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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Find the remainder upon division by $x^2-1$ of the determinant
$$ \begin{vmatrix}
x^3+3x & 2 & 1 & 0
x^2+5x & 3 & 0 & 2
x^4 +x^2+1 & 2 & 1 & 3
x^5 +1 & 1 & 2 & 3
\end{vmatrix} $$ | Alright, so I need to find the remainder when the determinant of this 4x4 matrix is divided by \( x^2 - 1 \). Hmm, okay. Let me think about how to approach this.
First, I remember that when you divide a polynomial by another polynomial, the remainder has a degree less than the divisor. Since the divisor here is \( x^... | To find the remainder when the determinant of the given 4x4 matrix is divided by \( x^2 - 1 \), we can use the polynomial remainder theorem. This theorem states that the remainder of a polynomial \( f(x) \) divided by \( x^2 - 1 \) is a linear polynomial \( ax + b \), which can be determined by evaluating \( f(x) \) at... | 1. First, we need to find the remainder of each element in the matrix when divided by \(x^2 - 1\). We know that \(x^2 \equiv 1 \mod{(x^2-1)}\), which implies:
\[
x^3 \equiv x \mod{(x^2-1)}, \quad x^4 \equiv 1 \mod{(x^2-1)}, \quad x^5 \equiv x \mod{(x^2-1)}
\]
2. Using these equivalences, we can reduce each el... |
Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$ , each with $3$ elements, so that no two of them are disjoint. | Okay, so I have this problem: Given a set X with n elements, find the largest number of subsets of X, each with 3 elements, such that no two of them are disjoint. Hmm, interesting. Let me try to figure this out step by step.
First, let's make sure I understand the problem correctly. We need to select as many 3-element... | To find the largest number of subsets of a set \( X \) with \( n \) elements, each with 3 elements, such that no two of them are disjoint, we analyze the problem using extremal set theory and the Erdos-Ko-Rado theorem.
1. **Case Analysis for Small \( n \)**:
- For \( n = 3 \), the only subset is the set itself, so ... | 1. **Understanding the Problem:**
We need to find the largest number of subsets of a set \( X \) with \( n \) elements, where each subset has 3 elements, and no two subsets are disjoint. This is a particular case of the Erdős–Ko–Rado theorem.
2. **Erdős–Ko–Rado Theorem:**
The Erdős–Ko–Rado theorem states that fo... |
Let $n$ be a positive integer. For each $4n$ -tuple of nonnegative real numbers $a_1,\ldots,a_{2n}$ , $b_1,\ldots,b_{2n}$ that satisfy $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$ , define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_i... | Okay, let's try to tackle this problem step by step. The problem is about finding the maximum possible minimum element m of the union of two sets A and B, which are defined based on certain sums involving a_i and b_j. The constraints are that the sums of all a_i and all b_j are equal to n, and all these a_i and b_j are... | To determine the maximum value of \( m \), the minimum element of \( A \cup B \), we need to consider the configuration of \( a_i \) and \( b_j \) that maximizes this minimum value.
Given the constraints that \(\sum_{i=1}^{2n} a_i = n\) and \(\sum_{j=1}^{2n} b_j = n\), we analyze the sets \( A \) and \( B \) defined ... | 1. **Define the problem and constraints:**
We are given two sets of nonnegative real numbers \(a_1, a_2, \ldots, a_{2n}\) and \(b_1, b_2, \ldots, b_{2n}\) such that:
\[
\sum_{i=1}^{2n} a_i = \sum_{j=1}^{2n} b_j = n
\]
We need to define the sets \(A\) and \(B\) as follows:
\[
A := \left\{\sum_{j=1}^... |
Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that
\[f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.\] | Okay, so I need to find all functions f from the non-negative integers to the non-negative integers excluding 1, such that for every natural number n, the equation f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375 holds. Hmm, let me start by understanding the problem.
First, the function is defined on N_0, which includes ... | To determine all functions \( f : \mathbb{N}_0 \rightarrow \mathbb{N}_0 \setminus \{1\} \) such that
\[ f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375 \quad \forall n \in \mathbb{N}, \]
we explored various approaches including constant functions, linear functions, quadratic functions, periodic functions, and piecewise f... | 1. Given the functional equation:
\[
f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb{N}.
\]
We need to determine all functions \( f : \mathbb{N}_0 \rightarrow \mathbb{N}_0 - \{1\} \) that satisfy this equation.
2. Consider the equation for \( n \) and \( n+2 \):
\[
f(n + 1... |
Find all triples of complex numbers $(x, y, z)$ for which $$ (x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0 $$ | Alright, so I need to find all triples of complex numbers (x, y, z) that satisfy both equations:
1. \((x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = 0\)
2. \(x^2(y + z) + y^2(z + x) + z^2(x + y) = 0\)
Hmm, okay. Let me start by analyzing each equation separately and then see how they might interact.
Fi... | To find all triples of complex numbers \((x, y, z)\) that satisfy both equations:
1. \((x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = 0\)
2. \(x^2(y + z) + y^2(z + x) + z^2(x + y) = 0\)
we start by analyzing each equation separately.
For equation (1), we use the identity \(a^3 + b^3 + c^3 - 3abc = (a +... | To solve the given problem, we need to find all triples of complex numbers \((x, y, z)\) that satisfy the following equations:
\[
(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = 0
\]
\[
x^2(y + z) + y^2(z + x) + z^2(x + y) = 0
\]
1. **Simplify the first equation:**
\[
(x + y)^3 + (y + z)^3 + (z + x)... |
Let $S$ be a set of $13$ distinct, pairwise relatively prime, positive integers. What is the smallest possible value of $\max_{s \in S} s- \min_{s \in S}s$ ?
[i]Proposed by James Lin | Okay, so I need to find the smallest possible difference between the largest and smallest numbers in a set S of 13 distinct, pairwise relatively prime positive integers. All numbers must be distinct, and every pair must be relatively prime. The goal is to minimize the difference between the maximum and minimum elements... | To find the smallest possible value of \(\max_{s \in S} s - \min_{s \in S} s\) for a set \(S\) of 13 distinct, pairwise relatively prime positive integers, we need to consider the following steps:
1. **Understanding Pairwise Relatively Prime**: Each pair of numbers in the set must share no common divisors other than 1... | 1. **Initial Consideration**: We need to find the smallest possible value of $\max_{s \in S} s - \min_{s \in S} s$ for a set $S$ of 13 distinct, pairwise relatively prime, positive integers.
2. **Lower Bound**: Since the integers are pairwise relatively prime, they must be distinct and cannot share any common factors... |
Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$ II- $a^3+b^3+1=(c-1)^3$ | Okay, let's tackle this problem step by step. So, we need to find all triples of positive integers (a, b, c) that satisfy both equations:
I. \( a^2 + b^2 = c^2 \)
II. \( a^3 + b^3 + 1 = (c - 1)^3 \)
First, equation I is the Pythagorean theorem, so (a, b, c) is a Pythagorean triple. That gives me some known structures... | To find all triples of positive integers \((a, b, c)\) that satisfy both equations:
I. \(a^2 + b^2 = c^2\)
II. \(a^3 + b^3 + 1 = (c - 1)^3\)
we start by recognizing that equation I is the Pythagorean theorem, meaning \((a, b, c)\) must be a Pythagorean triple. We then test small Pythagorean triples in equation II.
... | 1. **Assume without loss of generality that \(a \leq b\).**
2. **Given the equations:**
\[
\text{I. } a^2 + b^2 = c^2
\]
\[
\text{II. } a^3 + b^3 + 1 = (c-1)^3
\]
3. **We view \(c\) as constant and \(a, b\) as real variables. Under the condition \(a^2 + b^2 = c^2\), it is easy to show that the value... |
Let $n>1$ be a positive integer. Ana and Bob play a game with other $n$ people. The group of $n$ people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually.
Before B... | Okay, let's try to figure out this problem. So, we have Ana and Bob playing a game with n people arranged in a circle. Bob is going to assign each person a black or white hat. Each person can see everyone else's hat except their own. They all have to guess their own hat color based on some strategy that Ana comes up wi... | Ana and Bob are playing a game with \( n \) people arranged in a circle, where Ana aims to maximize the number of correct hat color guesses while Bob tries to minimize it. Each person can see all hats except their own and follows the same strategy provided by Ana.
Key steps and analysis:
1. **Strategy Analysis**: Ana... | 1. **Define the problem and setup:**
- We have \( n \) people in a circle, each wearing a hat that is either black or white.
- Each person can see all the hats except their own.
- Ana wants to maximize the number of correct guesses, while Bob wants to minimize it.
- Ana gives a strategy to the \( n \) peopl... |
The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD.
[asy]
pair A = (2,6);
pair B = (0,0);
pair C = (10,0);
pair D = (3.5,0) ;
pair E = (3.1,2);
draw(A--B);
draw(B--C);
draw(C--... | Okay, so I need to solve this geometry problem about triangle ABC with sides AB = 137, AC = 241, and BC = 200. There's a point D on BC such that both incircles of triangles ABD and ACD touch AD at the same point E. I have to find the length of CD. Hmm, let me think step by step.
First, let me visualize the triangle. A... | Given triangle \(ABC\) with sides \(AB = 137\), \(AC = 241\), and \(BC = 200\), we need to find the length of \(CD\) where point \(D\) on \(BC\) is such that both incircles of triangles \(ABD\) and \(ACD\) touch \(AD\) at the same point \(E\).
1. **Key Insight**: The distance from \(A\) to the touch point \(E\) on \(A... | 1. **Define Variables and Tangent Points:**
Let \( M \) be the tangent point on \( AB \), \( N \) be the tangent point on \( BD \), \( O \) be the tangent point on \( DC \), and \( P \) be the tangent point on \( AC \). Let \( E \) be the common tangent point on \( AD \) for both incircles of triangles \( ABD \) and... |
Let $m$ be an positive odd integer not divisible by $3$ . Prove that $\left[4^m -(2+\sqrt 2)^m\right]$ is divisible by $112.$ | Okay, so I need to prove that for a positive odd integer m that's not divisible by 3, the expression [4^m - (2 + √2)^m] is divisible by 112. Hmm, let's break this down step by step.
First, let's understand the problem. We have m as a positive odd integer, not divisible by 3. We need to show that when we compute 4^m mi... | Given a positive odd integer \( m \) not divisible by 3, we need to prove that \( \left[4^m - (2 + \sqrt{2})^m\right] \) is divisible by 112.
First, we recognize that the expression \( 4^m - (2 + \sqrt{2})^m \) is not an integer. However, considering the conjugate term \( (2 - \sqrt{2})^m \), we can form the integer ... | 1. **Define the sequence \(a_n\):**
We start by defining the sequence \(a_n\) as follows:
\[
a_1 = 0, \quad a_2 = 4, \quad a_3 = 24, \quad a_{n+3} = 8a_{n+2} - 18a_{n+1} + 8a_n \quad \forall n > 0
\]
2. **Express \(a_n\) in terms of \(4^n\) and \((2 \pm \sqrt{2})^n\):**
It is given that:
\[
a_n = ... |
For all positive integers $n$ the function $f$ satisfies $f(1) = 1, f(2n + 1) = 2f(n),$ and $f(2n) = 3f(n) + 2$ . For how many positive integers $x \leq 100$ is the value of $f(x)$ odd? $\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$ | Okay, so I have this problem here about a function f defined for all positive integers n. The function has some recursive rules: f(1) = 1, f(2n + 1) = 2f(n), and f(2n) = 3f(n) + 2. The question is asking how many positive integers x ≤ 100 result in f(x) being odd. The answer choices are from 4 to 10, so I need to figur... | The function \( f \) is defined for all positive integers \( n \) with the following rules: \( f(1) = 1 \), \( f(2n + 1) = 2f(n) \), and \( f(2n) = 3f(n) + 2 \). We need to determine how many positive integers \( x \leq 100 \) result in \( f(x) \) being odd.
1. **Base Case**: \( f(1) = 1 \) (odd).
2. **Odd Input**: Fo... | 1. We start by analyzing the given function properties:
- \( f(1) = 1 \)
- \( f(2n + 1) = 2f(n) \)
- \( f(2n) = 3f(n) + 2 \)
2. We need to determine for how many positive integers \( x \leq 100 \) the value of \( f(x) \) is odd.
3. First, let's consider the case when \( x \) is odd:
- If \( x \) is odd, t... |
Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$ .
*Demetres Christofides, Cyprus* | Okay, so I need to figure out the maximum possible value of r + s, where r is the number of common real roots (counting multiplicity) and s is the number of common coefficients of two distinct degree 2020 polynomials P(x) and Q(x) with non-zero coefficients. Let me start by understanding the problem step by step.
Firs... | Given two distinct polynomials \( P(x) \) and \( Q(x) \) of degree 2020 with non-zero coefficients, we need to determine the maximum possible value of \( r + s \), where \( r \) is the number of common real roots (counting multiplicity) and \( s \) is the number of common coefficients.
1. **Common Roots (r):** The num... | To determine the maximum possible value of \( r + s \) where \( P(x) \) and \( Q(x) \) are distinct polynomials of degree 2020 with non-zero coefficients, and they have \( r \) common real roots (counting multiplicity) and \( s \) common coefficients, we proceed as follows:
1. **Construction**:
- Let \( P(x) = (x^2... |
$39$ students participated in a math competition. The exam consisted of $6$ problems and each problem was worth $1$ point for a correct solution and $0$ points for an incorrect solution. For any $3$ students, there is at most $1$ problem that was not solved by any of the three. Let $B$ be the sum of all o... | Okay, so I need to find the smallest possible total score B for 39 students who took a math competition with 6 problems. Each problem is worth 1 point if solved, 0 otherwise. The key condition here is that for any 3 students, there's at most 1 problem that none of them solved. Hmm, that seems like a combinatorial const... | To find the smallest possible value of \( B \), the sum of all scores of 39 students in a math competition with 6 problems, we need to consider the constraint that for any 3 students, there is at most 1 problem that was not solved by any of the three.
Key steps:
1. **Understanding the Constraint**: For any three stud... | 1. **Define the sets and conditions:**
Let \( S_i \) be the set of students that did not solve problem \( i \). The given condition states that for any 3 students, there is at most 1 problem that none of them solved. This can be translated into the condition that for any two problems \( i \) and \( j \), the interse... |
When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$ . When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$ . If $ R_1 \equal{} R_2$ then $ m$ is:
$ \textbf{(A)}\ 0 \qquad \t... | Okay, let's see. I need to solve this problem where when the quadratic polynomial y² + my + 2 is divided by y - 1, the remainder is R₁, and when it's divided by y + 1, the remainder is R₂. Then, given that R₁ equals R₂, find the value of m from the options given.
First, I remember something called the Remainder Theor... | When the polynomial \( y^2 + my + 2 \) is divided by \( y - 1 \), the remainder \( R_1 \) can be found using the Remainder Theorem by substituting \( y = 1 \):
\[
R_1 = (1)^2 + m(1) + 2 = 1 + m + 2 = m + 3
\]
Similarly, when the polynomial \( y^2 + my + 2 \) is divided by \( y + 1 \), the remainder \( R_2 \) can be f... | 1. We start by using polynomial division to find the remainders when \( y^2 + my + 2 \) is divided by \( y - 1 \) and \( y + 1 \).
2. When \( y^2 + my + 2 \) is divided by \( y - 1 \):
\[
y^2 + my + 2 = (y - 1)Q_1(y) + R_1
\]
To find \( R_1 \), we substitute \( y = 1 \) into the polynomial:
\[
1^2 + ... |
Where is Tony's cheese? The bottom right corner of a 3x3 grid :0 We’re doing weekly posts of problems until the start of the in-person tournament on April 13th (Sign up **[here](https://www.stanfordmathtournament.com/competitions/smt-2024)**!) Registrations for our synchronous Online tournament are also open **[here](h... | Okay, let's tackle the easy problem first. The problem says we have two curves: the circle x² + y² = 1 and another quadratic curve 2x² + 2xy + y² - 2x - 2y = 0. They intersect at two points, one of which is (1, 0). We need to find the other intersection point.
First, I know that to find the intersection points of two ... | **Easy Problem Solution:**
To find the other intersection point of the curves \( x^2 + y^2 = 1 \) and \( 2x^2 + 2xy + y^2 - 2x - 2y = 0 \):
1. Subtract the circle equation from the second equation to eliminate quadratic terms:
\[
x^2 + 2xy - 2x - 2y = -1 \implies (x - 1)(x - 1 + 2y) = 0
\]
This gives two ... | To solve the hard problem, we need to calculate the probability that Tony the mouse reaches the cheese in the bottom right corner of a \(3 \times 3\) grid before he reaches the mousetrap in the center. We will label the points on the grid as follows:
\[
\begin{array}{ccc}
1 & 2 & 3 \\
4 & T & 5 \\
6 & 7 & C \\
\end{ar... |
For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order: $4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your... | Okay, so I need to figure out how many times the numbers 2015 and 2016 appear in the infinite sequence generated by the function f(n) for n > 1. The function f(n) is defined as the sum of the smallest factor of n (other than 1) and n itself. Let me start by understanding the problem thoroughly.
First, let's recap the ... | To determine how many times the numbers 2015 and 2016 appear in the sequence generated by the function \( f(n) \), we need to analyze the function \( f(n) = n + p \), where \( p \) is the smallest prime factor of \( n \).
### Analysis for 2015:
1. **Equation Setup**:
\[
n + p = 2015 \implies n = 2015 - p
\]
... | 1. **Understanding the function \( f(n) \):**
- For any integer \( n > 1 \), \( f(n) \) is defined as the sum of the smallest factor of \( n \) that is not 1 and \( n \) itself.
- If \( n \) is a prime number, the smallest factor other than 1 is \( n \) itself, so \( f(n) = n + n = 2n \).
- If \( n \) is compo... |
Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$ . | Alright, so I need to find the smallest positive multiple of 77 whose last four digits are 2020. Let me break this down. First, I know that a multiple of 77 is a number that can be expressed as 77 times some integer. The problem wants the smallest such number that ends with 2020. So essentially, I'm looking for the sma... | To find the smallest positive multiple of \(77\) whose last four digits are \(2020\), we need to solve the congruence \(77n \equiv 2020 \pmod{10000}\). This requires finding the modular inverse of \(77\) modulo \(10000\) using the Extended Euclidean Algorithm.
1. **Compute \(\gcd(77, 10000)\)**:
- Since \(77\) and ... | 1. We need to find the smallest positive multiple of \(77\) whose last four digits are \(2020\). Let \(N\) be such a number. We can express \(N\) in the form:
\[
N = 10000k + 2020
\]
where \(k\) is an integer.
2. Since \(N\) is a multiple of \(77\), we have:
\[
10000k + 2020 \equiv 0 \pmod{77}
\]
... |
In the unit squre For the given natural number $n \geq 2$ find the smallest number $k$ that from each set of $k$ unit squares of the $n$ x $n$ chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even | Okay, so I have this problem here: In an n x n chessboard, where n is a natural number greater than or equal to 2, I need to find the smallest number k such that from any set of k unit squares on the chessboard, one can choose a subset where the number of squares in each row and each column is even. Hmm, let me try to ... | To solve the problem of finding the smallest number \( k \) such that from any set of \( k \) unit squares on an \( n \times n \) chessboard, one can choose a subset with an even number of squares in each row and column, we can use the following reasoning:
1. **Problem Interpretation**: The goal is to ensure that any ... | 1. **Establishing the lower bound for \( k \):**
- We need to show that \( k > 2n - 1 \). Consider the \( 2n - 1 \) squares situated on the first row and the first column of the \( n \times n \) chessboard.
- If we select any subset of these \( 2n - 1 \) squares, it is impossible to find a subset such that the n... |
Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$ ? | Okay, let's see. The problem is to find the smallest positive integer N such that when you compute the sum N + 2N + 3N + ... + 9N, the result is a number where all digits are equal. Then, we need to find the sum of the digits of N.
First, let me break down the problem. The sum N + 2N + ... +9N can be simplified. Since... | To find the smallest positive integer \( N \) such that \( N + 2N + 3N + \ldots + 9N \) is a number with all digits equal, we start by simplifying the sum. The sum \( N + 2N + 3N + \ldots + 9N \) can be factored as \( N \times (1 + 2 + 3 + \ldots + 9) \). The sum of the first 9 positive integers is \( 45 \), so the tot... | 1. We start with the given expression \(N + 2N + 3N + \ldots + 9N\). This can be factored as:
\[
N(1 + 2 + 3 + \ldots + 9)
\]
2. The sum of the first 9 positive integers is:
\[
1 + 2 + 3 + \ldots + 9 = \frac{9 \cdot 10}{2} = 45
\]
Therefore, the expression simplifies to:
\[
45N
\]
3. We ne... |
Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations:
a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$ ;
b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$ .
| Okay, let's try to tackle these two functional equations. I need to find all functions f from integers to integers that satisfy each of the given conditions. Let me start with part a) first.
So, part a) says that applying the function f twice on any integer n gives 4n + 3. That is, f(f(n)) = 4n + 3 for all integers n.... | To determine all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) that satisfy the given functional equations, we analyze each part systematically.
### Part a) \( f(f(n)) = 4n + 3 \)
1. **Assume a linear function**: Let \( f(n) = an + b \).
2. **Substitute into the equation**:
\[
f(f(n)) = a(an + b) + b = a... | 1. **Injectivity of \( f \)**:
From \( P(n) \), we have \( f(f(n)) = 4n + 3 \). Suppose \( f(a) = f(b) \). Then:
\[
f(f(a)) = f(f(b)) \implies 4a + 3 = 4b + 3 \implies a = b
\]
Therefore, \( f \) is injective.
2. **Comparing \( P(n) \) and \( Q(2n) \)**:
From \( Q(2n) \), we have:
\[
f(f(2n) - ... |
Let $p$ be a prime number. We know that each natural number can be written in the form
\[\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)\]
Uniquely.
Now let $T$ be the set of all the sums of the form
\[\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).\]
(This means to allow numbers with an infini... | Okay, so I need to prove that if p is a prime number congruent to 1 modulo 4, then there exists an element x in the ring T (which consists of these infinite base-p expansions with coefficients between 0 and p-1) such that x squared plus 1 equals zero. Hmm, interesting. Let me try to unpack this step by step.
First, le... | Given a prime number \( p \) congruent to 1 modulo 4, we need to prove that there exists an element \( x \) in the ring \( T \) (the set of all sums of the form \(\sum_{i=0}^{\infty} a_i p^i\) with \( 0 \le a_i \le p-1 \)) such that \( x^2 + 1 = 0 \).
1. **Quadratic Residue Modulo \( p \)**:
- Since \( p \equiv 1 \... | To prove that there exists \( x \in T \) such that \( x^2 + 1 = 0 \) when \( p \equiv 1 \pmod{4} \), we can use Hensel's Lemma. Here is the detailed step-by-step solution:
1. **Existence of a solution modulo \( p \):**
Since \( p \equiv 1 \pmod{4} \), by Fermat's theorem on sums of two squares, there exists an inte... |
Find the smallest positive integer $m$ satisfying the following proposition:
There are $999$ grids on a circle. Fill a real number in each grid, such that for any grid $A$ and any positive integer $k\leq m$ , at least one of the following two propositions will be true: $\bullet$ The difference between the nu... | Okay, let me try to work through this problem step by step. The problem is about 999 grids arranged in a circle, each filled with a real number. The conditions are a bit complex, so I need to parse them carefully.
First, the proposition states that for any grid A and any positive integer k ≤ m, at least one of two thi... | To solve the problem, we need to find the smallest positive integer \( m \) such that if each grid on a circle of 999 grids satisfies the condition that for any grid \( A \) and any positive integer \( k \leq m \), at least one of the differences (clockwise or anticlockwise) is \( k \), then there must exist a grid \( ... | To solve this problem, we need to find the smallest positive integer \( m \) such that for any grid \( A \) and any positive integer \( k \leq m \), at least one of the following conditions holds:
1. The difference between the numbers in grid \( A \) and the \( k \)-th grid after \( A \) in the clockwise direction is \... |
Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$ .
Note that the slope of $l$ is greater than that of $m$ .
(1) Exress the slope of $l$ in terms of $a$ .
(2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the... | Okay, let's tackle these problems step by step. So, we have a point A(a, a-1) on the line y = x - 1. We need to find the tangent lines from A to the parabola y = x². Then, there are three parts to this problem. Let me start with part (1): Express the slope of l in terms of a.
First, I recall that the equation of a ta... | To solve the given problems, we start by analyzing the tangent lines from the point \( A(a, a-1) \) to the parabola \( y = x^2 \).
### (1) Slope of the Tangent Line \( l \)
The equation of a tangent line to the parabola \( y = x^2 \) is \( y = mx - \frac{m^2}{4} \). Since this line passes through \( A(a, a-1) \), subs... | ### Part (1)
To find the slope of the tangent line \( l \) in terms of \( a \):
1. The equation of the tangent line to the parabola \( y = x^2 \) at a point \( (x_0, y_0) \) is given by:
\[
y = 2x_0 x - x_0^2
\]
Here, \( y_0 = x_0^2 \).
2. Since the tangent line passes through the point \( A(a, a-1) \), w... |
There are two equilateral triangles with a vertex at $(0, 1)$ , with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$ . Find the area of the larger of the two triangles. | Alright, let's tackle this problem step by step. So, we have two equilateral triangles, each with one vertex at (0, 1). The other two vertices are on the line y = x + 1 and the parabola y = x² + 1. We need to find the area of the larger triangle.
First, let me visualize the setup. The point (0,1) is on both the line a... | Given two equilateral triangles with a vertex at \((0, 1)\), another vertex on the line \(y = x + 1\), and the final vertex on the parabola \(y = x^2 + 1\), we need to find the area of the larger triangle.
1. **Define Points**:
- Let \(P(a, a + 1)\) be a point on the line \(y = x + 1\).
- Let \(Q(b, b^2 + 1)\) b... | 1. **Identify the vertices of the equilateral triangle:**
We are given the points \((0,1)\), \((a, a+1)\), and \((b, b^2+1)\). We need to ensure that these points form an equilateral triangle.
2. **Set up the distance equations:**
For the triangle to be equilateral, the distances between all pairs of points must... |
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$ , then we call that $ L$ passes through $ A$ .
(1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for an... | Okay, so I need to tackle these two problems about dividing rational points into sets and lines passing through them. Let me start with part (1). The question says we need to divide all rational points into 100 pairwise non-intersecting (I guess that means disjoint) point sets, each with infinitely many elements. Moreo... | **Problem (1) Proof:**
To partition all rational points into 100 pairwise disjoint infinite sets such that any line containing two rational points intersects all sets, we use a cyclic enumeration:
1. **Enumerate Rational Points:** Since rational points are countable, list them as \( p_1, p_2, p_3, \ldots \).
2. **Ass... | ### Part (1)
1. **Define Rational Points and Sets**:
Let \( \mathbb{Q}^2 \) denote the set of all rational points on the plane, i.e., points \((x, y)\) where both \(x\) and \(y\) are rational numbers.
2. **Partitioning Rational Points**:
We need to partition \( \mathbb{Q}^2 \) into 100 pairwise non-intersecting... |
In an acute triangle $ABC$ with $AC > AB$ , let $D$ be the projection of $A$ on $BC$ , and let $E$ and $F$ be the projections of $D$ on $AB$ and $AC$ , respectively. Let $G$ be the intersection point of the lines $AD$ and $EF$ . Let $H$ be the second intersection point of the line $AD$ and the... | Okay, so I need to prove that AG multiplied by AH equals AD squared in this acute triangle ABC where AC is longer than AB. Let me try to visualize the problem first.
First, let's draw triangle ABC. Since it's acute, all altitudes are inside the triangle. AC > AB, so AB is shorter than AC. Point D is the projection of... | To prove that \( AG \cdot AH = AD^2 \) in an acute triangle \( ABC \) with \( AC > AB \), we start by considering the coordinates of the points involved.
1. **Coordinate Setup**:
- Place \( D \) at the origin \((0, 0)\), \( B \) at \((-b, 0)\), \( C \) at \((c, 0)\), and \( A \) at \((0, h)\).
- \( D \) is the ... | 1. **Identify the projections and intersections:**
- Let \( D \) be the foot of the perpendicular from \( A \) to \( BC \).
- Let \( E \) be the foot of the perpendicular from \( D \) to \( AB \).
- Let \( F \) be the foot of the perpendicular from \( D \) to \( AC \).
- Let \( G \) be the intersection of \... |
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$ | Okay, let's try to figure out this problem. So we need to find the maximum value of M(a) where M(a) is the number of natural numbers b such that a + b divides ab. And we need to do this for all a up to 1983. Hmm, interesting. Let me start by understanding the problem better.
First, the set is defined as { b ∈ N | a + ... | To solve the problem, we need to find the maximum value of \( M(a) \) for \( a \leq 1983 \), where \( M(a) \) is the number of natural numbers \( b \) such that \( a + b \) is a divisor of \( ab \).
1. **Key Insight**: The condition \( a + b \mid ab \) can be rewritten as \( (a - k)(b - k) = k^2 \) for some positive i... | 1. **Understanding the problem**: We need to find the maximum value of \( M(a) \) for \( a \leq 1983 \). Here, \( M(a) \) is defined as the number of elements \( b \in \mathbb{N} \) such that \( a + b \) is a divisor of \( ab \).
2. **Rewriting the condition**: The condition \( a + b \) is a divisor of \( ab \) can be... |
Compute the number of positive integers $b$ where $b \le 2013$ , $b \neq 17$ , and $b \neq 18$ , such that there exists some positive integer $N$ such that $\dfrac{N}{17}$ is a perfect $17$ th power, $\dfrac{N}{18}$ is a perfect $18$ th power, and $\dfrac{N}{b}$ is a perfect $b$ th power. | Okay, so I have this problem here where I need to compute the number of positive integers \( b \) such that \( b \leq 2013 \), \( b \neq 17 \), \( b \neq 18 \), and there exists a positive integer \( N \) satisfying three conditions:
1. \( \frac{N}{17} \) is a perfect 17th power.
2. \( \frac{N}{18} \) is a perfect 18t... | To solve the problem, we need to determine the number of positive integers \( b \leq 2013 \), \( b \neq 17 \), and \( b \neq 18 \), such that there exists a positive integer \( N \) where \( \frac{N}{17} \) is a perfect 17th power, \( \frac{N}{18} \) is a perfect 18th power, and \( \frac{N}{b} \) is a perfect \( b \)-t... | 1. We start by analyzing the given conditions. We need to find the number of positive integers \( b \) such that \( b \leq 2013 \), \( b \neq 17 \), and \( b \neq 18 \), and there exists some positive integer \( N \) such that:
\[
\frac{N}{17} \text{ is a perfect } 17 \text{th power},
\]
\[
\frac{N}{18} ... |
Find all integers $n\geq 3$ such that there exists a convex $n$ -gon $A_1A_2\dots A_n$ which satisfies the following conditions:
- All interior angles of the polygon are equal
- Not all sides of the polygon are equal
- There exists a triangle $T$ and a point $O$ inside the polygon such that the $n$ triang... | Okay, so I have this geometry problem to solve. Let me read it again to make sure I understand all the conditions.
We need to find all integers n ≥ 3 such that there exists a convex n-gon A₁A₂…Aₙ with the following properties:
1. All interior angles are equal. So it's an equiangular polygon, but not necessarily regul... | To solve the problem of finding all integers \( n \geq 3 \) such that there exists a convex \( n \)-gon \( A_1A_2 \dots A_n \) which satisfies the given conditions, we need to consider the following key points:
1. **Equiangular Polygon**: The polygon must have all interior angles equal. For a convex \( n \)-gon, each ... | To solve the problem, we need to find all integers \( n \geq 3 \) such that there exists a convex \( n \)-gon \( A_1A_2 \dots A_n \) with the following properties:
1. All interior angles of the polygon are equal.
2. Not all sides of the polygon are equal.
3. There exists a triangle \( T \) and a point \( O \) inside th... |
Hello Everyone,
i'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level
Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.
it will be pre... | Okay, so I have this problem here: Let f(n) denote the sum of the digits of n. We need to find f(f(f(N))) where N is 4444 raised to the power of 4444. Hmm, that seems pretty big. Alright, let's start breaking it down step by step.
First off, N is 4444^4444. That's a massive number. Obviously, calculating it directly i... | To solve the problem, we need to find \( f(f(f(N))) \) where \( N = 4444^{4444} \) and \( f(n) \) denotes the sum of the digits of \( n \).
1. **Understanding the Digital Root**: The sum of the digits of a number \( n \) (denoted \( f(n) \)) is congruent to \( n \mod 9 \). Repeatedly applying the digit sum until a sin... | To solve the problem, we need to find \( f(f(f(N))) \) where \( N = 4444^{4444} \) and \( f(n) \) denotes the sum of the digits of \( n \).
1. **Understanding the Sum of Digits Function \( f(n) \)**:
The function \( f(n) \) represents the sum of the digits of \( n \). A key property of this function is that \( f(n)... |
Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint *regions*. Find the number of such regions. | Okay, so I need to figure out how many regions a dodecahedron's planes divide the space into. Let me start by recalling what a dodecahedron is. It's one of the Platonic solids, right? Specifically, it has 12 regular pentagonal faces, 20 vertices, and 30 edges. Each face is a plane, so there are 12 planes in total. The ... | The number of regions formed by the 12 face planes of a regular dodecahedron can be determined by analyzing its symmetry properties and their correspondence to the arrangement's combinatorial structure.
1. **Symmetry Group Consideration**: The regular dodecahedron has icosahedral symmetry, with a symmetry group of or... | To solve the problem of finding the number of regions into which the planes of a dodecahedron divide the space, we can use the Euler characteristic for three-dimensional space. The Euler characteristic for \(\mathbb{R}^3\) is \(-1\). The formula we use is:
\[ V - E + F - S = -1 \]
where \(V\) is the number of vertice... |
Find all real numbers $ a$ and $ b$ such that
\[ 2(a^2 \plus{} 1)(b^2 \plus{} 1) \equal{} (a \plus{} 1)(b \plus{} 1)(ab \plus{} 1).
\]
*Valentin Vornicu* | Okay, so I have this problem: Find all real numbers a and b such that 2(a² + 1)(b² + 1) = (a + 1)(b + 1)(ab + 1). Hmm, let's see. I need to find all real pairs (a, b) that satisfy this equation. The equation looks a bit complicated, but maybe I can expand both sides and simplify. Alternatively, perhaps there's a symmet... | To find all real numbers \(a\) and \(b\) such that
\[
2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1),
\]
we start by expanding both sides of the equation.
**Left-Hand Side (LHS):**
\[
2(a^2 + 1)(b^2 + 1) = 2(a^2b^2 + a^2 + b^2 + 1)
\]
**Right-Hand Side (RHS):**
\[
(a + 1)(b + 1)(ab + 1)
\]
First, expand \((a + 1)(b + ... | To find all real numbers \(a\) and \(b\) such that
\[ 2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1), \]
we will start by expanding and simplifying both sides of the equation.
1. **Expand the left-hand side:**
\[
2(a^2 + 1)(b^2 + 1) = 2(a^2b^2 + a^2 + b^2 + 1).
\]
2. **Expand the right-hand side:**
\[
(... |
$2016$ bugs are sitting in different places of $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of cont... | Okay, so there's this problem about 2016 bugs on a 1-meter stick. Each bug is at a different spot, and they all move either left or right with the same speed. When two bugs meet each other, they both turn around but keep moving at the same speed. If a bug reaches the end of the stick, it flies away. The question is ask... | To determine the greatest number of contacts (meetings) between 2016 bugs on a 1-meter stick, we need to consider the following key points:
1. **Bug Movement and Meetings**: Each bug moves either left or right with the same speed. When two bugs meet, they reverse direction but continue moving at the same speed. If a b... | 1. **Reinterpret the Problem**: We can simplify the problem by imagining that the bugs pass through each other without changing direction. This reinterpretation does not change the number of contacts because the bugs' interactions are symmetric.
2. **Define Variables**: Let \( a \) be the number of bugs facing one dir... |
Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$ . | Okay, let's try to tackle this problem: Find the smallest prime which is not the difference (in some order) of a power of 2 and a power of 3. Hmm, so I need to check primes one by one and see if they can be written as either 2^a - 3^b or 3^c - 2^d where a, b, c, d are non-negative integers. The question is asking for t... | To find the smallest prime which is not the difference (in some order) of a power of 2 and a power of 3, we need to check each prime number sequentially to see if it can be expressed as either \(2^a - 3^b\) or \(3^c - 2^d\) where \(a, b, c, d\) are non-negative integers.
1. **Prime 2**: \(3^1 - 2^0 = 3 - 1 = 2\)
2. **... | To find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$, we need to check each prime number and see if it can be expressed as $2^x - 3^y$ or $3^y - 2^x$ for non-negative integers $x$ and $y$.
1. **Check small primes:**
- $2 = 3^1 - 2^0$
- $3 = 2^2 - 3^0$
- $... |
Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number.
*I. Tonov* | Okay, so I need to find all pairs of natural numbers (m, n), both bigger than 1, such that 2^m + 3^n is a perfect square. Hmm, let's start by understanding the problem.
First, natural numbers bigger than 1 means m and n are at least 2. So 2^m and 3^n are both going to be at least 4 and 9 respectively. Their sum needs ... | To find all pairs of natural numbers \((m, n)\) bigger than 1 for which \(2^m + 3^n\) is a perfect square, we start by noting that both \(m\) and \(n\) must be at least 2. We need to solve the equation \(2^m + 3^n = k^2\) for some integer \(k\).
### Key Steps:
1. **Initial Checks for Small Values**:
- For small va... | 1. **Modulo 3 Analysis:**
- Consider the equation \(2^m + 3^n = k^2\) where \(k\) is a whole number.
- Taking modulo 3, we have:
\[
2^m + 3^n \equiv k^2 \pmod{3}
\]
- Since \(3^n \equiv 0 \pmod{3}\), we get:
\[
2^m \equiv k^2 \pmod{3}
\]
- The possible values of \(2^m \pmod{3}\... |
(a) Prove that given constants $a,b$ with $1<a<2<b$ , there is no partition of the set of positive integers into two subsets $A_0$ and $A_1$ such that: if $j \in \{0,1\}$ and $m,n$ are in $A_j$ , then either $n/m <a$ or $n/m>b$ .
(b) Find all pairs of real numbers $(a,b)$ with $1<a<2<b$ for which the... | Alright, so I need to solve both parts (a) and (b) of this problem. Let me start with part (a). The problem states that given constants a and b with 1 < a < 2 < b, there is no partition of the set of positive integers into two subsets A₀ and A₁ such that if two numbers m and n are in the same subset Aⱼ, then either n/m... | (a) To prove that there is no partition of the set of positive integers into two subsets \( A_0 \) and \( A_1 \) such that if \( j \in \{0,1\} \) and \( m, n \) are in \( A_j \), then either \( n/m < a \) or \( n/m > b \):
1. Assume such a partition exists. Start with 1 in \( A_0 \).
2. Since \( 1 < a < 2 \), the inte... | ### Part (a)
1. **Assume a partition exists:**
Suppose there exists a partition of the set of positive integers into two subsets \( A_0 \) and \( A_1 \) such that if \( j \in \{0,1\} \) and \( m, n \) are in \( A_j \), then either \( \frac{n}{m} < a \) or \( \frac{n}{m} > b \).
2. **Coloring scheme:**
Color a p... |
On rectangular coordinates, point $A = (1,2)$ , $B = (3,4)$ . $P = (a, 0)$ is on $x$ -axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$ . | Okay, so I need to figure out the value of a for the point P = (a, 0) on the x-axis that minimizes the sum of the distances AP + PB, where A is (1,2) and B is (3,4). Then, once I find a, I have to compute 60 times that a. Hmm, let me start by recalling some geometry concepts.
First, I know that minimizing the sum of ... | To find the value of \(a\) for the point \(P = (a, 0)\) on the x-axis that minimizes the sum of the distances \(AP + PB\), where \(A = (1, 2)\) and \(B = (3, 4)\), we use the reflection principle. Reflecting point \(B\) over the x-axis gives the point \(B' = (3, -4)\). The optimal point \(P\) is the intersection of the... | 1. **Reflect Point \( B \) Across the \( x \)-Axis:**
- Given points \( A = (1, 2) \) and \( B = (3, 4) \), reflect point \( B \) across the \( x \)-axis to get \( B' = (3, -4) \). This reflection helps in minimizing the path \( AP + PB \) by considering the straight-line distance \( AP + PB' \).
2. **Minimize the ... |
The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$ , $y$ , erases them and then writes down $x + y + xy$ . He continues to do this until only one number is left on the blackboard. What is this number? | Okay, so I have this problem where the numbers 1/1, 1/2, ..., up to 1/2010 are written on a blackboard. A student repeatedly picks any two numbers, say x and y, erases them, and then writes down x + y + xy. This process continues until only one number remains. The question is asking what that final number is.
Hmm, let... | The numbers \(\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}\) are written on a blackboard. A student repeatedly chooses two numbers \(x\) and \(y\), erases them, and writes down \(x + y + xy\). This process continues until only one number remains. We need to determine this final number.
First, we observe that the o... | 1. **Initial Setup**: We start with the numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}$ on the blackboard.
2. **Transformation Rule**: The student chooses any two numbers, say $x$ and $y$, erases them, and writes down $x + y + xy$. This can be rewritten using the identity:
\[
x + y + xy = (x+1)(y+1) -... |
Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set $S$ of $n>1$ collinear points. Then, for every unordered pair of points $\{X,Y\}$ in $S$ , Elmo draws the circle with diameter $XY$ so that each pair of circles which intersect at two distinct points are drawn in different colors. C... | Okay, so I need to figure out the minimum number of colors Elmo used when drawing circles with diameters between every pair of points in a collinear set S of n > 1 points. The key is that each pair of circles that intersect at two distinct points must be drawn in different colors. The question is asking for the minimum... | Elmo is drawing circles with diameters between every pair of points in a collinear set \( S \) of \( n > 1 \) points. The goal is to determine the minimum number of colors needed such that any two circles intersecting at two distinct points are colored differently.
Key steps and reasoning:
1. **Circle Intersection Co... | To determine the minimum number of colors Elmo could have used, we need to analyze the problem in detail. Let's break it down step-by-step.
1. **Understanding the Problem:**
- Elmo marks a set \( S \) of \( n > 1 \) collinear points.
- For every unordered pair of points \(\{X, Y\}\) in \( S \), Elmo draws a circ... |
Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle. | Okay, so I need to find the largest value of the expression (p/(R)) * (1 - (r)/(3R)), where p is the perimeter, R is the radius of the circumscribed circle, and r is the radius of the inscribed circle of a triangle. Hmm, let me start by recalling some formulas related to triangles, their perimeters, inradius, and circu... | To find the largest value of the expression \(\frac{p}{R}\left( 1- \frac{r}{3R}\right)\) where \(p\), \(R\), and \(r\) are the perimeter, circumradius, and inradius of a triangle respectively, we start by expressing the key variables in terms of trigonometric functions for a triangle.
1. **Perimeter and Circumradius**... | To find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$, where $p$, $R$, and $r$ are the perimeter, the radius of the circumscribed circle, and the radius of the inscribed circle of a triangle respectively, we will proceed as follows:
1. **Express the given expression in terms of known t... |
There are $17$ students in Marek's class, and all of them took a test. Marek's score was $17$ points higher than the arithmetic mean of the scores of the other students. By how many points is Marek's score higher than the arithmetic mean of the scores of the entire class? Justify your answer. | Okay, let me try to figure out this problem step by step. So, Marek is in a class with 17 students total, right? That means there are 16 other students besides him. They all took a test, and Marek's score is 17 points higher than the arithmetic mean of the other students' scores. The question is asking by how many poin... | Given that there are 17 students in Marek's class, and his score is 17 points higher than the arithmetic mean of the scores of the other 16 students, we need to determine by how many points Marek's score is higher than the arithmetic mean of the entire class.
1. Let \( S \) be the total score of the other 16 students.... | 1. Let the scores of the 16 other students be \( s_1, s_2, \ldots, s_{16} \).
2. Let \( S \) be the sum of the scores of these 16 students. Thus, \( S = s_1 + s_2 + \cdots + s_{16} \).
3. The arithmetic mean of the scores of the other students is given by:
\[
\text{Mean}_{\text{others}} = \frac{S}{16}
\]
4. Le... |
Let $x_1$ , $x_2$ , …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$ ? | Alright, let's tackle this problem. So, we have 10 numbers: x₁, x₂, ..., x₁₀. The condition given is that for each i from 1 through 9, x_i + 2x_{i+1} = 1. And we need to find the value of x₁ + 512x_{10}. Hmm, okay. Let me try to understand what's going on here.
First, let me write down the equations given. For each i ... | Given 10 numbers \( x_1, x_2, \ldots, x_{10} \) such that \( x_i + 2x_{i+1} = 1 \) for each \( i \) from 1 through 9, we need to find the value of \( x_1 + 512x_{10} \).
We start by expressing each \( x_i \) in terms of the subsequent variable \( x_{i+1} \):
1. \( x_1 = 1 - 2x_2 \)
2. \( x_2 = 1 - 2x_3 \)
3. \( x_3 =... | 1. We start with the given equations:
\[
x_i + 2x_{i+1} = 1 \quad \text{for} \quad i = 1, 2, \ldots, 9
\]
2. To solve for \( x_1 + 512x_{10} \), we multiply each equation by \( (-2)^{i-1} \) and then sum them up. This approach leverages the telescoping series property.
3. Multiply each equation by \( (-2)^{i... |
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
| Okay, so I need to figure out how many subsets of the set {1,2,3,...,12} have the property that no two elements differ by more than 5. That means in any such subset, the largest number minus the smallest number is at most 5. For example, {2,5,7} is okay because 7 - 2 = 5, but {1,3,5,7} isn't because 7 - 1 = 6, which is... | The problem requires counting subsets of {1,2,…,12} where no two elements differ by more than 5.
**Key Steps:**
1. **Smallest Element Analysis**: For each element \( i \), consider subsets where \( i \) is the smallest element. The largest element in such subsets can be at most \( i + 5 \).
2. **Subset Count per Elem... | 1. Define the set \( S = \{1, 2, 3, \ldots, 12\} \).
2. Let \( a_n \) be the number of subsets of \( \{1, 2, \ldots, n\} \) such that no two elements differ by more than 5.
3. Consider the element \( n \). If \( n \) is included in the subset, then the remaining elements must come from \( \{n-1, n-2, n-3, n-4, n-5\} \)... |
Given an odd number $n >1$ , let
\begin{align*} S =\{ k \mid 1 \le k < n , \gcd(k,n) =1 \} \end{align*}
and let \begin{align*} T = \{ k \mid k \in S , \gcd(k+1,n) =1 \} \end{align*}
For each $k \in S$ , let $r_k$ be the remainder left by $\frac{k^{|S|}-1}{n}$ upon division by $n$ . Prove
\begin{align*} \prod _{k... | Okay, let's try to tackle this problem step by step. First, I need to understand what is given and what needs to be proven.
We have an odd number \( n > 1 \). The set \( S \) is defined as all integers \( k \) between 1 and \( n-1 \) such that \( \gcd(k, n) = 1 \). So \( S \) is essentially the multiplicative group mo... | Given an odd number \( n > 1 \), we define the sets \( S \) and \( T \) as follows:
\[
S = \{ k \mid 1 \le k < n, \gcd(k, n) = 1 \}
\]
\[
T = \{ k \mid k \in S, \gcd(k+1, n) = 1 \}
\]
For each \( k \in S \), \( r_k \) is the remainder when \( \frac{k^{|S|} - 1}{n} \) is divided by \( n \). We aim to prove:
\[
\prod_{k ... | 1. **Define the sets \( S \) and \( T \):**
\[
S = \{ k \mid 1 \le k < n, \gcd(k, n) = 1 \}
\]
\[
T = \{ k \mid k \in S, \gcd(k+1, n) = 1 \}
\]
2. **Determine the size of \( S \):**
\[
|S| = \phi(n)
\]
where \(\phi(n)\) is Euler's totient function, which counts the number of integers up t... |
What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$ ? | Okay, so I need to find the largest integer n such that n is divisible by every integer less than the cube root of n. Hmm, let me start by understanding the problem better.
First, the problem says "every integer less than the cube root of n". That means if I take the cube root of n, let's call that number k, then n h... | To find the largest integer \( n \) such that \( n \) is divisible by every integer less than \( \sqrt[3]{n} \), we need to consider the following steps:
1. **Understanding the Problem**: The integer \( n \) must be divisible by all integers less than \( \sqrt[3]{n} \). This means if \( \sqrt[3]{n} \) is not an intege... | 1. We need to find the largest integer \( n \) such that \( n \) is divisible by every integer less than \( \sqrt[3]{n} \). Let's denote \( k = \sqrt[3]{n} \). This means \( n = k^3 \).
2. For \( n \) to be divisible by every integer less than \( k \), \( n \) must be divisible by the least common multiple (LCM) of all... |
There is a hexagon $ABCDEF$ inscribed in a circle. Show that if sides $AB$ and $DE$ are parallel, and sides $BC$ and $EF$ are parallel, then sides $CD$ and $FA$ are also parallel. | Okay, so I need to show that in a cyclic hexagon ABCDEF, if AB is parallel to DE and BC is parallel to EF, then CD must be parallel to FA. Hmm, cyclic hexagons have all their vertices on a circle, right? So maybe properties related to cyclic polygons can help here. Also, the fact that certain sides are parallel might r... | Given a cyclic hexagon \(ABCDEF\) inscribed in a circle, we need to show that if \(AB \parallel DE\) and \(BC \parallel EF\), then \(CD \parallel FA\).
1. **Central Angles and Parallel Sides**:
- Assign central angles \(\alpha, \beta, \gamma, \delta, \epsilon, \zeta\) to sides \(AB, BC, CD, DE, EF, FA\) respectivel... | 1. **Identify the given conditions and the goal:**
- We have a hexagon \(ABCDEF\) inscribed in a circle.
- Sides \(AB\) and \(DE\) are parallel.
- Sides \(BC\) and \(EF\) are parallel.
- We need to show that sides \(CD\) and \(FA\) are also parallel.
2. **Use Pascal's Theorem:**
- Pascal's Theorem state... |
For positive reals $p$ and $q$ , define the *remainder* when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and ... | Okay, let's tackle this problem step by step. First, I need to understand the definitions and what exactly is being asked here.
So, we have positive real numbers p and q, and the "remainder" when p is divided by q is defined as the smallest nonnegative real number r such that (p - r)/q is an integer. That makes sense... | To solve the problem, we need to find the number of ordered pairs \((a, b)\) of positive integers such that \(a, b \leq 20\) and the sum of the remainders \(r_1\) and \(r_2\) when \(a\sqrt{2} + b\sqrt{3}\) is divided by \(\sqrt{2}\) and \(\sqrt{3}\) respectively equals \(\sqrt{2}\).
1. **Understanding the Remainders**... | 1. Define the remainders \( r_1 \) and \( r_2 \) as follows:
\[
r_1 = a\sqrt{2} + b\sqrt{3} - k_1\sqrt{2}
\]
where \( k_1 \) is an integer such that \( 0 \leq a\sqrt{2} + b\sqrt{3} - k_1\sqrt{2} < \sqrt{2} \). Simplifying, we get:
\[
r_1 = b\sqrt{3} - (k_1 - a)\sqrt{2}
\]
Similarly, for \( r_2 \... |
Suppose that there are 100 seats in a saloon for 100 students. All students except one know their seat. First student (which is the one who doesn't know his seat) comes to the saloon and sits randomly somewhere. Then others enter the saloon one by one. Every student that enters the saloon and finds his seat vacant, sit... | Okay, so there's this problem about 100 students and 100 seats in a saloon. All the students except the first one know their assigned seats. The first student randomly picks a seat. Then each subsequent student goes to their assigned seat. If it's taken, they pick a random seat. The question is asking for the probabili... | The probability that the last two students sit on their seats is **1/3**.
**Step-by-Step Explanation:**
1. **Problem Setup:**
There are 100 seats and 100 students. All students except the first know their assigned seats. The first student chooses randomly. Subsequent students take their seat if available; otherw... | 1. **Labeling and Initial Conditions**:
- Label the seats from 1 to 100.
- The first student does not know his seat and sits randomly in one of the 100 seats.
- Each subsequent student (from 2 to 100) knows their seat number and will sit in their seat if it is vacant. If their seat is occupied, they will sit i... |
We have an $n \times n$ square divided into unit squares. Each side of unit square is called unit segment. Some isoceles right triangles of hypotenuse $2$ are put on the square so all their vertices are also vertices of unit squares. For which $n$ it is possible that every unit segment belongs to exactly one tria... | Okay, so I need to figure out for which values of n it's possible to tile an n x n square grid with isosceles right triangles of hypotenuse 2, such that every unit segment is part of exactly one triangle. Let me try to break this down step by step.
First, let me visualize the problem. The grid is n x n, divided into u... | To determine for which values of \( n \) it is possible to tile an \( n \times n \) square grid with isosceles right triangles of hypotenuse 2 such that every unit segment belongs to exactly one triangle, we analyze the problem as follows:
1. **Understanding the Triangles and Grid**:
- Each isosceles right triangle... | 1. **Counting the Unit Segments:**
- Consider an \( n \times n \) square grid. Each side of the unit square is called a unit segment.
- The total number of unit segments in the grid can be calculated as follows:
- Each row has \( n \) horizontal unit segments, and there are \( n+1 \) such rows (including the... |
Determine all positive integers $k$ , $\ell$ , $m$ and $n$ , such that $$ \frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$ | Okay, so I need to find all positive integers k, ℓ, m, and n such that 1/k! + 1/ℓ! + 1/m! = 1/n!. Hmm, factorials can get really big very quickly, so maybe the numbers involved aren't too large? Let me start by thinking about small factorials and see if I can find any solutions.
First, factorials of 1, 2, 3, 4, 5, 6 a... | To determine all positive integers \( k \), \( \ell \), \( m \), and \( n \) such that
\[
\frac{1}{k!} + \frac{1}{\ell!} + \frac{1}{m!} = \frac{1}{n!},
\]
we start by considering the factorials of small integers and analyze the possible values for \( n \).
1. **Case \( n = 1 \)**:
- The equation becomes \( \frac... | To determine all positive integers \( k, \ell, m, \) and \( n \) such that
\[ \frac{1}{k!} + \frac{1}{\ell!} + \frac{1}{m!} = \frac{1}{n!}, \]
we start by noting that \( n \) must be less than \( k, \ell, \) and \( m \) because the sum of three positive terms on the left-hand side must be greater than any single term ... |
Let be nonnegative real numbers $ a,b,c, $ holding the inequality: $ \sum_{\text{cyc}} \frac{a}{b+c+1} \le 1. $ Prove that $ \sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1. $ | Okay, let's tackle this problem. We have nonnegative real numbers a, b, c such that the cyclic sum of a/(b+c+1) is less than or equal to 1. We need to prove that the cyclic sum of 1/(b+c+1) is at least 1. Hmm, interesting. So, given that the sum of these fractions involving a, b, c in the numerators is bounded by 1, we... | Given nonnegative real numbers \(a, b, c\) such that the cyclic sum \(\sum_{\text{cyc}} \frac{a}{b+c+1} \le 1\), we need to prove that \(\sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1\).
First, we denote the denominators as \(x = b + c + 1\), \(y = a + c + 1\), and \(z = a + b + 1\). The given condition translates to:
\[
\fr... | 1. **Given Inequality and Applying Cauchy-Schwarz:**
We start with the given inequality:
\[
\sum_{\text{cyc}} \frac{a}{b+c+1} \le 1.
\]
By the Cauchy-Schwarz inequality in the form:
\[
\left( \sum_{\text{cyc}} \frac{a}{b+c+1} \right) \left( \sum_{\text{cyc}} a(b+c+1) \right) \geq (a+b+c)^2,
\]
... |
Let $PA_1A_2...A_{12} $ be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $ .
Find the volume of the pyramid. | Okay, so I need to find the volume of this regular pyramid PA₁A₂...A₁₂. The base is a regular 12-sided polygon, right? The problem gives me the area S of triangle PA₁A₅ and the angle α between the base and the plane of triangle PA₁A₅. Hmm, volume of a pyramid is generally (1/3)*base area*height, so if I can figure out ... | To find the volume of the regular pyramid \( PA_1A_2 \ldots A_{12} \) with base \( A_1A_2 \ldots A_{12} \) (a regular dodecagon), given the area \( S \) of triangle \( PA_1A_5 \) and the angle \( \alpha \) between the planes of the base and triangle \( PA_1A_5 \):
1. **Identify Key Components**:
- The base is a reg... | 1. **Define Variables and Relationships:**
- Let \( H \) be the height of the pyramid.
- Let \( R \) be the radius of the circumcircle of the base.
- Let \( h \) be the height of the triangle \( PA_1A_5 \).
- Let \( a \) be the length of the segment \( A_1A_5 \).
- Let \( A \) be the midpoint of \( A_1A_... |
The measure of the angle $\angle A$ of the acute triangle $ABC$ is $60^o$ , and $HI = HB$ , where $I$ and $H$ are the incenter and the orthocenter of the triangle $ABC$ . Find the measure of the angle $\angle B$ . | Alright, let's try to tackle this geometry problem. So, we have an acute triangle ABC with angle A being 60 degrees. The incenter is I, and the orthocenter is H. It's given that HI = HB, and we need to find the measure of angle B. Hmm, okay. Let me break this down step by step.
First, let me recall what an incenter an... | Given an acute triangle \(ABC\) with \(\angle A = 60^\circ\), where \(H\) is the orthocenter and \(I\) is the incenter, and \(HI = HB\), we need to find the measure of \(\angle B\).
1. **Initial Setup and Analysis**:
- \(\angle A = 60^\circ\), so the remaining angles \(\angle B\) and \(\angle C\) sum to \(120^\circ... | 1. Given that $\angle A = 60^\circ$ in the acute triangle $ABC$, and $HI = HB$, where $I$ is the incenter and $H$ is the orthocenter of the triangle $ABC$.
2. Since $\angle A = 60^\circ$, triangle $ABC$ is not only acute but also has a special property: $B, H, I, C$ are concyclic. This means that points $B, H, I, C$ l... |
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$ . Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$ . Find $100p+q$ .
*Proposed by Michael Ren* | Alright, let's dive into this problem. So, we have three real numbers a, b, c such that a < b < c, and each of them satisfies the equation x³ - 3x + 1 = 0. Then, we need to find the value of the expression 1/(a² + b) + 1/(b² + c) + 1/(c² + a), express it as a reduced fraction p/q, and compute 100p + q.
First, let me ... | To solve the problem, we start by recognizing that \(a\), \(b\), and \(c\) are roots of the cubic equation \(x^3 - 3x + 1 = 0\). Using Vieta's formulas, we know:
- \(a + b + c = 0\)
- \(ab + bc + ca = -3\)
- \(abc = -1\)
We need to evaluate the expression \(S = \frac{1}{a^2 + b} + \frac{1}{b^2 + c} + \frac{1}{c^2 + a}... | 1. Given the equations \(a^3 - 3a + 1 = 0\), \(b^3 - 3b + 1 = 0\), and \(c^3 - 3c + 1 = 0\), we need to find the value of \(\frac{1}{a^2 + b} + \frac{1}{b^2 + c} + \frac{1}{c^2 + a}\).
2. Notice that the polynomial \(x^3 - 3x + 1 = 0\) has roots \(a\), \(b\), and \(c\). We can use trigonometric identities to solve for... |
Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying
\[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \] | Okay, let's tackle this problem step by step. The question is asking for all positive 3-digit integers \(\overline{abc}\) such that \(b \cdot \overline{ac} = c \cdot \overline{ab} + 10\).
First, let's parse the notation. \(\overline{abc}\) is a 3-digit number where \(a\), \(b\), and \(c\) are its digits. So, mathemat... | Given the equation \( b \cdot \overline{ac} = c \cdot \overline{ab} + 10 \) for a 3-digit integer \(\overline{abc}\), we start by expressing the terms algebraically. The 3-digit number \(\overline{abc}\) is \(100a + 10b + c\), \(\overline{ac}\) is \(10a + c\), and \(\overline{ab}\) is \(10a + b\). Substituting these in... | To find all positive 3-digit integers $\overline{abc}$ that satisfy the equation
\[ b \cdot \overline{ac} = c \cdot \overline{ab} + 10, \]
we will start by expressing $\overline{abc}$, $\overline{ac}$, and $\overline{ab}$ in terms of $a$, $b$, and $c$.
1. **Express the numbers in terms of $a$, $b$, and $c$:**
- $\... |
Find the least positive integers $m,k$ such that
a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube.
b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square.
The author is Vasile Suceveanu | Alright, so I need to find the least positive integers m and k such that there exist 2m+1 consecutive natural numbers whose sum of cubes (for part a) and sum of squares (for part b) are themselves a cube or a square, respectively. Let me start by understanding the problem step by step.
First, let's tackle part a). The... | To find the least positive integers \( m \) and \( k \) such that there exist \( 2m+1 \) consecutive natural numbers whose sum of cubes is a cube (part a) and \( 2k+1 \) consecutive natural numbers whose sum of squares is a square (part b):
### Part a)
We need to find the smallest \( m \) such that the sum of cubes o... | ### Part (a)
We need to find the least positive integer \( m \) such that there exist \( 2m+1 \) consecutive natural numbers whose sum of cubes is also a cube.
1. Consider the sequence of three consecutive natural numbers: \( 3, 4, 5 \).
2. Calculate the sum of their cubes:
\[
3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 2... |
Regular tetrahedron $ABCD$ has center $O$ and side length $1$ . Points $A'$ , $B'$ , $C'$ , and $D'$ are defined by reflecting $A$ , $B$ , $C$ , and $D$ about $O$ . Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$ . | Okay, so I need to compute the volume of the polyhedron formed by the vertices of a regular tetrahedron ABCD and their reflections A', B', C', D' about the center O. The side length of the original tetrahedron is 1. Hmm, let me try to visualize this first.
First, a regular tetrahedron has four triangular faces, each a... | To compute the volume of the polyhedron with vertices \(ABCDA'B'C'D'\) formed by reflecting a regular tetrahedron \(ABCD\) with side length 1 about its center \(O\), we follow these steps:
1. **Coordinates of the Original Tetrahedron**:
- The vertices of the regular tetrahedron \(ABCD\) are placed in a coordinate s... | 1. **Understanding the Problem:**
- We have a regular tetrahedron \(ABCD\) with center \(O\) and side length 1.
- Points \(A', B', C', D'\) are reflections of \(A, B, C, D\) about \(O\).
- We need to compute the volume of the polyhedron formed by vertices \(A, B, C, D, A', B', C', D'\).
2. **Reflection and Sy... |
(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$ , then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$ .
(b) Show that there exists a closed bound... | Alright, let's tackle part (a) first. The problem states that if we have a closed bounded interval I and a non-constant monic polynomial f defined on I with the maximum absolute value less than 2, then there exists another non-constant monic polynomial g on I with maximum absolute value less than 1. Hmm, so we need to ... | (a) Let $I$ be a closed bounded interval and $f: I \to \mathbb{R}$ a non-constant monic polynomial with $\max_{x \in I} |f(x)| < 2$. Consider the interval $I$ scaled to $[-1, 1]$ via an affine transformation. The Chebyshev polynomials $T_n(x)$, which minimize the maximum norm on $[-1, 1]$, provide monic polynomials whe... | ### Part (a)
1. **Given**: \( I \subset \mathbb{R} \) is a closed bounded interval, and \( f : I \to \mathbb{R} \) is a non-constant monic polynomial function such that \( \max_{x \in I} |f(x)| < 2 \).
2. **Objective**: Show that there exists a non-constant monic polynomial function \( g : I \to \mathbb{R} \) such th... |
Let
\[f(x)=\cos(x^3-4x^2+5x-2).\]
If we let $f^{(k)}$ denote the $k$ th derivative of $f$ , compute $f^{(10)}(1)$ . For the sake of this problem, note that $10!=3628800$ . | Okay, so I need to compute the 10th derivative of the function f(x) = cos(x³ - 4x² + 5x - 2) evaluated at x = 1. That sounds pretty intense. Let me start by breaking down the problem step by step.
First, let me recall some calculus rules related to derivatives of composite functions, especially trigonometric function... | To compute the 10th derivative of \( f(x) = \cos(x^3 - 4x^2 + 5x - 2) \) evaluated at \( x = 1 \), we start by analyzing the inner function \( g(x) = x^3 - 4x^2 + 5x - 2 \).
First, we evaluate \( g(x) \) and its derivatives at \( x = 1 \):
- \( g(1) = 0 \)
- \( g'(1) = 0 \)
- \( g''(1) = -2 \)
- \( g'''(1) = 6 \)
- A... | 1. **Substitute \( u = x - 1 \):**
We start by substituting \( u = x - 1 \) into the function \( f(x) \). This gives us:
\[
f(x) = \cos(x^3 - 4x^2 + 5x - 2) \implies f(u + 1) = \cos((u + 1)^3 - 4(u + 1)^2 + 5(u + 1) - 2)
\]
Simplifying the expression inside the cosine function:
\[
(u + 1)^3 - 4(u +... |
The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers $ n $ . Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $ . | Okay, let me try to figure out this problem step by step. So, we have a sequence defined by a₀ = 2, a₁ = 4, and then the recurrence relation aₙ₊₁ = (aₙ * aₙ₋₁)/2 + aₙ + aₙ₋₁ for all n ≥ 1. We need to find all prime numbers p such that there exists some positive integer m where p divides aₘ - 1. Hmm, interesting.
First... | The sequence \( \{ a_n \} \) is defined by \( a_0 = 2 \), \( a_1 = 4 \), and the recurrence relation:
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers \( n \). We need to determine all prime numbers \( p \) for which there exists a positive integer \( m \) such that \( p \) divides \( a_m... | 1. **Define a new sequence \( b_n \):**
We start by defining a new sequence \( b_n \) such that \( b_n = a_n + 2 \). Given the initial conditions \( a_0 = 2 \) and \( a_1 = 4 \), we have:
\[
b_0 = a_0 + 2 = 2 + 2 = 4
\]
\[
b_1 = a_1 + 2 = 4 + 2 = 6
\]
2. **Derive the recurrence relation for \( b_n... |
Let $\triangle ABC$ be an acute-scalene triangle, and let $N$ be the center of the circle wich pass trough the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ . Prove that $A$ , $D$ and $N$ are collinear iff $\measuredangle BAC = 45º$ . | Okay, so I have this geometry problem here. Let me read it again to make sure I understand.
We have an acute-scalene triangle ABC. N is the center of the circle that passes through the feet of the altitudes. That circle is probably the nine-point circle, right? Because the nine-point circle passes through the feet of ... | To solve the problem, we need to prove that points \( A \), \( D \), and \( N \) are collinear if and only if \( \angle BAC = 45^\circ \) in an acute-scalene triangle \( \triangle ABC \).
### Key Steps and Analysis:
1. **Definitions and Properties**:
- \( N \) is the nine-point center, which is the midpoint of the... | 1. **Claim 1: \(AH \parallel OD\)**
*Proof:* Let \(X\) be the perpendicular from \(D\) to line \(BC\). By equal tangents, we have \(BD = DC\). By HL similarity, we have \(\triangle BXD \sim \triangle CXD\) so \(BX = XC\). Hence, \(D\) lies on the perpendicular bisector of \(BC\). Similarly, we can show that \(O\) l... |
Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$ . Compute the sum of all bases and exponents in the prime factorization of $A$ . For example, if $A=7\cdot 11^5$ , the answer would be $7+11+5=23$ . | Okay, so I need to find the greatest possible value of a product of positive integers that sum up to 2014. Then, once I have that maximum product A, I need to factor it into primes and sum all the bases and exponents in its prime factorization.
First, the problem is about breaking down the number 2014 into positive i... | To find the greatest possible value of a product of positive integers that sum to 2014, we use the strategy of dividing the number by 3 and using the quotient and remainder to determine the optimal product.
1. Divide 2014 by 3:
- Quotient: 671
- Remainder: 1
2. Since the remainder is 1, we replace one 3 and th... | 1. To maximize the product of positive integers that sum to 2014, we need to consider the properties of numbers. Specifically, the product of numbers is maximized when the numbers are as close to each other as possible. For positive integers, this means using as many 3's as possible, since $3 \times 3 = 9$ is greater t... |
Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$ . | Okay, so I need to find the smallest positive integer m such that 529^n + m·132^n is divisible by 262417 for all odd positive integers n. Hmm, let's start by breaking down the problem step by step.
First, let's note that 262417 is a prime number? Wait, maybe I should check if it's prime first. If it's not prime, facto... | To determine the smallest positive integer \( m \) such that \( 529^n + m \cdot 132^n \) is divisible by \( 262417 \) for all odd positive integers \( n \):
1. **Factorize \( 262417 \)**:
- \( 262417 = 661 \times 397 \), where \( 661 \) and \( 397 \) are primes.
2. **Solve the congruence modulo \( 661 \)**:
- C... | To determine the smallest positive integer \( m \) such that \( 529^n + m \cdot 132^n \) is divisible by \( 262417 \) for all odd positive integers \( n \), we start by factoring \( 262417 \).
1. **Factorize \( 262417 \):**
\[
262417 = 397 \times 661
\]
Both \( 397 \) and \( 661 \) are prime numbers.
2. *... |
$30$ same balls are put into four boxes $ A$ , $ B$ , $ C$ , $ D$ in such a way that sum of number of balls in $ A$ and $ B$ is greater than sum of in $ C$ and $ D$ . How many possible ways are there? $\textbf{(A)}\ 2472 \qquad\textbf{(B)}\ 2600 \qquad\textbf{(C)}\ 2728 \qquad\textbf{(D)}\ 2856 \qqua... | Okay, so I need to figure out how many ways there are to distribute 30 identical balls into four boxes: A, B, C, D. The condition is that the sum of balls in A and B is greater than the sum in C and D. The answer choices are given, so maybe after working through it, I can check which one matches. Let me start by unders... | To determine the number of ways to distribute 30 identical balls into four boxes \( A, B, C, D \) such that the sum of the balls in \( A \) and \( B \) is greater than the sum in \( C \) and \( D \), we start by noting that the total number of ways to distribute 30 balls into 4 boxes without any constraints is given by... | 1. **Calculate the total number of ways to distribute 30 balls into 4 boxes:**
The number of ways to distribute \(30\) identical balls into \(4\) distinct boxes can be calculated using the stars and bars theorem. The formula for distributing \(n\) identical items into \(k\) distinct groups is given by:
\[
\bin... |
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed? | Okay, so I need to figure out the minimum number of problems required to create ten test papers for a National Olympiad. Each paper has 4 problems, and the condition is that no two papers share more than 1 problem. Hmm, this seems like a combinatorial problem, maybe related to block designs or something? Let me try to ... | To determine the minimum number of problems needed for ten test papers, each with 4 problems and no two sharing more than one problem, we analyze the constraints and apply combinatorial design principles.
1. **Pair Counting Argument**:
Each test paper has \( \binom{4}{2} = 6 \) unique problem pairs. For 10 papers... | 1. **Define the problem and variables:**
Let \( n \) be the minimal number of problems needed. Let \( P_1, P_2, \ldots, P_n \) be the problems. Let \( A_1, A_2, \ldots, A_{10} \) be the sets of problems in each test paper. Each set \( A_i \) contains 4 problems, and no two sets \( A_i \) and \( A_j \) (for \( i \neq... |
The sports ground has the shape of a rectangle $ABCD$ , with the angle between the diagonals $AC$ and $BD$ is equal to $60^o$ and $AB >BC$ . The trainer instructed Andriyka to go first $10$ times on the route $A-C-B-D-A$ , and then $15$ more times along the route $A-D-A$ . Andriyka performed the task, mo... | Okay, let's see. I need to find the length of diagonal AC in this rectangular sports ground. The problem says that the angle between the diagonals AC and BD is 60 degrees, and that AB is longer than BC. So, the rectangle is longer along AB, which makes sense since AB > BC.
First, let me visualize the rectangle. ABCD ... | Given a rectangular sports ground \(ABCD\) with \(AB > BC\) and the angle between the diagonals \(AC\) and \(BD\) equal to \(60^\circ\), we need to determine the length of diagonal \(AC\). Andriyka's total running distance of 4.5 km includes 10 laps around the route \(A-C-B-D-A\) and 15 laps around the route \(A-D-A\).... | 1. **Understanding the problem**: We are given a rectangle \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at an angle of \(60^\circ\). Andriyka travels along two routes: \(A-C-B-D-A\) and \(A-D-A\), and the total distance covered is \(4.5\) km. We need to find the length of the diagonal \(AC\).
2. **Analyzing ... |
In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$... | Okay, let's see. I need to find the distance between the rose bush and the jasmine in this forest with five trees A, B, C, D, E arranged in a straight line. The positions of the flowers are at the midpoints of specific tree pairs. Let me try to visualize this first.
So, the trees are in the order A, B, C, D, E on a st... | Given five trees \( A, B, C, D, E \) arranged in a straight line with the total distance from \( A \) to \( E \) being 28 meters, and the distance between the daisy (midpoint of \( AB \)) and the carnation (midpoint of \( DE \)) being 20 meters, we need to find the distance between the rose bush (midpoint of \( BC \)) ... | 1. Let the distances between the trees be as follows:
- \( AB = 2a \)
- \( BC = 2b \)
- \( CD = 2c \)
- \( DE = 2d \)
2. The total distance between \( A \) and \( E \) is given as \( 28 \) meters. Therefore, we have:
\[
AB + BC + CD + DE = 2a + 2b + 2c + 2d = 28
\]
Simplifying, we get:
\[
... |
Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$ . | Alright, let's tackle this problem step by step. The goal is to find all real numbers \( a \) such that there exists a non-constant function \( f: \mathbb{R} \to \mathbb{R} \) satisfying the functional equation \( f(f(x)) = x f(x) - a x \) for all real \( x \), and also \( f \) takes the value \( a \).
First, let me ... | To find all real numbers \( a \) such that there exists a non-constant function \( f: \mathbb{R} \to \mathbb{R} \) satisfying the functional equation \( f(f(x)) = x f(x) - ax \) and taking the value \( a \), we proceed as follows:
1. **Substitution and Key Relations**:
- Substitute \( x = c \) where \( f(c) = a \).... | 1. **Assume the existence of a function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the given conditions.**
- Given conditions:
- \( f(f(x)) = x f(x) - ax \) for all \( x \in \mathbb{R} \)
- \( f \) is not a constant function
- \( f \) takes the value \( a \)
2. **Let \( f(x_0) = a \) for some ... |
Let $ABF$ be a right-angled triangle with $\angle AFB = 90$ , a square $ABCD$ is externally to the triangle. If $FA = 6$ , $FB = 8$ and $E$ is the circumcenter of the square $ABCD$ , determine the value of $EF$ | Okay, so I need to solve this geometry problem. Let me read it again to make sure I understand all the details.
We have a right-angled triangle ABF where angle AFB is 90 degrees. There's a square ABCD constructed externally on the triangle. The lengths FA is 6 units and FB is 8 units. E is the circumcenter of the squ... | Given a right-angled triangle \(ABF\) with \(\angle AFB = 90^\circ\), where \(FA = 6\) and \(FB = 8\). A square \(ABCD\) is constructed externally on the triangle. We need to determine the value of \(EF\), where \(E\) is the circumcenter of the square \(ABCD\).
1. **Calculate the hypotenuse \(AB\)**:
\[
AB = \sq... | 1. **Identify the given information and the goal:**
- We have a right-angled triangle \( \triangle ABF \) with \( \angle AFB = 90^\circ \).
- A square \( ABCD \) is constructed externally to the triangle.
- \( FA = 6 \), \( FB = 8 \), and \( E \) is the circumcenter of the square \( ABCD \).
- We need to de... |
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$ . Prove that the sequence $\{x_{n}\}$ converges and find its limit. | Okay, so I have this sequence defined by x₁ = a, where a is at least 1, and then each subsequent term is x_{n+1} = 1 + ln(x_n² / (1 + ln x_n)). I need to prove that this sequence converges and find its limit. Alright, let's start by understanding the problem step by step.
First, let's write down the recursion formula ... | To prove that the sequence \(\{x_n\}\) converges and find its limit, we start by analyzing the recursion formula \(x_{n+1} = 1 + \ln\left(\frac{x_n^2}{1 + \ln x_n}\right)\) with \(x_1 = a \geq 1\).
1. **Fixed Point Analysis**:
- Suppose the sequence converges to a limit \(L\). Then \(L\) must satisfy the equation:
... | 1. **Define the function and sequence:**
Let \( f(x) = 1 + \ln\left(\frac{x^2}{1 + \ln x}\right) \). The sequence is defined as \( x_1 = a \) and \( x_{n+1} = f(x_n) \) for \( n \geq 1 \).
2. **Check the derivative of \( f(x) \):**
Compute the derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx} \left( 1 + \ln\... |
Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$ . | Okay, so I need to find the number of integers c between -2007 and 2007 (inclusive) such that there exists an integer x where x² + c is a multiple of 2²⁰⁰⁷. Hmm, that's a pretty high power of 2. Let me try to break this down step by step.
First, the problem is asking for values of c where x² ≡ -c mod 2²⁰⁰⁷ has a solut... | To determine the number of integers \( c \) within \(-2007 \leq c \leq 2007\) such that there exists an integer \( x \) where \( x^2 + c \) is a multiple of \( 2^{2007} \), we analyze quadratic residues modulo \( 2^{2007} \).
1. **Quadratic Residues Modulo \( 2^{2007} \)**:
- **Even Residues**: Must be divisible by... | To solve the problem, we need to find the number of integers \( c \) such that \( -2007 \leq c \leq 2007 \) and there exists an integer \( x \) such that \( x^2 + c \) is a multiple of \( 2^{2007} \). This means we need \( x^2 + c \equiv 0 \pmod{2^{2007}} \), or equivalently, \( x^2 \equiv -c \pmod{2^{2007}} \).
1. **... |
Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$ , $f(p)$ and $f(q)$ are relatively prime. | Okay, so I need to find all polynomials with integer coefficients such that for any two distinct primes p and q, f(p) and f(q) are relatively prime. Hmm, let's start by thinking about what this condition really means.
First, let's consider constant polynomials. If f is a constant polynomial, say f(x) = c for some inte... | To determine all polynomials \( f \) with integer coefficients such that for any two distinct primes \( p \) and \( q \), \( f(p) \) and \( f(q) \) are relatively prime, we analyze the problem through several key steps:
1. **Constant Polynomials**:
- If \( f \) is a constant polynomial, it must be either \( 1 \) o... | 1. **Assume the form of the polynomial \( f \):**
Let \( f(x) = x^m g(x) \) for some \( g \in \mathbb{Z}[x] \) with \( g(0) \neq 0 \) and \( m \in \mathbb{Z}_+ \). This is a general form where \( f(x) \) is factored into a monomial \( x^m \) and another polynomial \( g(x) \).
2. **Consider the case when \( \deg g =... |
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$ s and $1'$ s such that there are equal number of $00$ and $11$ blocks in each string is equal to
\[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\] | Okay, so I need to show that the number of binary strings of length n (with n ≥ 2) that have an equal number of 00 and 11 blocks is equal to 2 times the binomial coefficient of (n-2) choose floor((n-2)/2). Hmm, let me try to think through this step by step.
First, let's make sure I understand the problem correctly. We... | To show that the number of binary strings of length \( n \) such that there are an equal number of \( 00 \) and \( 11 \) blocks is given by \( 2\binom{n-2}{\left\lfloor \frac{n-2}{2} \right\rfloor} \), we can analyze the structure of these strings and their properties.
1. **Understanding Blocks**: A block is a maximal... | To show that the number of strings of length \( n \) consisting of \( 0 \)'s and \( 1 \)'s such that there are equal numbers of \( 00 \) and \( 11 \) blocks in each string is equal to
\[ 2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}, \]
we will proceed with the following steps:
1. **Replace \( 0 \)'s with \(... |
What is the least positive integer $m$ such that the following is true?
*Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$* \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm ... | Alright, so I have this problem here: I need to find the least positive integer m such that if I have m integers between 1 and 2023 inclusive, there must exist two of them, a and b, where 1 < a/b ≤ 2. The options given are a.10, b.11, c.12, d.13, e.1415. Let me try to work through this step by step.
First, I recall th... | To determine the least positive integer \( m \) such that any \( m \) integers between 1 and 2023 inclusive must contain two integers \( a \) and \( b \) with \( 1 < \frac{a}{b} \le 2 \), we use the Pigeonhole Principle and interval partitioning.
1. **Interval Partitioning**:
- We partition the numbers from 1 to 20... | To solve this problem, we need to find the smallest positive integer \( m \) such that among any \( m \) integers chosen from the set \(\{1, 2, \ldots, 2023\}\), there exist two integers \( a \) and \( b \) satisfying \( 1 < \frac{a}{b} \leq 2 \).
1. **Understanding the Problem:**
- We need to ensure that for any \... |
In a board of $2021 \times 2021$ grids, we pick $k$ unit squares such that every picked square shares vertice(s) with at most $1$ other picked square. Determine the maximum of $k$ . | Okay, let me try to figure out this problem. So, we have a 2021x2021 grid, and we need to pick k unit squares such that each picked square shares vertices (corners) with at most 1 other picked square. We need to find the maximum possible k. Hmm, interesting. Let me start by understanding the problem better.
First, let... | To determine the maximum number \( k \) of unit squares that can be picked on a \( 2021 \times 2021 \) grid such that each picked square shares vertices with at most 1 other picked square, we need to consider the constraints and possible configurations.
1. **Understanding the Problem**:
- Each picked square can sha... | 1. **Generalization for any \( n \equiv 1 \pmod{4} \):**
We are given a \( n \times n \) grid where \( n \equiv 1 \pmod{4} \). We need to determine the maximum number of unit squares \( k \) that can be picked such that each picked square shares vertices with at most one other picked square.
2. **Shapes and their a... |
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