⁿ - nP₁P₂...Pₙ is at least (n - 2) times the maximum degree among the P_i's. Also, I need to fi... | Given \( n \ge 2 \) real coefficient polynomials \( P_1(x), P_2(x), \ldots, P_n(x) \) with positive leading coefficients, not all the same, we need to prove that
\[ \deg(P_1^n + P_2^n + \cdots + P_n^n - nP_1P_2 \cdots P_n) \ge (n-2) \max_{1 \le i \le n}(\deg P_i) \]
and determine when equality holds.
### Analysis and ... | 1. **Assume the degrees of the polynomials are equal:**
Let \( \deg(P_i) = m \) for all \( i = 1, 2, \ldots, n \). Denote the leading coefficient of \( P_i \) by \( a_i \), where \( a_i > 0 \).
2. **Consider the leading term of the polynomial:**
The leading term of \( P_i^n \) is \( a_i^n x^{mn} \). Therefore, t... |
Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$ a!b!c!d!=24! $$ $\textbf{(A)}~4$ $\textbf{(B)}~4!$ $\textbf{(C)}~4^4$ $\textbf{(D)}~\text{None of the above}$ | Okay, so I need to find the number of ordered quadruples (a, b, c, d) of positive integers such that the product of their factorials equals 24!. The answer choices are given, but the correct one is supposedly not listed, so I have to check each step carefully.
First, let me recall that factorials grow very quickly, so... | To solve the problem of finding the number of ordered quadruples \((a, b, c, d)\) such that \(a!b!c!d! = 24!\), we need to consider the prime factorization of \(24!\) and the structure of factorials.
1. **Prime Factorization of \(24!\)**:
- The prime factors of \(24!\) are \(2^{22}, 3^{10}, 5^4, 7^3, 11^1, 13^1, 17... | To solve the problem, we need to find the number of ordered quadruples \((a, b, c, d)\) of positive integers such that \(a!b!c!d! = 24!\).
1. **Factorial Decomposition**:
First, we note that \(24! = 24 \times 23 \times 22 \times \cdots \times 1\). Since \(24!\) is a very large number, we need to consider the possib... |
A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$ , preserves the one with mass $b$ , and creates a new omon whose mass is $\frac 12 (a+b)$ . The physicist can then repeat th... | Okay, so there's this problem about a physicist who has these particles called omons. The machine he uses takes two omons with masses a and b, destroys the one with mass a, keeps the one with mass b, and creates a new omon with mass (a + b)/2. The goal is to find the maximum number of times he can use this machine with... | The problem involves a physicist who can use a machine to entangle two omons, destroying one and creating a new omon with a mass that is the average of the two original masses. The goal is to determine the maximum number of times this machine can be used without producing a non-integer mass, starting with two distinct ... | 1. **Initial Setup**: We start with two omons of masses \(a\) and \(b\), where \(a\) and \(b\) are distinct positive integers less than 1000. Without loss of generality, assume \(a < b\).
2. **Entanglement Process**: The machine destroys the omon with mass \(a\), preserves the omon with mass \(b\), and creates a new o... |
Let $ f$ be a function with the following properties:
(i) $f(1) \equal{} 1$ , and
(ii) $ f(2n) \equal{} n\times f(n)$ , for any positive integer $ n$ .
What is the value of $ f(2^{100})$ ? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2^{99} \qquad \textbf{(C)}\ 2^{100} \qquad \textbf{(D)}\ 2^{4950} \qquad \textbf{(... | Alright, let's tackle this problem step by step. So, we have a function f with two properties: first, f(1) = 1, and second, for any positive integer n, f(2n) = n * f(n). We need to find the value of f(2^100). The answer choices are given, so hopefully, after working through the problem, one of them will match.
Let me ... | Given a function \( f \) with the properties:
1. \( f(1) = 1 \)
2. \( f(2n) = n \times f(n) \) for any positive integer \( n \),
we need to find the value of \( f(2^{100}) \).
First, we compute \( f \) for small powers of 2 to identify a pattern:
- \( f(2^1) = f(2) = 1 \)
- \( f(2^2) = f(4) = 2 \times f(2) = 2 \times... | To solve for \( f(2^{100}) \), we need to use the given properties of the function \( f \):
1. \( f(1) = 1 \)
2. \( f(2n) = n \times f(n) \) for any positive integer \( n \).
We will start by calculating \( f \) for some small powers of 2 to identify a pattern.
1. \( f(1) = 1 \)
2. \( f(2) = 1 \times f(1) = 1 \)
3. \... |
On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$ . If $\angle OEB = 140^o$ then determine with proof the magnitude of $\ang... | Okay, so I need to find the measure of angle AOB given that angle OEB is 140 degrees. Let me start by drawing a diagram to visualize the problem. There's a circle with center O. Chord AB is extended to a point X, from which two tangents XC and XD are drawn, touching the circle at points C and D. E is the midpoint of CD... | Given a circle with center \( O \) and a chord \( AB \) extended to a point \( X \). Tangents \( XC \) and \( XD \) are drawn from \( X \) to the circle, touching at points \( C \) and \( D \). \( E \) is the midpoint of \( CD \), and it is given that \( \angle OEB = 140^\circ \). We need to determine the magnitude of ... | 1. **Identify the given elements and relationships:**
- \( O \) is the center of the circle.
- \( AB \) is a chord of the circle.
- \( X \) is a point on the extension of \( AB \).
- \( XC \) and \( XD \) are tangents to the circle from \( X \), with \( C \) and \( D \) lying on the circle.
- \( E \) is ... |
Let $ ABC$ be a triangle with circumradius $ R$ , perimeter $ P$ and area $ K$ . Determine the maximum value of: $ \frac{KP}{R^3}$ . | Alright, so I have this geometry problem here: given a triangle ABC with circumradius R, perimeter P, and area K, I need to determine the maximum value of the expression (KP)/R³. Hmm, okay. Let me start by recalling some formulas related to triangles that might be useful here.
First, the area K of a triangle can be ex... | Given a triangle \( ABC \) with circumradius \( R \), perimeter \( P \), and area \( K \), we need to determine the maximum value of the expression \( \frac{KP}{R^3} \).
First, we express the area \( K \) and perimeter \( P \) in terms of the sides \( a, b, c \) and the circumradius \( R \). Using the formulas:
- \( K... | 1. **Establishing the inequality \(3\sqrt{3}R \ge P\):**
- We start by noting that the perimeter \(P\) of the triangle can be expressed in terms of the sides \(a, b, c\) as \(P = a + b + c\).
- Using the sine rule, we have \(a = 2R \sin A\), \(b = 2R \sin B\), and \(c = 2R \sin C\).
- Therefore, \(P = 2R (\sin... |
Given that $ \minus{} 4\le x\le \minus{} 2$ and $ 2\le y\le 4$ , what is the largest possible value of $ (x \plus{} y)/x$ ? $ \textbf{(A)}\ \minus{}\!1\qquad
\textbf{(B)}\ \minus{}\!\frac {1}{2}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac {1}{2}\qquad
\textbf{(E)}\ 1$ | Okay, let's see. The problem is asking for the largest possible value of (x + y)/x given that x is between -4 and -2, and y is between 2 and 4. The answer choices are from A to E, with some fractions and zero. Hmm. Alright, let me think step by step.
First, I need to understand the expression (x + y)/x. Let me rewrite... | Given the problem of finding the largest possible value of \( \frac{x + y}{x} \) within the ranges \( -4 \le x \le -2 \) and \( 2 \le y \le 4 \), we start by rewriting the expression as \( 1 + \frac{y}{x} \). Since \( x \) is negative and \( y \) is positive, \( \frac{y}{x} \) will be negative. To maximize \( 1 + \frac... | 1. We are given the ranges for \( x \) and \( y \):
\[
-4 \leq x \leq -2 \quad \text{and} \quad 2 \leq y \leq 4
\]
2. We need to find the largest possible value of the expression \(\frac{x + y}{x}\).
3. Rewrite the expression:
\[
\frac{x + y}{x} = \frac{x}{x} + \frac{y}{x} = 1 + \frac{y}{x}
\]
The... |
Given $$ b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1, $$ prove that $$ b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2. $$ Hence, as a corollary, show $$ \lim_{n \to \infty} b_n =2. $$ | Okay, let me try to tackle this problem. So, I need to prove that for the sequence defined by \( b_n = \sum_{k=0}^{n} \frac{1}{\binom{n}{k}} \), there's a recurrence relation \( b_n = \frac{n+1}{2n} b_{n-1} +1 \) for \( n \geq 2 \). Then, using this recurrence, I have to show that the limit as \( n \) approaches infini... | To prove the recurrence relation \( b_n = \frac{n+1}{2n} b_{n-1} +1 \) for \( n \geq 2 \), we start by expressing \( b_n \) as a sum involving binomial coefficients and manipulate it using symmetry and combinatorial identities. By splitting the sum and adjusting indices, we relate terms of \( b_n \) to those of \( b_{n... | To prove the given recurrence relation and the limit, we will proceed step-by-step.
1. **Expression for \( b_n \):**
\[
b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}
\]
This can be rewritten using the definition of binomial coefficients:
\[
b_n = \sum_{k=0}^{n} \frac{1}{\binom{n}{k}} = \sum_{k=0}^{n} \frac{... |
A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$ , $BE=4$ , $AD=5$ and $AC=2\sqrt{7} $ .
Find the angle $ \angle BDC$ . | Alright, let's try to tackle this geometry problem step by step. First, I need to understand the problem clearly and visualize the given information. So, we have triangle ABC. A circle passes through points A and C of this triangle. This circle intersects sides AB and BC at points D and E, respectively. The given ratio... | To solve the problem, we start by using the Power of a Point theorem at point \( B \). Given \( \frac{BD}{CE} = \frac{3}{2} \), let \( BD = 3k \) and \( CE = 2k \). Using the Power of a Point theorem for point \( B \) with respect to the circle passing through \( A, C, D, E \):
\[
BD \cdot BA = BE \cdot BC
\]
Substit... | 1. **Given Information and Setup:**
- A circle passes through points \( A \) and \( C \) of triangle \( ABC \).
- The circle intersects sides \( AB \) and \( BC \) at points \( D \) and \( E \) respectively.
- Given: \( \frac{BD}{CE} = \frac{3}{2} \), \( BE = 4 \), \( AD = 5 \), and \( AC = 2\sqrt{7} \).
2. *... |
Given positive integer $ m \geq 17$ , $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m\minus{}1$ r... | Okay, so I need to find the minimal positive integer n such that after n rounds of the competition, for any group of 4 contestants, either none of them have played against each other, or there have been at least 2 games among them. The competition has 2m contestants, each round divides them into m groups of two, and th... | To solve the problem of finding the least possible positive integer \( n \) such that after \( n \) rounds, for any 4 contestants, none of them has played with the others or there have been at least 2 games played within those 4, we need to consider the structure of the competition and the properties of the graph forme... | 1. **Constructing the competition schedule for even \( m \):**
- Label the contestants as \( A_1, A_2, \ldots, A_m, B_1, B_2, \ldots, B_m \).
- For \( 1 \le i < m \), on day \( i \), pair \( A_j \) with \( A_{i + j} \) for each \( 1 \le j \le m \), where indices are modulo \( m \).
- On day \( m \), pair \( A_... |
Let $x$ be the largest root of $x^4 - 2009x + 1$ . Find the nearest integer to $\frac{1}{x^3-2009}$ . | Okay, so I need to find the largest root of the equation x⁴ - 2009x + 1 = 0, and then compute the nearest integer to 1/(x³ - 2009). Hmm, that seems a bit complicated, but let's break it down step by step.
First, let me recall that finding roots of quartic equations can be tricky. Since it's a fourth-degree polynomial,... | Given the polynomial \( x^4 - 2009x + 1 \), we need to find the largest root \( x \) and then determine the nearest integer to \( \frac{1}{x^3 - 2009} \).
1. **Identifying the largest root**:
- The polynomial \( x^4 - 2009x + 1 \) is analyzed for its roots. By the Intermediate Value Theorem, we find that the larges... | 1. Let \( f(x) = x^4 - 2009x + 1 \). We need to find the largest root of this polynomial and then determine the nearest integer to \( \frac{1}{x^3 - 2009} \).
2. If \( x \) is a root of \( f(x) \), then:
\[
x^4 - 2009x + 1 = 0 \implies x^4 - 2009x = -1 \implies x(x^3 - 2009) = -1 \implies \frac{1}{x^3 - 2009} = ... |
10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form $$ \frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)} $$ where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$ . ... | Okay, let's try to figure out how to solve this sum: the summation from k=1 to 2020 of k times cosine of (4kπ/4041). The problem says it can be written in a specific form, and we need to find a, b, c, p, q such that they meet the given conditions and then compute a+b+c+p+q.
First, let's write down the sum explicitly:... | To solve the sum \(\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)\), we start by expressing the sum in terms of complex exponentials and using the formula for the sum of an arithmetic-geometric series.
We denote \(\theta = \frac{4\pi}{4041}\) and use Euler's formula to rewrite the cosine terms. After seve... | 1. We start with the sum \(\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)\). To simplify this, we use the complex exponential form of the cosine function: \(\cos x = \frac{e^{ix} + e^{-ix}}{2}\).
2. Let \(z = e^{i \frac{4 \pi}{4041}}\). Then, \(\cos \left(\frac{4 k \pi}{4041}\right) = \frac{z^k + z^{-k}}{2... |
$x$ is a base- $10$ number such that when the digits of $x$ are interpreted as a base- $20$ number, the resulting number is twice the value as when they are interpreted as a base- $13$ number. Find the sum of all possible values of $x$ . | Okay, let me try to figure out this problem. So, the problem says that there's a base-10 number x. When we take the digits of x and interpret them as a base-20 number, the result is twice the value when those same digits are interpreted as a base-13 number. We need to find all possible x and sum them up.
First, let me... | Given a base-10 number \( x \), we need to find all such \( x \) such that when its digits are interpreted as a base-20 number, the resulting value is twice the value when interpreted as a base-13 number. We then sum all possible values of \( x \).
1. **Formulating the Problem**:
- Let \( x \) be a base-10 number w... | 1. We start by interpreting the problem statement. We need to find a base-10 number \( x \) such that when its digits are interpreted as a base-20 number, the resulting number is twice the value of the number when its digits are interpreted as a base-13 number.
2. Let's denote \( x \) as a 3-digit number \( abc \) in ... |
Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$ . | Okay, so I need to find all polynomials f(x) with integer coefficients such that for every natural number n, f(n) and f(2^n) are coprime. Hmm, let's start by understanding the problem.
First, co-prime means that the greatest common divisor (gcd) of f(n) and f(2^n) is 1. So, for every natural number n, gcd(f(n), f(2^n)... | To find all polynomials \( f(x) \) with integer coefficients such that \( f(n) \) and \( f(2^n) \) are coprime for all natural numbers \( n \), we start by considering the simplest cases and gradually build up to more complex scenarios.
1. **Constant Polynomials**:
- If \( f(x) \) is a constant polynomial, say \( c... | 1. **Identify the problem and initial conditions:**
We need to find all polynomials \( f(x) \) with integer coefficients such that \( f(n) \) and \( f(2^n) \) are co-prime for all natural numbers \( n \).
2. **Consider the case when \( f(x) \) is a constant polynomial:**
- If \( f(x) \) is a constant polynomial... |
We have a set of ${221}$ real numbers whose sum is ${110721}$ . It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$ .
| Okay, let me try to work through this problem step by step. So, we have 221 real numbers arranged in a rectangular table. The total sum of all these numbers is 110721. The conditions are that every row is an arithmetic progression, and also the first and last columns are arithmetic progressions. Each of these progressi... | Given a set of 221 real numbers arranged in a rectangular table such that every row, the first column, and the last column form arithmetic progressions with more than one element, we need to prove that the sum of the elements in the four corners is 2004. The total sum of all numbers is 110721.
1. **Identify the dimens... | 1. **Define the variables and the structure of the table:**
Let \( a^j_i \) denote the element in the \( i \)-th row and the \( j \)-th column of the rectangle. Let \( n \) be the number of rows and \( m \) be the number of columns. The total number of elements in the table is \( n \times m = 221 \). The sum of all ... |
Determine the number of positive integers $N = \overline{abcd}$ , with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$ . | Okay, so I need to find the number of positive integers N = abcd, where each digit a, b, c, d is non-zero, that satisfy the equation (2a -1)(2b -1)(2c -1)(2d -1) = 2abcd -1. Hmm, let me start by understanding what the problem is asking.
First, N is a four-digit number with all digits from 1 to 9 (since they are non-ze... | To determine the number of positive integers \( N = \overline{abcd} \) with nonzero digits \( a, b, c, d \) that satisfy the equation \((2a - 1)(2b - 1)(2c - 1)(2d - 1) = 2abcd - 1\), we start by analyzing the equation and exploring various approaches.
1. **Initial Checks and Parity Analysis**:
- Both sides of the ... | To determine the number of positive integers \( N = \overline{abcd} \) with \( a, b, c, d \) being nonzero digits that satisfy the equation \((2a - 1)(2b - 1)(2c - 1)(2d - 1) = 2abcd - 1\), we need to analyze the given equation step by step.
1. **Rewrite the equation:**
\[
(2a - 1)(2b - 1)(2c - 1)(2d - 1) = 2abc... |
The circle $x^2 + y^2 = 25$ intersects the abscissa in points $A$ and $B$ . Let $P$ be a point that lies on the line $x = 11$ , $C$ is the intersection point of this line with the $Ox$ axis, and the point $Q$ is the intersection point of $AP$ with the given circle. It turned out that the area of the tri... | Alright, let me try to figure out this problem step by step. So, we have the circle x² + y² = 25. That means it's centered at the origin (0,0) with a radius of 5. The circle intersects the x-axis (abscissa) at points A and B. Since it's a circle centered at the origin, intersecting the x-axis, those points should be wh... | The circle \( x^2 + y^2 = 25 \) intersects the x-axis at points \( A \) and \( B \), which are \((-5, 0)\) and \((5, 0)\) respectively. Point \( P \) lies on the line \( x = 11 \), and point \( C \) is the intersection of this line with the x-axis, which is \((11, 0)\). Point \( Q \) is the intersection of line \( AP \... | 1. **Identify the points of intersection of the circle with the x-axis:**
The equation of the circle is \(x^2 + y^2 = 25\). To find the points where the circle intersects the x-axis, set \(y = 0\):
\[
x^2 + 0^2 = 25 \implies x^2 = 25 \implies x = \pm 5
\]
Therefore, the points of intersection are \(A(-5,... |
Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square. | Okay, so I need to find all positive integers m, n and prime numbers p such that the expression (5^m + 2^n p)/(5^m - 2^n p) is a perfect square. Hmm, let's see. Alright, let's start by understanding the problem.
First, we have a fraction where both the numerator and denominator involve 5 raised to the power m, and 2 ... | To find all positive integers \( m, n \) and prime numbers \( p \) for which \( \frac{5^m + 2^n p}{5^m - 2^n p} \) is a perfect square, we start by setting the ratio equal to \( k^2 \) where \( k \) is a positive integer. This gives us the equation:
\[
\frac{5^m + 2^n p}{5^m - 2^n p} = k^2
\]
Cross-multiplying and re... | To find all positive integers \( m, n \) and prime numbers \( p \) for which \(\frac{5^m + 2^n p}{5^m - 2^n p}\) is a perfect square, we start by analyzing the given expression.
1. **GCD Analysis**:
\[
\gcd(5^m + 2^n p, 5^m - 2^n p) = \gcd(5^m + 2^n p, 2 \cdot 5^m)
\]
This is because:
\[
\gcd(a, b) =... |
Let $p$ is a prime and $p\equiv 2\pmod 3$ . For $\forall a\in\mathbb Z$ , if $$ p\mid \prod\limits_{i=1}^p(i^3-ai-1), $$ then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time) | Okay, so I need to figure out how many "GuGu" numbers there are in the set {1, 2, ..., p} where p is a prime and p ≡ 2 mod 3. The problem states that a number a is called a "GuGu" number if p divides the product of (i³ - a i - 1) for i from 1 to p.
First, let me make sure I understand the problem correctly. For each ... | Given a prime \( p \) such that \( p \equiv 2 \pmod{3} \), we need to determine the number of "GuGu" numbers in the set \(\{1, 2, \ldots, p\}\). A number \( a \) is called a "GuGu" number if \( p \) divides the product \(\prod_{i=1}^p (i^3 - ai - 1)\).
Key steps and reasoning:
1. **Understanding the Problem**: For \(... | 1. **Initial Assumptions and Simplifications:**
- Given \( p \) is a prime such that \( p \equiv 2 \pmod{3} \).
- We need to find the number of "GuGu" numbers in the set \(\{1, 2, \ldots, p\}\).
- A number \( a \) is called a "GuGu" number if \( p \mid \prod_{i=1}^p (i^3 - ai - 1) \).
2. **Considering the Pol... |
Let $ABC$ be an isosceles triangle with $AB = BC$ . Point $E$ lies on the side $AB$ , and $ED$ is the perpendicular from $E$ to $BC$ . It is known that $AE = DE$ . Find $\angle DAC$ . | Alright, let's tackle this geometry problem step by step. First, let me parse the problem again to make sure I understand all the given information and what needs to be found.
We have an isosceles triangle ABC with AB = BC. Wait, hold on, if AB = BC, then the triangle has two equal sides: AB and BC. But in a triangle,... | To solve the problem, we start by analyzing the given isosceles triangle \(ABC\) with \(AB = BC\). We need to find \(\angle DAC\) given that point \(E\) on \(AB\) satisfies \(AE = DE\), where \(ED\) is the perpendicular from \(E\) to \(BC\).
1. **Coordinate Setup**:
- Place \(B\) at \((0, 0)\), \(C\) at \((c, 0)\),... | 1. **Identify the given information and set up the problem:**
- Triangle \(ABC\) is isosceles with \(AB = BC\).
- Point \(E\) lies on side \(AB\).
- \(ED\) is the perpendicular from \(E\) to \(BC\).
- \(AE = DE\).
- We need to find \(\angle DAC\).
2. **Introduce angles and use properties of isosceles tr... |
Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$ .
*Alternative formulation:* Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$ , $ y$ , $ z$ . | Okay, so I need to find all non-negative integers x, y, z such that 7^x + 1 = 3^y + 5^z. Let me start by understanding the problem and thinking through possible approaches.
First, since we're dealing with exponents, it might help to check small values of x, y, z and see if any solutions pop up. Maybe there are only a ... | To solve the equation \(7^x + 1 = 3^y + 5^z\) for non-negative integers \(x, y, z\), we start by checking small values and then use modular arithmetic and other constraints to narrow down possible solutions.
1. **Checking Small Values:**
- For \(x = 0\): \(7^0 + 1 = 2\). The equation becomes \(3^y + 5^z = 2\), whic... | We need to find all nonnegative integer solutions \((x, y, z)\) to the equation:
\[ 7^x + 1 = 3^y + 5^z. \]
1. **Initial Observations:**
- If \(x = 0\), then \(7^x + 1 = 1 + 1 = 2\). Thus, \(3^y + 5^z = 2\). The only solution in nonnegative integers is \(y = 0\) and \(z = 0\). Therefore, \((x, y, z) = (0, 0, 0)\) i... |
Find all postitive integers n such that $$ \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2 $$ where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$ . | Okay, so I need to find all positive integers n where the product of the floor of n/2, floor of n/3, and floor of n/4 equals n squared. That is, floor(n/2) * floor(n/3) * floor(n/4) = n². Hmm, let's start by understanding what the floor function does here. For any real number x, floor(x) is the greatest integer less th... | To find all positive integers \( n \) such that \( \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2 \), we start by testing small values of \( n \) and analyzing the behavior of the product of the floor functions compared to \( n^2 \)... | To find all positive integers \( n \) such that
\[ \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2, \]
we will proceed step-by-step.
1. **Initial Observation:**
We start by noting that for any integer \( n \), the floor function... |
A soccer coach named $C$ does a header drill with two players $A$ and $B$ , but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach $C$ heads the ball to $A$ , who heads the ball back to $C$ , who then heads t... | Okay, let's try to figure out this sunscreen problem. So, there's a coach C and two players A and B. They're doing headers in a repeating pattern: C heads to A, A heads back to C, then C heads to B, B heads back to C, and this continues forever. Each time someone heads the ball, 1/10 of the sunscreen left on the ball t... | To determine the fraction of the sunscreen originally on the ball that ends up on Coach C's forehead, we analyze the repeating header pattern \(CACBCACB\ldots\). Each time a person headers the ball, \(1/10\) of the sunscreen on the ball transfers to their forehead, leaving \(9/10\) on the ball.
1. **Header Sequence an... | 1. Let's denote the initial amount of sunscreen on the ball as \( S \).
2. Each time a person heads the ball, \( \frac{1}{10} \) of the remaining sunscreen on the ball is transferred to the person's forehead, and \( \frac{9}{10} \) of the sunscreen remains on the ball.
3. We need to find the fraction of the original su... |
For a positive integer $n$ , an *$n$ -branch* $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$ . An integer $x$ is said to *appear* in $B$ if it is an element of the last set... | Okay, let me try to work through this problem step by step. The problem is about counting the number of perfect n-plants, denoted as T_n, and then using that to find x such that a certain generating function equals e^(6/29). Then, x is given as a reduced fraction m/n, and we need to find m + n.
First, let's parse the... | To solve the problem, we need to determine the number of distinct perfect \( n \)-plants \( T_n \) and use this to find \( x \) such that the given generating function converges to \( e^{6/29} \).
1. **Understanding Perfect \( n \)-Plants**:
- A perfect \( n \)-plant is a set of branches where each integer from 1 ... | 1. **Define the necessary terms and functions:**
- Let \( B_n \) be the number of \( n \)-branches.
- Let \( C_n \) be the number of \( n \)-branches which contain \(\{1, 2, \ldots, n\}\). It can be seen that \( C_n = \frac{B_n + 1}{2} \).
- Let \( T_n \) be the number of distinct perfect \( n \)-plants.
2. *... |
Let $K(O,R)$ be a circle with center $O$ and radious $R$ and $(e)$ to be a line thst tangent to $K$ at $A$ . A line parallel to $OA$ cuts $K$ at $B, C$ , and $(e)$ at $D$ , ( $C$ is between $B$ and $D$ ). Let $E$ to be the antidiameric of $C$ with respect to $K$ . $EA$ cuts $BD$ at $F$ ... | Okay, let me try to work through these geometry problems step by step. So, we have a circle K with center O and radius R. There's a tangent line (e) at point A. A line parallel to OA cuts the circle at B and C, and the tangent line (e) at D, with C between B and D. E is the antidiametric point of C with respect to K, m... | Given a circle \( K(O, R) \) with center \( O \) and radius \( R \), and a tangent line \( (e) \) at point \( A \). A line parallel to \( OA \) intersects the circle at \( B \) and \( C \), and the tangent line \( (e) \) at \( D \) (with \( C \) between \( B \) and \( D \)). The antidiametric point of \( C \) with resp... | ### Part (i): Examine if \( \triangle CEF \) is isosceles.
1. **Identify the geometric relationships:**
- \( K(O, R) \) is a circle with center \( O \) and radius \( R \).
- \( (e) \) is a tangent to \( K \) at point \( A \).
- A line parallel to \( OA \) intersects \( K \) at points \( B \) and \( C \), and ... |
How many two digit numbers have exactly $4$ positive factors? $($ Here $1$ and the number $n$ are also considered as factors of $n. )$ | Okay, let's see. The question is asking how many two-digit numbers have exactly 4 positive factors. Factors include 1 and the number itself, so I need to find two-digit numbers where there are exactly two more factors besides these. Hmm, I remember something about prime numbers and factors. Wait, primes have exactly tw... | To determine how many two-digit numbers have exactly 4 positive factors, we need to consider numbers that are either the product of two distinct primes (semiprimes) or the cube of a prime.
### Key Steps:
1. **Semiprimes (Product of Two Distinct Primes)**:
- List all two-digit numbers that are the product of two d... | To determine how many two-digit numbers have exactly 4 positive factors, we need to understand the structure of numbers with exactly 4 factors.
A number \( n \) has exactly 4 positive factors if and only if it can be expressed in one of the following forms:
1. \( n = p^3 \), where \( p \) is a prime number.
2. \( n =... |
**p1.** Three real numbers $a, b, c$ are such that $ac + bc + c^2 < 0$ . Prove that $b^2 > 4ac$ .**p2.** Prove that the number $n^4 + 4^n$ is prime if an only if $n = 1$ (here $n$ is an integer).**p3.** You are given three equal coins and a pen. A circle with the diameter equal to the one of coins is drawn on... | Okay, let's dive into problem p1. So, we have three real numbers a, b, c such that ac + bc + c² < 0. We need to prove that b² > 4ac. Hmm, let's start by understanding the given inequality.
First, let me write down the given condition: ac + bc + c² < 0. Maybe I can factor out a c from some terms. Let's see:
ac + bc +... | Given three real numbers \(a, b, c\) such that \(ac + bc + c^2 < 0\), we need to prove that \(b^2 > 4ac\).
First, we factor the given inequality:
\[
ac + bc + c^2 = c(a + b + c) < 0
\]
This implies that either \(c > 0\) and \(a + b + c < 0\), or \(c < 0\) and \(a + b + c > 0\).
We introduce \(k = \frac{a}{c}\) and \(... | **p1.** Three real numbers \( a, b, c \) are such that \( ac + bc + c^2 < 0 \). Prove that \( b^2 > 4ac \).
1. Consider the quadratic equation \( ax^2 + bx + c = 0 \).
2. The discriminant of this quadratic equation is given by \( \Delta = b^2 - 4ac \).
3. For the quadratic equation to have real roots, the discriminant... |
Find the number of unordered pairs $\{ A,B \}$ of subsets of an n-element set $X$ that satisfies the following:
(a) $A \not= B$
(b) $A \cup B = X$ | Alright, so I need to find the number of unordered pairs {A, B} of subsets of an n-element set X such that A ≠ B and A ∪ B = X. Let me start by understanding the problem better.
First, let's parse the requirements. We have a set X with n elements. We need to consider all possible pairs of subsets A and B of X. These p... | To find the number of unordered pairs \(\{A, B\}\) of subsets of an \(n\)-element set \(X\) that satisfy \(A \neq B\) and \(A \cup B = X\), we start by considering the total number of ordered pairs \((A, B)\) such that \(A \cup B = X\).
For each element in \(X\), there are three possibilities: it can be in \(A\) only... | To find the number of unordered pairs $\{A, B\}$ of subsets of an $n$-element set $X$ that satisfy the conditions:
(a) $A \neq B$
(b) $A \cup B = X$
we can use a recursive approach. Let's denote $a_n$ as the number of such pairs for an $n$-element set $X$.
1. **Base Case:**
For $n = 1$, the set $X$ has only one el... |
For an integer $n > 0$ , denote by $\mathcal F(n)$ the set of integers $m > 0$ for which the polynomial $p(x) = x^2 + mx + n$ has an integer root. [list=a] [*] Let $S$ denote the set of integers $n > 0$ for which $\mathcal F(n)$ contains two consecutive integers. Show that $S$ is infinite but \[ \sum_... | Okay, so I need to tackle this problem about quadratic polynomials with integer coefficients and integer roots. Let me start by understanding what the problem is asking.
First, for a given integer n > 0, F(n) is the set of integers m > 0 such that the polynomial p(x) = x² + m x + n has an integer root. The first part ... | To address the problem, we analyze the sets $\mathcal{F}(n)$ and demonstrate the required properties for parts (a) and (b).
### Part (a)
**Showing $S$ is infinite:**
For an integer $n > 0$, $\mathcal{F}(n)$ contains sums of pairs of divisors of $n$. We construct $n$ such that $\mathcal{F}(n)$ includes two consecuti... | ### Part (a)
1. **Claim:** We can fully classify \( S \) as follows:
\[
S = \{x(x+1)y(y+1) : x, y \in \mathbb{Z}_{>0}\}.
\]
**Proof:**
- **(Implication \(\impliedby\)):** Given \( n = x(x+1)y(y+1) \), choosing \( m = 2xy + x + y \) works:
\[
m = x(y+1) + y(x+1)
\]
\[
m+1 = xy + (... |
Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$ tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ l... | Okay, so I need to prove that n has to be 23 given these conditions. Let me start by understanding the problem step by step.
First, the setup: n is defined as 2k - 1, where k is an integer greater than or equal to 6. So n is an odd number starting from 11 (when k=6) up. The set T consists of all n-tuples of 0s and 1s.... | Given \( n = 2k - 1 \) where \( k \geq 6 \) is an integer, we need to prove that \( n = 23 \) under the conditions that there exists a subset \( S \) of \( T \) with \( 2^k \) elements such that every element \( x \in T \) is within Hamming distance 3 of exactly one element \( y \in S \).
1. **Understanding the Proble... | 1. **Define the problem and given conditions:**
- Let \( n = 2k - 1 \) where \( k \geq 6 \) is an integer.
- \( T \) is the set of all \( n \)-tuples \( (x_1, x_2, \ldots, x_n) \) where \( x_i \in \{0,1\} \) for all \( i = 1, 2, \ldots, n \).
- For \( x = (x_1, x_2, \ldots, x_n) \in T \) and \( y = (y_1, y_2, ... |
Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$
Show that $m_n \to \frac{1}{2}$ , as $n \to \infty.$ | Okay, so I need to show that the minimum value of the function f_n(x) = sum_{k=0}^{2n} x^k approaches 1/2 as n goes to infinity. Hmm, let's start by understanding the function itself.
First, the function f_n(x) is a finite geometric series. The sum of a geometric series from k=0 to m is (1 - x^{m+1})/(1 - x) when x ≠... | To show that \( m_n \to \frac{1}{2} \) as \( n \to \infty \), we start by analyzing the function \( f_n(x) = \sum_{k=0}^{2n} x^k \), which is a finite geometric series. The sum can be expressed as:
\[
f_n(x) = \frac{1 - x^{2n+1}}{1 - x} \quad \text{for} \quad x \neq 1
\]
To find the minimum value \( m_n \), we take t... | 1. Consider the function \( f_n(x) = \sum_{k=0}^{2n} x^k \). We need to find the smallest value of this function, denoted as \( m_n \), and show that \( m_n \to \frac{1}{2} \) as \( n \to \infty \).
2. First, we take the derivative of \( f_n(x) \) to find the critical points:
\[
f_n'(x) = \sum_{k=1}^{2n} k x^{k-... |
The product of $10$ integers is $1024$ . What is the greatest possible sum of these $10$ integers? | Okay, so I need to figure out the greatest possible sum of 10 integers whose product is 1024. Let me start by understanding the problem properly.
First, the product of 10 integers is 1024. I need to find such integers that their sum is as large as possible. Hmm. Alright, let's break this down.
First, 1024 is a powe... | The problem requires finding the greatest possible sum of 10 integers whose product is 1024. Here's the systematic breakdown:
1. **Prime Factorization**: \(1024 = 2^{10}\), so the integers must multiply to \(2^{10}\).
2. **Key Insight**: For a fixed product, the sum is maximized when the numbers are as unequal as po... | 1. We start by noting that the product of 10 integers is 1024. We need to find the greatest possible sum of these 10 integers.
2. Since 1024 is a power of 2, we can express it as \(1024 = 2^{10}\).
3. To maximize the sum of the integers, we should use the smallest possible integers that multiply to 1024. The smallest i... |
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$ , then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in ... | Okay, so I have this problem here where I need to find the minimum positive integer N such that if there's a set S of integers with size N that contains a complete residue system modulo m, then there must be a nonempty subset A of S where the sum of the elements in A is divisible by n. Hmm, let me try to unpack this st... | To find the minimum positive integer \( N \) such that any set \( S \) of integers with \( |S| = N \) containing a complete residue system modulo \( m \) has a nonempty subset \( A \subseteq S \) with \( n \mid \sum_{x \in A} x \), we need to consider the interaction between the residues modulo \( m \) and the modulus ... | 1. **Interpreting the Problem Statement:**
The problem asks for the minimum positive integer \( N \) such that if a set \( S \) of integers contains a complete residue system modulo \( m \) and \( |S| \geq N \), then there exists a nonempty subset \( A \subseteq S \) such that \( n \mid \sum_{x \in A} x \).
2. **Un... |
Let $a_1, a_2, \ldots$ be a sequence of real numbers such that $a_1=4$ and $a_2=7$ such that for all integers $n$ , $\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}$ forms an arithmetic progression, and $a_{2n}, a_{2n+1}, a_{2n+2}$ forms an arithmetic progression. Find, with proof, the prime factori... | Okay, let's see. I need to find the prime factorization of a_{2019} in this sequence. The sequence starts with a_1=4 and a_2=7. There are two different arithmetic progression conditions given for odd and even terms. Hmm. Let me break down the problem step by step.
First, let's parse the problem again to make sure I un... | Given the sequence \(a_1, a_2, \ldots\) with initial terms \(a_1 = 4\) and \(a_2 = 7\), we need to determine the prime factorization of \(a_{2019}\). The sequence follows two arithmetic progression (AP) conditions:
1. For all integers \(n\), \(\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}\) forms an AP.
2. F... | 1. Given the sequence \(a_1, a_2, \ldots\) with \(a_1 = 4\) and \(a_2 = 7\), we know that \(\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}\) forms an arithmetic progression. This implies:
\[
\frac{1}{a_{2n+1}} = 2 \cdot \frac{1}{a_{2n}} - \frac{1}{a_{2n-1}}
\]
Additionally, \(a_{2n}, a_{2n+1}, a_{... |
A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$ , $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$ . Find the measure of the angle $\angle ACE$ . | Okay, let me try to figure out how to solve this problem. So, we have a pentagon ABCDE that's circumscribed about a circle. That means there's a circle inside the pentagon that's tangent to all five sides. The angles at vertices A, C, and E are each 100 degrees. We need to find the measure of angle ACE. Hmm, angle ACE ... | Given a pentagon \(ABCDE\) circumscribed about a circle with angles at vertices \(A\), \(C\), and \(E\) each equal to \(100^\circ\), we need to find the measure of angle \(\angle ACE\).
1. **Understanding the Tangential Pentagon**:
- A tangential pentagon has an incircle tangent to all its sides.
- The sum of th... | 1. **Identify the given information and the goal:**
- Pentagon \(ABCDE\) is circumscribed about a circle.
- Angles at vertices \(A\), \(C\), and \(E\) are \(100^\circ\).
- We need to find the measure of \(\angle ACE\).
2. **Understand the properties of a circumscribed pentagon:**
- The circle is tangent to... |
Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and
\[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\]
Determine $ f \left( \frac{1}{7} \rig... | Okay, so I need to determine the value of f(1/7) given these conditions. Let me try to parse the problem step by step.
First, we have a function f that's continuous on [0,1], with f(0) = 0 and f(1) = 1. Then there's this functional equation: for any x and y in [0,1] where x ≤ y, the function evaluated at the midpoint... | Given a continuous function \( f \) on \([0, 1]\) satisfying \( f(0) = 0 \), \( f(1) = 1 \), and the functional equation
\[ f \left( \frac{x + y}{2} \right) = (1 - a) f(x) + a f(y) \quad \forall x, y \in [0, 1] \text{ with } x \leq y, \]
we need to determine \( f \left( \frac{1}{7} \right) \).
1. **Functional Equation... | 1. Given the function \( f \) is continuous on \([0, 1]\) and satisfies the conditions \( f(0) = 0 \), \( f(1) = 1 \), and
\[
f \left( \frac{x+y}{2} \right) = (1-a) f(x) + a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.
\]
We need to determine \( f \left( \frac{1}{7} \right) \).
2. Let's consid... |
Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$ . If $2000|a_{1999}$ , find the smallest $n\ge 2$ such that $2000|a_n$ . | Okay, so I have this problem here about a sequence of integers defined by a recurrence relation, and I need to find the smallest n ≥ 2 such that 2000 divides a_n, given that 2000 divides a_1999. Let me try to break this down step by step.
First, let's parse the problem statement again. The sequence {a_n} satisfies the... | Given the sequence \(\{a_n\}\) of integers satisfying the recurrence relation \((n-1)a_{n+1} = (n+1)a_n - 2(n-1)\) for all \(n \ge 1\), and given that \(2000 \mid a_{1999}\), we need to find the smallest \(n \ge 2\) such that \(2000 \mid a_n\).
1. **Recurrence Relation Analysis**:
- The recurrence relation is rearr... | 1. Given the sequence $\{a_n\}$ of integers satisfying the recurrence relation:
\[
(n-1)a_{n+1} = (n+1)a_n - 2(n-1) \quad \forall n \ge 1
\]
we start by making a substitution to simplify the recurrence. Let:
\[
a_n = b_n + 2(n-1)
\]
Substituting $a_n = b_n + 2(n-1)$ into the recurrence relation,... |
Let $R$ be the rectangle on the cartesian plane with vertices $(0,0)$ , $(5,0)$ , $(5,7)$ , and $(0,7)$ . Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$ .
*Proposed by Boyan Litchev*
<details><summary>Solution</summary>*... | Okay, so I need to find the number of squares with sides parallel to the axes and vertices that are lattice points within the rectangle R with vertices (0,0), (5,0), (5,7), and (0,7). Let me try to visualize this. The rectangle is 5 units wide along the x-axis and 7 units tall along the y-axis. All the squares we're lo... | To find the number of squares with sides parallel to the axes and vertices that are lattice points within the rectangle \( R \) with vertices \((0,0)\), \((5,0)\), \((5,7)\), and \((0,7)\), we consider the following steps:
1. **Determine the dimensions of the rectangle**: The rectangle is 5 units wide (along the x-axi... | To find the number of squares with sides parallel to the axes and vertices that are lattice points within the rectangle \( R \) with vertices \((0,0)\), \((5,0)\), \((5,7)\), and \((0,7)\), we need to count the number of such squares for each possible side length \( n \).
1. **Identify the dimensions of the rectangle:... |
Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$ ,
(i) $(x + 1)\star 0 = (0\star x) + 1$ (ii) $0\star (y + 1) = (y\star 0) + 1$ (iii) $(x + 1)\star (y + 1) = (x\star y) + 1$ .
If $123\star 456 = 789$ , find $246\star... | Okay, so I need to solve this problem where there's a special operation called "star" defined on nonnegative integers. The problem gives three properties for this operation, and then tells me that 123 ⋆ 456 = 789. Using that information, I have to find what 246 ⋆ 135 is. Alright, let's try to break this down step by st... | Given the operation \(\star\) defined on the set of nonnegative integers with the properties:
1. \((x + 1) \star 0 = (0 \star x) + 1\)
2. \(0 \star (y + 1) = (y \star 0) + 1\)
3. \((x + 1) \star (y + 1) = (x \star y) + 1\)
We need to find \(246 \star 135\) given that \(123 \star 456 = 789\).
First, we deduced that th... | 1. **Define the operation for base cases:**
Let $0 \star 0 = c$. We need to determine the value of $c$ and understand the operation $\star$ for other values.
2. **Prove $n \star 0 = 0 \star n = n + c$ by induction:**
- **Base case:** For $n = 0$, we have $0 \star 0 = c$ which is given.
- **Induction hypothesi... |
Let $a\# b$ be defined as $ab-a-3$ . For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$ . | Okay, so I have this problem here where I need to compute (2#0)#(1#4), and the operation a#b is defined as ab - a - 3. Let me try to figure this out step by step. First, I should probably make sure I understand how the operation works. The example they gave is 4#5 = 20 - 4 - 3 = 13. Let me break that down. So, a is 4, ... | To compute \((2\#0)\#(1\#4)\) where the operation \(a \# b\) is defined as \(ab - a - 3\):
1. **Compute \(2 \# 0\):**
\[
2 \# 0 = (2 \cdot 0) - 2 - 3 = 0 - 2 - 3 = -5
\]
2. **Compute \(1 \# 4\):**
\[
1 \# 4 = (1 \cdot 4) - 1 - 3 = 4 - 1 - 3 = 0
\]
3. **Compute \((-5) \# 0\):**
\[
(-5) \# 0 = ... | 1. First, compute \(2 \# 0\):
\[
2 \# 0 = 2 \cdot 0 - 2 - 3 = 0 - 2 - 3 = -5
\]
2. Next, compute \(1 \# 4\):
\[
1 \# 4 = 1 \cdot 4 - 1 - 3 = 4 - 1 - 3 = 0
\]
3. Now, use the results from steps 1 and 2 to compute \((-5) \# 0\):
\[
-5 \# 0 = (-5) \cdot 0 - (-5) - 3 = 0 + 5 - 3 = 2
\]
The fin... |
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) | Okay, so I need to find the largest positive integer n that's not divisible by 10, and it has to be a multiple of each number formed by deleting two consecutive digits from it. Also, those two digits can't be in the first or last position. Hmm, let me try to break this down.
First, let's understand the problem. The nu... | The largest positive integer \( n \) not divisible by 10 that satisfies the given conditions is **12625**.
**Step-by-Step Explanation:**
1. **Problem Understanding:** We need a number \( n \) such that deleting any two consecutive middle digits results in a number that divides \( n \). \( n \) must not end with 0.
... | 1. **Identify the problem constraints:**
We need to find the largest positive integer \( n \) not divisible by \( 10 \) which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of \( n \).
2. **Consider the structure of \( n \):**
Let ... |
Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers. $ \textbf{(A)}\ 55 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 37$ | Alright, so I need to find the sum of distinct residues of the number \(2012^n + m^2\) modulo 11, where \(m\) and \(n\) are positive integers. The answer choices are given, so maybe after some calculations, I can match my result to one of them. Let me start by breaking down the problem.
First, I should simplify \(2012... | To find the sum of distinct residues of \(2012^n + m^2 \mod 11\) where \(m\) and \(n\) are positive integers, we start by simplifying \(2012 \mod 11\).
First, we calculate:
\[
2012 \div 11 = 182 \text{ remainder } 10 \implies 2012 \equiv 10 \equiv -1 \mod 11
\]
Thus, \(2012^n \equiv (-1)^n \mod 11\). This means \(201... | 1. First, we need to simplify \(2012^n \mod 11\). Notice that:
\[
2012 \equiv -1 \mod 11
\]
Therefore,
\[
2012^n \equiv (-1)^n \mod 11
\]
This means \(2012^n \mod 11\) will be either \(1\) or \(-1\) depending on whether \(n\) is even or odd.
2. Next, we need to consider \(m^2 \mod 11\). The pos... |
Let $ a_0$ , $ a_1$ , $ a_2$ , $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$ . Prove that $ a_n\ge 2^n$ for all $ n\ge 0$ .
*Proposed by Morteza... | Okay, so I need to prove that for this sequence of positive integers a₀, a₁, a₂, ..., where the gcd of any two consecutive terms is greater than the previous term, each term aₙ is at least 2ⁿ. Hmm. Let me start by understanding the problem statement again.
The condition given is that for each i ≥ 1, gcd(aᵢ, aᵢ₊₁) > aᵢ... | To prove that \( a_n \geq 2^n \) for all \( n \geq 0 \) given the sequence of positive integers \( a_0, a_1, a_2, \ldots \) where the greatest common divisor of any two consecutive terms is greater than the preceding term, we use mathematical induction.
**Base Case**:
For \( n = 0 \), \( a_0 \) is a positive integer, ... | We are given a sequence of positive integers \(a_0, a_1, a_2, \ldots\) such that the greatest common divisor (gcd) of any two consecutive terms is greater than the preceding term, i.e., \(\gcd(a_i, a_{i+1}) > a_{i-1}\). We need to prove that \(a_n \ge 2^n\) for all \(n \ge 0\).
1. **Base Case:**
- For \(n = 0\), \(... |
Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$ . What is the maximum possible value of $b^2$ ? | Okay, let's tackle this problem step by step. The problem states that a and b are integers, and a + b is a root of the quadratic equation x² + a x + b = 0. We need to find the maximum possible value of b².
First, let me parse the problem again to make sure I understand it correctly. We have two integers a and b. The ... | Given that \(a\) and \(b\) are integers and \(a + b\) is a root of the quadratic equation \(x^2 + ax + b = 0\), we need to find the maximum possible value of \(b^2\).
First, substituting \(x = a + b\) into the equation, we get:
\[
(a + b)^2 + a(a + b) + b = 0
\]
Expanding and simplifying this equation:
\[
a^2 + 2ab + ... | 1. We are given that \(a + b\) is a root of the quadratic equation \(x^2 + ax + b = 0\). Therefore, we can substitute \(x = a + b\) into the equation:
\[
(a + b)^2 + a(a + b) + b = 0
\]
Expanding and simplifying, we get:
\[
a^2 + 2ab + b^2 + a^2 + ab + b = 0
\]
\[
2a^2 + 3ab + b^2 + b = 0
... |
The operation $*$ is defined by $a*b=a+b+ab$ , where $a$ and $b$ are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\]
*2017 CCA Math Bonanza Team Round #3* | Okay, so I need to solve this problem where the operation * is defined by a*b = a + b + ab. The expression given is a nested operation starting from 1/2 up to 1/11. The question is to find the value of this nested operation. Let me try to break this down step by step.
First, let me understand the operation *. The defi... | The operation \( * \) is defined by \( a * b = a + b + ab \). We need to find the value of the nested operation:
\[
\frac{1}{2} * \left( \frac{1}{3} * \left( \cdots * \left( \frac{1}{10} * \frac{1}{11} \right) \right) \right).
\]
First, we observe that the operation \( * \) can be rewritten using the identity \( a * ... | 1. The operation \( * \) is defined by \( a * b = a + b + ab \). We need to find the value of
\[
\frac{1}{2} * \left( \frac{1}{3} * \left( \cdots * \left( \frac{1}{9} * \left( \frac{1}{10} * \frac{1}{11} \right) \right) \right) \right).
\]
2. First, let's explore the properties of the operation \( * \). We c... |
Let $k$ and $n$ be integers such that $k\ge 2$ and $k \le n \le 2k-1$ . Place rectangular tiles, each of size $1 \times k$ , or $k \times 1$ on a $n \times n$ chessboard so that each tile covers exactly $k$ cells and no two tiles overlap. Do this until no further tile can be placed in this way. For each s... | Okay, so I need to figure out the minimum number of 1×k or k×1 tiles needed to cover an n×n chessboard such that no more tiles can be placed. The constraints are that k is at least 2 and n is between k and 2k-1, inclusive. Hmm, let's break this down step by step.
First, let's make sure I understand the problem correct... | To determine the minimum number of \(1 \times k\) or \(k \times 1\) tiles needed to cover an \(n \times n\) chessboard such that no further tiles can be placed, we analyze the problem for integers \(k \geq 2\) and \(k \leq n \leq 2k - 1\).
1. **Understanding the Constraints**: The board size \(n\) ranges from \(k\) to... | 1. **Case \( n = k \):**
- When \( n = k \), the \( n \times n \) chessboard is a \( k \times k \) square.
- We can place \( k \) tiles of size \( 1 \times k \) or \( k \times 1 \) to cover the entire board without any overlap.
- Therefore, the minimum number of tiles required is \( k \).
2. **Case \( k < n <... |
A finite non-empty set of integers is called $3$ -*good* if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\{0,1,2,\ldots,9\}$ . | Alright, so I need to find the number of 3-good subsets of the set {0,1,2,...,9}. A 3-good subset is one where the sum of its elements is divisible by 3. Hmm, okay. Let me start by understanding the problem better.
First, the set in question is {0,1,2,...,9}. So it's the first ten non-negative integers. A subset is 3-... | To find the number of 3-good subsets of the set \(\{0, 1, 2, \ldots, 9\}\), we start by categorizing the elements based on their residues modulo 3:
- **Residue 0**: \(0, 3, 6, 9\) (4 elements)
- **Residue 1**: \(1, 4, 7\) (3 elements)
- **Residue 2**: \(2, 5, 8\) (3 elements)
A subset is 3-good if the sum of its elem... | 1. **Define the generating function:**
Consider the generating function \( P(x) = 2 \prod_{k=1}^9 (1 + x^k) \). This function encodes the subsets of \(\{1, 2, \ldots, 9\}\) where each term \(1 + x^k\) represents the choice of either including or not including the element \(k\) in a subset.
2. **Expand the generatin... |
What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$ ? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$ | Okay, so I need to find the sum of the real roots of the equation \( x^4 - 7x^3 + 14x^2 - 14x + 4 = 0 \). The answer choices are from A to E, with numbers 1 to 5. Hmm, let me think step by step.
First, I know that for any polynomial equation, the sum of all its roots (real and complex) can be found using Vieta's formu... | To find the sum of the real roots of the equation \( x^4 - 7x^3 + 14x^2 - 14x + 4 = 0 \), we start by factoring the polynomial.
First, we consider factoring the quartic polynomial into two quadratic polynomials:
\[
(x^2 + ax + b)(x^2 + cx + d)
\]
Expanding and comparing coefficients with the original polynomial, we s... | 1. **Factor the polynomial**: We start with the polynomial equation \(x^4 - 7x^3 + 14x^2 - 14x + 4 = 0\). We are given that it can be factored as \((x^2 - 5x + 2)(x^2 - 2x + 2) = 0\).
2. **Solve each quadratic equation**:
- For the first quadratic equation \(x^2 - 5x + 2 = 0\):
\[
x = \frac{-b \pm \sqrt{b... |
Three numbers have sum $k$ (where $k\in \mathbb{R}$ ) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression.
Find those numbers? | Okay, let's see. I need to find three numbers that form an arithmetic progression, and their sum is k. Then, if I keep the first two numbers the same and add k/6 to the third number, they form a geometric progression. Hmm. Let me break this down step by step.
First, let's recall what an arithmetic progression (AP) is.... | Given three numbers in arithmetic progression (AP) with sum \( k \), we denote these numbers as \( a - d \), \( a \), and \( a + d \). The sum of these numbers is \( 3a = k \), hence \( a = \frac{k}{3} \). Therefore, the numbers are \( \frac{k}{3} - d \), \( \frac{k}{3} \), and \( \frac{k}{3} + d \).
When the first tw... | 1. Let the three numbers in arithmetic progression be \(a - d\), \(a\), and \(a + d\). Given that their sum is \(k\), we have:
\[
(a - d) + a + (a + d) = k
\]
Simplifying, we get:
\[
3a = k \implies a = \frac{k}{3}
\]
Therefore, the three numbers are:
\[
\frac{k}{3} - d, \frac{k}{3}, \frac... |
You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m/s throughout the race. Meanwhile, Al starts running at a constant speed of $ 1$ m/s. Thereafter, when exactly $ 1$ second has passed from when Al last changed its speed, Al’s speed instantaneousl... | Okay, let's try to figure out how long this race was. So, the problem is about me racing against this AI robot called Al. I can run at a constant speed of 10 m/s the whole time. Al, on the other hand, starts at 1 m/s and every second, its speed increases by 1 m/s. That means in the first second, it's going 1 m/s, in th... | To determine the distance of the race, we analyze the performance of both the runner and the AI robot, Al. The runner maintains a constant speed of 10 m/s, while Al's speed increases by 1 m/s every second starting from 1 m/s. Al beats the runner by exactly 1 second.
1. **Distance Covered by Al**:
- Al's speed in th... | 1. Let's denote the length of the race by \( d \) meters.
2. You run at a constant speed of \( 10 \) m/s, so the time it takes for you to complete the race is \( \frac{d}{10} \) seconds.
3. Al's speed increases by \( 1 \) m/s every second. In the \( k \)-th second, Al runs at \( k \) m/s.
4. The distance Al covers in t... |
Given that $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$ , the value of $$ \sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} $$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ . | Okay, so I need to compute the sum from n = 3 to n = 10 of (C(n,2) divided by [C(n,3) times C(n+1,3)]), and then express the result as a reduced fraction m/n and find m + n. Alright, let's start by understanding each part of the problem.
First, let's recall that the binomial coefficient C(n, k) is equal to n! / [k!(n ... | To solve the given sum, we start by expressing the binomial coefficients in terms of factorials and simplifying the term inside the sum:
\[
\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} = \frac{\frac{n(n-1)}{2}}{\frac{n(n-1)(n-2)}{6} \cdot \frac{(n+1)n(n-1)}{6}} = \frac{18}{n(n-1)(n-2)(n+1)}
\]
Next, we decompose t... | 1. **Express the given sum in terms of binomial coefficients:**
\[
\sum_{n=3}^{10} \frac{\binom{n}{2}}{\binom{n}{3} \binom{n+1}{3}}
\]
2. **Simplify the binomial coefficients:**
Recall the binomial coefficient formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
Therefore,
\[
\binom{n}{2} = ... |
Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions:
(i) For any positive integer $n$ , $b_n \leq c_n$ ;
(ii) For any positive integer $n$ , $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$ . | Okay, let's see. I need to find all real number sequences {b_n} and {c_n} such that for every positive integer n, b_n ≤ c_n, and for each n, b_{n+1} and c_{n+1} are the two roots of the quadratic equation x² + b_n x + c_n = 0. Hmm. So each subsequent pair (b_{n+1}, c_{n+1}) are roots of the previous equation. Let me tr... | To find all real number sequences \(\{b_n\}\) and \(\{c_n\}\) that satisfy the given conditions, we start by analyzing the recurrence relations derived from the quadratic equation roots:
1. For any positive integer \(n\), the roots \(b_{n+1}\) and \(c_{n+1}\) of the equation \(x^2 + b_n x + c_n = 0\) must satisfy:
... | To find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the given conditions, we will proceed step-by-step.
1. **Initial Conditions and Roots**:
- Given that for any positive integer $n$, $b_n \leq c_n$.
- For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ are the two ... |
Hey,
This problem is from the VTRMC 2006.
3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$ , $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$ . Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6).
As,... | Okay, so I need to figure out the last digit of F(2006), which is the same as finding F(2006) mod 10. The Fibonacci sequence is defined by F(0) = 0, F(1) = 1, and each subsequent term is the sum of the two previous ones. Since we’re dealing with mod 10, the last digits of Fibonacci numbers repeat in a cycle called the ... | To determine the last digit of \( F(2006) \), we need to find \( F(2006) \mod 10 \). The Fibonacci sequence modulo 10 repeats in a cycle known as the Pisano period. For modulus 10, the Pisano period is 60. This means the last digits of the Fibonacci numbers repeat every 60 terms.
First, we reduce 2006 modulo 60:
\[
20... | To determine the last digit of \( F(2006) \), we need to find \( F(2006) \mod 10 \). We will use the periodicity of the Fibonacci sequence modulo 10.
1. **Establish the periodicity of Fibonacci numbers modulo 10:**
- The Fibonacci sequence modulo 10 is periodic. This means that after a certain number of terms, the ... |
A point $P$ is interior to the triangle $ABC$ such that $AP \perp BC$ . Let $E, F$ be the projections of $CA, AB$ . Suppose that the tangents at $E, F$ of the circumcircle of triangle $AEF$ meets at a point on $BC$ . Prove that $P$ is the orthocenter of triangle $ABC$ .
Do Thanh Son, High School of Nat... | Okay, so I need to prove that P is the orthocenter of triangle ABC given these conditions. Let me start by understanding the problem step by step.
First, the problem states that P is an interior point of triangle ABC such that AP is perpendicular to BC. Then, E and F are the projections of CA and AB. Wait, projection... | To prove that \( P \) is the orthocenter of triangle \( ABC \) given the conditions, we start by noting that \( P \) is an interior point of \( ABC \) such that \( AP \perp BC \). The projections \( E \) and \( F \) of \( P \) onto \( CA \) and \( AB \) respectively lie on the circumcircle of \( \triangle AEF \). The t... | 1. **Given Information and Initial Setup:**
- Point \( P \) is interior to triangle \( ABC \) such that \( AP \perp BC \).
- Let \( E \) and \( F \) be the projections of \( C \) and \( B \) onto \( AB \) and \( AC \) respectively.
- The tangents at \( E \) and \( F \) of the circumcircle of triangle \( AEF \)... |
**p1.** A man is walking due east at $2$ m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to $4$ m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)?**p2.** Prove that any tri... | Okay, let me try to work through problem p1. So, the problem is about a man walking east and perceiving the wind direction, and we need to find the actual speed of the wind. Hmm, okay.
First, let me parse the problem again. The man is walking due east at 2 mph, and the wind appears to be coming from the north. When h... | To solve the problem of determining the speed of the wind, we analyze the relative wind velocity as perceived by the man when he is walking at different speeds.
1. **First Case (Walking East at 2 mph):**
- The man's velocity is \( \mathbf{v}_1 = (2, 0) \).
- The wind appears to come from the north, meaning the ... | To show that the common difference of three prime numbers, each greater than \(3\), in arithmetic progression is a multiple of \(6\), we proceed as follows:
1. **Representation of Primes Greater than 3**:
Any prime number greater than \(3\) can be written in the form \(6k \pm 1\) for some integer \(k\). This is bec... |
Given a natural number $k$ , find the smallest natural number $C$ such that $$ \frac C{n+k+1}\binom{2n}{n+k} $$ is an integer for every integer $n\ge k$ . | Okay, so I need to find the smallest natural number C such that the expression C divided by (n + k + 1) multiplied by the binomial coefficient (2n choose n + k) is an integer for every integer n ≥ k. Hmm, let's break this down step by step.
First, let's understand the problem. For a given natural number k, we need to ... | Given a natural number \( k \), we need to find the smallest natural number \( C \) such that \( \frac{C}{n+k+1} \binom{2n}{n+k} \) is an integer for every integer \( n \ge k \).
To solve this, we start by analyzing the expression \( \frac{C}{n+k+1} \binom{2n}{n+k} \). We need this to be an integer for all \( n \ge k ... | 1. **Claim and Initial Check:**
We claim that the smallest natural number \( C \) such that
\[
\frac{C}{n+k+1} \binom{2n}{n+k}
\]
is an integer for every integer \( n \ge k \) is \( C = 2k+1 \).
2. **Lower Bound for \( C \):**
To establish a lower bound for \( C \), consider the case when \( n = k \... |
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$ . Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$ . If $AB=10$ and $CD=15$ , compute the maximum possible value of $XY$ . | Alright, let's tackle this trapezoid problem. Hmm, okay, so we have trapezoid ABCD with AB parallel to CD, and AB is perpendicular to BC. That means AB is not only parallel to CD but also perpendicular to BC, so angle ABC is 90 degrees. Let me visualize this: AB is the top base, BC is the vertical side, CD is the botto... | Given a trapezoid \(ABCD\) with \(AB \parallel CD\) and \(AB \perp BC\), where \(AB = 10\) and \(CD = 15\). We need to find the maximum possible value of \(XY\) where \(X\) is a point on segment \(\overline{AD}\) such that \(AD\) externally bisects \(\angle BXC\), and \(Y\) is the intersection of diagonals \(AC\) and \... | 1. **Identify the given information and setup the problem:**
- $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$.
- $X$ is a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally.
- $Y$ is the intersection of $AC$ and $BD$.
- $AB = 10$ and $CD = 15$.
2. **Introduce po... |
The ordered pair of four-digit numbers $(2025, 3136)$ has the property that each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the fi rst number. Find, with proof, all ordered pairs of fi ve-digit numbers and ordered pairs of six-digit numbers... | Okay, so I need to find all ordered pairs of five-digit numbers and six-digit numbers where each number is a perfect square, and each digit of the second number is exactly 1 more than the corresponding digit of the first number. The example given is (2025, 3136), which are both four-digit squares (45² and 56²) and each... | To find ordered pairs of five-digit and six-digit numbers where each number is a perfect square and each digit of the second number is 1 more than the corresponding digit of the first number, we start by analyzing the given example (2025, 3136). This pair shows that the difference between the two squares is 1111, which... | To solve this problem, we need to find ordered pairs of five-digit and six-digit numbers \((a^2, b^2)\) such that each number in the pair is a perfect square and each digit of \(b^2\) is 1 more than the corresponding digit of \(a^2\).
### Five-Digit Numbers
1. **Set up the equation:**
\[
a^2 - b^2 = 11111
\]... |
Find the area of inscribed convex octagon, if the length of four sides is $2$ , and length of other four sides is $ 6\sqrt {2}$ . $\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$ | Okay, so I need to find the area of an inscribed convex octagon where four of its sides are length 2 and the other four sides are 6√2. The answer choices are given, so maybe I can use some properties of cyclic octagons to figure this out. Let me start by recalling what I know about regular octagons, but this one isn't ... | To find the area of an inscribed convex octagon with four sides of length 2 and four sides of length \(6\sqrt{2}\), we start by noting it is a cyclic octagon. The central angles for the sides of length 2 are denoted as \(\theta\) and for the sides of length \(6\sqrt{2}\) as \(\phi\). Since the sum of all central angles... | 1. **Identify the structure of the octagon:**
The octagon is inscribed, meaning it is inside a circle. The sides are arranged in the pattern \(2, 6\sqrt{2}, 2, 6\sqrt{2}, 2, 6\sqrt{2}, 2, 6\sqrt{2}\).
2. **Break down the octagon into simpler shapes:**
- The octagon can be divided into four \(45^\circ-45^\circ-90... |
Let $x$ , $y$ , $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$ . Find the greatest possible value of $x+y+z$ . | Okay, let's tackle this problem step by step. So, we have three positive integers x, y, z with x < y < z, and they satisfy the equation x + xy + xyz = 37. We need to find the greatest possible value of x + y + z. Hmm, let's break this down.
First, the equation given is x + xy + xyz = 37. Let me factor out x from the f... | Given the equation \(x + xy + xyz = 37\) where \(x\), \(y\), and \(z\) are positive integers such that \(x < y < z\), we need to find the greatest possible value of \(x + y + z\).
First, we factor out \(x\) from the equation:
\[
x(1 + y + yz) = 37
\]
Since 37 is a prime number, the divisors of 37 are 1 and 37. Given \... | 1. Given the equation \( x + xy + xyz = 37 \), we start by factoring out \( x \):
\[
x(1 + y + yz) = 37
\]
Since \( x \), \( y \), and \( z \) are positive integers and \( x < y < z \), \( x \) must be a divisor of 37. The divisors of 37 are 1 and 37, but since \( x < y < z \), \( x \) must be 1 because 37 ... |
**interesting sequence** $n$ is a natural number and $x_1,x_2,...$ is a sequence of numbers $1$ and $-1$ with these properties:
it is periodic and its least period number is $2^n-1$ . (it means that for every natural number $j$ we have $x_{j+2^n-1}=x_j$ and $2^n-1$ is the least number with this property.... | Okay, so I need to prove that for every natural number s less than 2^n - 1, the sum of x_i multiplied by x_{i+s} over one period (which is 2^n - 1) is equal to -1. The sequence x_i is periodic with least period 2^n - 1, and it satisfies a recurrence relation involving products of certain terms. Hmm, this seems related ... | To prove that for every natural number \( s < 2^n - 1 \), the sum \(\sum_{i=1}^{2^n-1} x_i x_{i+s} = -1\), we leverage the connection between the given sequence and maximal-length linear feedback shift register (LFSR) sequences (m-sequences). Here's the step-by-step reasoning:
1. **Binary Correspondence**: Define a bi... | 1. **Understanding the Problem:**
We are given a sequence \( x_1, x_2, \ldots \) consisting of numbers \( 1 \) and \( -1 \) that is periodic with the least period \( 2^n - 1 \). This means:
\[
x_{j + 2^n - 1} = x_j \quad \text{for all natural numbers } j
\]
Additionally, there exist distinct integers \( ... |
A positive integer $n$ is called $\textit{un-two}$ if there does not exist an ordered triple of integers $(a,b,c)$ such that exactly two of $$ \dfrac{7a+b}{n},\;\dfrac{7b+c}{n},\;\dfrac{7c+a}{n} $$ are integers. Find the sum of all un-two positive integers.
*Proposed by **stayhomedomath*** | Okay, so I need to solve this problem where a positive integer n is called "un-two" if there doesn't exist an ordered triple (a, b, c) such that exactly two out of the three fractions (7a + b)/n, (7b + c)/n, and (7c + a)/n are integers. Then, I have to find the sum of all such un-two positive integers. Hmm, that's a bi... | To determine the sum of all un-two positive integers, we need to find integers \( n \) such that there does not exist an ordered triple \((a, b, c)\) where exactly two of the fractions \(\frac{7a + b}{n}\), \(\frac{7b + c}{n}\), and \(\frac{7c + a}{n}\) are integers.
1. **Analyzing Conditions**:
- For \( n \) to be... | To determine the sum of all un-two positive integers, we need to analyze the given condition and find the integers \( n \) for which there does not exist an ordered triple \((a, b, c)\) such that exactly two of the expressions \(\frac{7a+b}{n}\), \(\frac{7b+c}{n}\), and \(\frac{7c+a}{n}\) are integers.
1. **Case Analy... |
On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$ . What is the perpendicular distance from $C$ to the circle?
| Okay, let me try to figure out this geometry problem. So, we have a circle with radius 30 cm, and two points A and B on the circle such that the distance between them is 16 cm. Point C is the midpoint of AB, and we need to find the perpendicular distance from C to the circle. Hmm, that seems interesting.
First, let me... | Given a circle with radius 30 cm, two points \( A \) and \( B \) on the circle are 16 cm apart, and \( C \) is the midpoint of \( AB \). We need to find the perpendicular distance from \( C \) to the circle.
1. **Key Steps and Analysis:**
- The circle has radius 30 cm, so the distance from the center \( O \) to any... | 1. Given a circle with radius \( R = 30 \) cm, and two points \( A \) and \( B \) on the circle such that \( AB = 16 \) cm. Let \( C \) be the midpoint of \( AB \).
2. Since \( C \) is the midpoint of \( AB \), we have \( AC = CB = \frac{AB}{2} = \frac{16}{2} = 8 \) cm.
3. The perpendicular distance from \( C \) to t... |
Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$ . | Okay, so I need to find the least value of f(e) such that the inequality a³ + b³ + c³ + d³ ≤ e²(a² + b² + c² + d²) + f(e)(a⁴ + b⁴ + c⁴ + d⁴) holds for all real numbers a, b, c, d. Hmm, let's start by understanding what the problem is asking. We have four variables, a, b, c, d, and we need to ensure that the left-hand s... | To find the least value of \( f(e) \) such that the inequality \( a^{3} + b^{3} + c^{3} + d^{3} \leq e^{2}(a^{2} + b^{2} + c^{2} + d^{2}) + f(e)(a^{4} + b^{4} + c^{4} + d^{4}) \) holds for all real numbers \( a, b, c, d \), we analyze the problem by considering different cases and optimizing the expression.
1. **Singl... | 1. **Apply the AM-GM Inequality:**
The Arithmetic Mean-Geometric Mean (AM-GM) Inequality states that for any non-negative real numbers \( x_1, x_2, \ldots, x_n \),
\[
\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}
\]
with equality if and only if \( x_1 = x_2 = \cdots = x_n \).
2. *... |
Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$ . Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon. | Okay, let's try to figure out the area of the convex hexagon formed by the roots of the equation \( x^6 + 6x^3 - 216 = 0 \). Hmm, complex roots and their geometric interpretation... Alright, let's start by understanding the equation.
First, the equation is a sixth-degree polynomial: \( x^6 + 6x^3 - 216 = 0 \). I notic... | Given the equation \( x^6 + 6x^3 - 216 = 0 \), we start by substituting \( y = x^3 \), which transforms the equation into the quadratic \( y^2 + 6y - 216 = 0 \). Solving this quadratic equation using the quadratic formula, we find the roots \( y = 12 \) and \( y = -18 \).
These roots correspond to the equations \( x^... | 1. **Substitute \( y = x^3 \) into the given equation:**
\[
x^6 + 6x^3 - 216 = 0 \implies y^2 + 6y - 216 = 0
\]
2. **Solve the quadratic equation for \( y \):**
\[
y^2 + 6y - 216 = 0
\]
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
y = \frac{-6 \pm \sqrt{6^2... |
Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$ ), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root.
*Proposed by Hesam Rajabzadeh* | Okay, so I need to find all injective functions f from the non-negative integers to themselves such that for every natural number n and real numbers a_0, a_1, ..., a_n (not all zero), the polynomial P(x) = sum_{i=0}^n a_i x^i has a real root if and only if the polynomial Q(x) = sum_{i=0}^n a_i x^{f(i)} has a real root.... | To find all injective functions \( f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0} \) such that for every natural number \( n \) and real numbers \( a_0, a_1, \ldots, a_n \) (not all zero), the polynomial \( \sum_{i=0}^{n} a_i x^i \) has a real root if and only if the polynomial \( \sum_{i=0}^{n} a_i x^{f(i)} \) has a r... | 1. **Define the problem and the function properties:**
We need to find all injective functions \( f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0} \) such that for every natural number \( n \) and real numbers \( a_0, a_1, \ldots, a_n \) (not all zero), the polynomial \( \sum_{i=0}^{n} a_i x^i \) has a real root if an... |
Let $n$ be an integer number greater than $2$ , let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that
\[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\]
and let $k$ be a real number greater than $1$ . Show that:
\[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\]
and determine the cases of eq... | Okay, so I need to prove this inequality involving sums of reciprocals. Let me start by understanding the problem statement again. We have an integer n > 2, and n positive real numbers x₁, x₂, ..., xₙ such that the sum of 1/(x_i + 1) from i=1 to n is equal to 1. Then, for a real number k > 1, we need to show that the s... | Given an integer \( n > 2 \) and positive real numbers \( x_1, x_2, \ldots, x_n \) such that
\[
\sum_{i=1}^{n} \frac{1}{x_i + 1} = 1,
\]
we need to show that for a real number \( k > 1 \),
\[
\sum_{i=1}^{n} \frac{1}{x_i^k + 1} \ge \frac{n}{(n-1)^k + 1},
\]
and determine the cases of equality.
### Key Steps and An... | 1. **Lemma 1:** We start by proving the inequality:
\[
n^{k-1}\sum_{i=1}^{n} a_i^k \ge (\sum_{i=1}^{n} a_i)^k
\]
**Proof:** By Hölder's Inequality, we have:
\[
\left( \sum_{i=1}^{n} 1^{\frac{k}{k-1}} \right)^{\frac{k-1}{k}} \left( \sum_{i=1}^{n} a_i^k \right)^{\frac{1}{k}} \ge \sum_{i=1}^{n} a_i
\]... |
We have $2n$ lights in two rows, numbered from $1$ to $n$ in each row. Some (or none) of the lights are on and the others are off, we call that a "state". Two states are distinct if there is a light which is on in one of them and off in the other. We say that a state is good if there is the same number of lights ... | Okay, so I need to prove that the ratio of good states to total states is equal to that product of odd numbers divided by 2^n times n factorial. Let me start by understanding the problem.
We have two rows of n lights each, so 2n total lights. Each light can be on or off. A state is good if the number of lights on in t... | To determine the ratio of good states to total states, we start by calculating the number of good states. A state is good if the number of lights on in the first row equals the number in the second row. For each \( k \) (from 0 to \( n \)), there are \( \binom{n}{k} \) ways to choose \( k \) lights in each row. Thus, t... | 1. **Total Number of States:**
Each light can be either on or off independently. Since there are \(2n\) lights, the total number of states is:
\[
2^{2n}
\]
2. **Number of Good States:**
A state is good if the number of lights turned on in the first row is equal to the number of lights turned on in the s... |
Triangle $A_1 A_2 A_3$ has a right angle at $A_3$ . A sequence of points is now defined by the following iterative process, where $n$ is a positive integer. From $A_n$ ( $n \geq 3$ ), a perpendicular line is drawn to meet $A_{n-2}A_{n-1}$ at $A_{n+1}$ .
(a) Prove that if this process is continued indefinitel... | Okay, so I need to solve part (a) and (b) of this problem about the triangle and the iterative process. Let me start with part (a).
First, let me try to visualize the setup. There's a right-angled triangle A1A2A3 with the right angle at A3. Then, from each An (starting from n=3), we draw a perpendicular line to the l... | **Problem Analysis and Solution:**
**Part (a):**
We start with a right-angled triangle \( A_1A_2A_3 \) with the right angle at \( A_3 \). The iterative process involves drawing a perpendicular from \( A_n \) (for \( n \geq 3 \)) to the line segment \( A_{n-2}A_{n-1} \), obtaining \( A_{n+1} \).
1. **Nested Triangles... | ### Part (a)
1. **Define the Skew-Center:**
For a right triangle \( \triangle ABC \) with a right angle at \( C \), the skew-center is defined as the second intersection of the circles with diameters \( AC \) and \( BD \), where \( D \) is the foot of the altitude from \( C \).
2. **Lemma:**
The skew-center of ... |
Let $ABCD$ be a square of side length $13$ . Let $E$ and $F$ be points on rays $AB$ and $AD$ respectively, so that the area of square $ABCD$ equals the area of triangle $AEF$ . If $EF$ intersects $BC$ at $X$ and $BX = 6$ , determine $DF$ . | Okay, let's try to tackle this geometry problem step by step. So, we have a square ABCD with side length 13. Points E and F are on the rays AB and AD, respectively. The area of the square is equal to the area of triangle AEF. Then, EF intersects BC at point X, and we know that BX = 6. We need to find the length of DF. ... | Given a square \(ABCD\) with side length 13, points \(E\) and \(F\) are on rays \(AB\) and \(AD\) respectively such that the area of the square equals the area of triangle \(AEF\). The line \(EF\) intersects \(BC\) at \(X\) with \(BX = 6\). We need to determine the length of \(DF\).
1. **Coordinates and Area Calculati... | 1. **Identify the given information and set up the problem:**
- The side length of square \(ABCD\) is \(13\).
- Points \(E\) and \(F\) are on rays \(AB\) and \(AD\) respectively.
- The area of square \(ABCD\) is equal to the area of triangle \(AEF\).
- \(EF\) intersects \(BC\) at \(X\) and \(BX = 6\).
2. *... |
If $a,b,c$ are the roots of $x^3+20x^2+1x+5$ , compute $(a^2+1)(b^2+1)(c^2+1)$ .
*2015 CCA Math Bonanza Tiebreaker Round #2* | Okay, so I need to compute the value of (a² + 1)(b² + 1)(c² + 1) where a, b, c are the roots of the cubic polynomial x³ + 20x² + 1x + 5. Hmm, let me think about how to approach this.
First, I remember that if a polynomial has roots a, b, c, then we can use Vieta's formulas to relate the coefficients of the polynomial ... | To compute \((a^2 + 1)(b^2 + 1)(c^2 + 1)\) where \(a, b, c\) are the roots of the polynomial \(x^3 + 20x^2 + x + 5\), we use Vieta's formulas and properties of polynomials.
Given the polynomial \(x^3 + 20x^2 + x + 5\), the roots \(a, b, c\) satisfy:
\[
\begin{align*}
a + b + c &= -20, \\
ab + bc + ca &= 1, \\
abc &= -... | 1. Given that \(a, b, c\) are the roots of the polynomial \(x^3 + 20x^2 + x + 5\), we need to compute \((a^2 + 1)(b^2 + 1)(c^2 + 1)\).
2. Notice that \((a^2 + 1)(b^2 + 1)(c^2 + 1)\) can be rewritten using complex numbers. Specifically, we can use the fact that \(a^2 + 1 = (a - i)(a + i)\), \(b^2 + 1 = (b - i)(b + i)\)... |
In triangle $ABC$ we have $|AB| \ne |AC|$ . The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$ , respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$ . | Okay, so I have this geometry problem here. Let me read it again to make sure I understand.
"In triangle ABC, |AB| ≠ |AC|. The bisectors of angles ABC and ACB meet AC and AB at E and F, respectively, and intersect at I. If |EI| = |FI|, find the measure of angle BAC."
Hmm. So we have triangle ABC, with AB not equal to... | Given triangle \(ABC\) with \( |AB| \ne |AC| \), the angle bisectors of \(\angle ABC\) and \(\angle ACB\) meet \(AC\) and \(AB\) at \(E\) and \(F\) respectively, and intersect at the incenter \(I\). We need to find the measure of \(\angle BAC\) given \( |EI| = |FI| \).
1. **Coordinate System Setup**:
- Place \(A\) ... | 1. **Applying the Angle Bisector Theorem:**
By the Angle Bisector Theorem, we know that the angle bisectors of a triangle divide the opposite sides in the ratio of the adjacent sides. Therefore, for $\triangle ABC$:
\[
\frac{BE}{EC} = \frac{AB}{AC} \quad \text{and} \quad \frac{CF}{FB} = \frac{AC}{AB}
\]
2.... |
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$ , the sum of the digits of $n$ is $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 19 \qquad\te... | Alright, let's try to tackle this problem. It's about a recursive sequence where t₁ = 1, and for n > 1, the terms are defined differently depending on whether n is even or odd. If n is even, then tₙ = 1 + tₙ/₂, and if n is odd, then tₙ = 1 / tₙ₋₁. We are told that tₙ = 19/87, and we need to find the sum of the digits o... | Given the recursive sequence defined by \( t_1 = 1 \) and for \( n > 1 \):
- \( t_n = 1 + t_{n/2} \) when \( n \) is even,
- \( t_n = \frac{1}{t_{n-1}} \) when \( n \) is odd,
we need to find the sum of the digits of \( n \) given \( t_n = \frac{19}{87} \).
1. Since \( t_n = \frac{19}{87} \) is a fraction less than 1... | 1. We start with the given recursive sequence defined by:
\[
t_1 = 1
\]
For \( n > 1 \):
\[
t_n = 1 + t_{\frac{n}{2}} \quad \text{if } n \text{ is even}
\]
\[
t_n = \frac{1}{t_{n-1}} \quad \text{if } n \text{ is odd}
\]
2. We are given that \( t_n = \frac{19}{87} \). Since \( \frac{19}{87... |
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$ 's in their binary representations, for any $x,y \in \mathbb{N}$ . | Okay, so I need to find all functions \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that for any positive integers \( x \) and \( y \), the numbers \( f(x) + y \) and \( f(y) + x \) have the same number of 1's in their binary representations. Hmm, that's an interesting functional equation problem. Let me try to brea... | To find all functions \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(x) + y \) and \( f(y) + x \) have the same number of 1's in their binary representations for any \( x, y \in \mathbb{N} \), we start by considering simple functions and verifying their validity.
1. **Identity Function**:
- The identit... | 1. **Define the function and the problem statement:**
Let $\mathbb{N}$ be the set of all positive integers. We need to find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations for any $x, y \in \mathbb{N}$.
2. **Intr... |
Find the number of positive integers less than $2018$ that are divisible by $6$ but are not divisible by at least one of the numbers $4$ or $9$ . | Alright, let me try to work through this problem step by step. The question is asking for the number of positive integers less than 2018 that are divisible by 6 but are not divisible by at least one of the numbers 4 or 9. Hmm, okay. Let's break this down.
First, the numbers we are interested in are those divisible by ... | To find the number of positive integers less than 2018 that are divisible by 6 but not divisible by at least one of the numbers 4 or 9, we can follow these steps:
1. **Count numbers divisible by 6 less than 2018**:
\[
\left\lfloor \frac{2017}{6} \right\rfloor = 336
\]
This gives us 336 numbers.
2. **Count... | 1. **Find the total number of integers less than $2018$ that are divisible by $6$:**
The number of integers less than $2018$ that are divisible by $6$ is given by:
\[
\left\lfloor \frac{2018}{6} \right\rfloor = 336
\]
2. **Find the number of integers less than $2018$ that are divisible by $6$ and also div... |
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