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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If both $ x$ and $ y$ are both integers, how many pairs of solutions are there of the equation $ (x\minus{}8)(x\minus{}10) \equal{} 2^y?$
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{more than 3}$ | Okay, so I need to solve this problem where both x and y are integers, and the equation is (x - 8)(x - 10) = 2^y. The question is asking how many pairs (x, y) satisfy this equation, and the answer choices are from 0 to more than 3. Let me start by understanding the problem step by step.
First, the equation is (x - 8)(... | Given the equation \((x - 8)(x - 10) = 2^y\) where both \(x\) and \(y\) are integers, we need to find the number of pairs \((x, y)\) that satisfy this equation.
1. **Analyze the product \((x - 8)(x - 10)\)**:
- The product must be a power of 2, i.e., \(2^y\).
- Since \(2^y\) is positive, both factors \((x - 8)\)... | To solve the problem, we need to find all integer pairs \((x, y)\) that satisfy the equation \((x-8)(x-10) = 2^y\).
1. **Expand and simplify the equation:**
\[
(x-8)(x-10) = x^2 - 18x + 80
\]
Therefore, we need to find integer values of \(x\) such that \(x^2 - 18x + 80 = 2^y\).
2. **Analyze the quadratic ... |
Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit. | Okay, so I need to find all possible leftmost digits that 2^n and 3^n can share when n is a natural number. Hmm, let me start by understanding the problem.
First, the leftmost digit of a number in its decimal representation depends on its magnitude. For example, if a number is between 100 and 999, the leftmost digit i... | To determine all possible leftmost digits shared by \(2^n\) and \(3^n\) for natural numbers \(n\), we analyze the fractional parts of \(n \log_{10} 2\) and \(n \log_{10} 3\). The leading digit of a number \(x\) is determined by \(10^{\{ \log_{10} x \}}\), where \(\{ \cdot \}\) denotes the fractional part. For \(2^n\) a... | To solve the problem, we need to find all possible values of the leftmost digit in the decimal representations of \(2^n\) and \(3^n\) such that they are the same. We will use a brute force approach to verify the possible values.
1. **Understanding the Problem:**
- We need to find \(n\) such that the leftmost digit ... |
For real constant numbers $ a,\ b,\ c,\ d,$ consider the function $ f(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ such that $ f( \minus{} 1) \equal{} 0,\ f(1) \equal{} 0,\ f(x)\geq 1 \minus{} |x|$ for $ |x|\leq 1.$
Find $ f(x)$ for which $ \int_{ \minus{} 1}^1 \{f'(x) \minus{} x\}^2\ dx$ is minim... | Okay, so I need to find the cubic function f(x) = ax³ + bx² + cx + d that satisfies the given conditions and minimizes the integral of {f'(x) - x}² from -1 to 1. Let me start by breaking down the problem step by step.
First, the problem states that f(-1) = 0 and f(1) = 0. So, plugging those x-values into the cubic equ... | To find the cubic function \( f(x) = ax^3 + bx^2 + cx + d \) that satisfies \( f(-1) = 0 \), \( f(1) = 0 \), \( f(x) \geq 1 - |x| \) for \( |x| \leq 1 \), and minimizes the integral \( \int_{-1}^1 (f'(x) - x)^2 \, dx \), we proceed as follows:
1. **Equality Constraints**:
- \( f(-1) = -a + b - c + d = 0 \)
- \( ... | 1. Given the function \( f(x) = ax^3 + bx^2 + cx + d \) with the conditions \( f(-1) = 0 \) and \( f(1) = 0 \), we can write:
\[
f(-1) = -a + b - c + d = 0
\]
\[
f(1) = a + b + c + d = 0
\]
Adding these two equations, we get:
\[
(-a + b - c + d) + (a + b + c + d) = 0 \implies 2b + 2d = 0 \imp... |
Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient? | Okay, so there's this problem about two thieves who stole an open chain with 2k white beads and 2m black beads. They want to split the loot equally by cutting the chain so that each gets k white beads and m black beads. The question is asking for the minimal number of cuts that is always sufficient, no matter how the b... | To solve the problem of determining the minimal number of cuts required to divide an open chain with \(2k\) white beads and \(2m\) black beads into two equal parts each containing \(k\) white beads and \(m\) black beads, we need to consider the following steps:
1. **Understanding the Problem**: The goal is to divide t... | 1. **Labeling the Beads:**
- We label the beads modulo $2(k+m)$ as $0, 1, 2, \ldots, 2(k+m)-1$.
2. **Considering Intervals:**
- We consider intervals of length $k+m$ of the form $[x, x+(k+m)-1]$.
3. **Analyzing the Number of White Beads:**
- As $x$ changes by one, the number of white beads in the interval ch... |
A natural number $n$ was alternately divided by $29$ , $41$ and $59$ . The result was three nonzero remainders, the sum of which equals $n$ . Find all such $n$ | Okay, let's tackle this problem step by step. The problem states that a natural number n is alternately divided by 29, 41, and 59, resulting in three nonzero remainders whose sum equals n. We need to find all such n. Hmm, interesting. Let me try to understand what this means.
First, when the problem says "alternately ... | Given a natural number \( n \) that, when divided by 29, 41, and 59, results in three nonzero remainders whose sum equals \( n \), we need to find all such \( n \).
We start by noting that the remainders \( r_1 \), \( r_2 \), and \( r_3 \) when \( n \) is divided by 29, 41, and 59 respectively, must satisfy:
\[
n = r_... | 1. Let \( n \equiv a \pmod{29} \), \( n \equiv b \pmod{41} \), and \( n \equiv c \pmod{59} \). We are given that \( a + b + c = n \).
2. Since \( a, b, c \) are remainders, we have \( 0 < a < 29 \), \( 0 < b < 41 \), and \( 0 < c < 59 \). Therefore, \( n = a + b + c \leq 28 + 40 + 58 = 126 \).
3. We also know that \(... |
Let $P$ and $Q$ be points on the plane $ABC$ such that $m(\widehat{BAC})=90^\circ$ , $|AB|=1$ , $|AC|=\sqrt 2$ , $|PB|=1=|QB|$ , $|PC|=2=|QC|$ , and $|PA|>|QA|$ . What is $|PA|/|QA|$ ? $ \textbf{(A)}\ \sqrt 2 +\sqrt 3
\qquad\textbf{(B)}\ 5-\sqrt 6
\qquad\textbf{(C)}\ \sqrt 6 -\sqrt 2
\qquad\textbf{(D)}\ \s... | Alright, let me try to figure out this geometry problem. Hmm, okay, so we have a right triangle ABC with a right angle at A. The sides AB are length 1, AC are sqrt(2), so BC must be the hypotenuse. Let me confirm that using Pythagoras: AB² + AC² = BC². So 1² + (√2)² = 1 + 2 = 3, so BC is sqrt(3). Got that.
Now, points... | Given a right triangle \(ABC\) with \(\angle BAC = 90^\circ\), \( |AB| = 1 \), \( |AC| = \sqrt{2} \), we need to find the ratio \( \frac{|PA|}{|QA|} \) where points \(P\) and \(Q\) lie on the plane of \(ABC\) such that \( |PB| = |QB| = 1 \) and \( |PC| = |QC| = 2 \), with \( |PA| > |QA| \).
1. **Coordinates Setup**:
... | 1. **Identify the given information and set up the problem:**
- $\angle BAC = 90^\circ$
- $|AB| = 1$
- $|AC| = \sqrt{2}$
- $|PB| = 1$
- $|QB| = 1$
- $|PC| = 2$
- $|QC| = 2$
- $|PA| > |QA|$
2. **Use the given distances to place points $A$, $B$, and $C$ in a coordinate system:**
- Place $A$ at... |
In scalene $\triangle ABC$ , $I$ is the incenter, $I_a$ is the $A$ -excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$ , and $M$ is the midpoint of side $BC$ . Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$ . Let $Q$ be the intersection of $DP$... | Okay, let's try to tackle this geometry problem step by step. It's a bit complex with several points and conditions, so I need to take it slowly and visualize everything first. Let me start by listing out all the given information and what needs to be found.
We have a scalene triangle ABC. The incenter is I, and the A... | Given a scalene triangle \( \triangle ABC \) with incenter \( I \), \( A \)-excenter \( I_a \), midpoint \( D \) of arc \( BC \) of the circumcircle not containing \( A \), and midpoint \( M \) of side \( BC \). Extending ray \( IM \) past \( M \) to point \( P \) such that \( IM = MP \), we need to find the ratio \( \... | 1. **Define the angles and relationships:**
Let \( \angle BAC = A \), \( \angle ABC = B \), and \( \angle ACB = C \). We know that \( D \) is the midpoint of the arc \( BC \) not containing \( A \), so \( D \) is equidistant from \( B \) and \( C \). This implies that \( \angle BDC = \angle BIC = 90^\circ + \frac{A}... |
$ What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?$ I'm posting this problem for people to discuss | Okay, so I have this problem: What is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101? Hmm, interesting. Let me try to figure this out step by step.
First, I need to understand the problem clearly. We need a sequence of distinct integers where eve... | The maximum number of distinct integers in such a sequence is **22**.
**Construction:**
1. **First 11 terms:** \(-6, -3, 0, 3, 6, 9, 12, 15, 18, 21, 25\) (sum = 100).
2. **Next 11 terms:** Generated by alternating \(+1\) and \(-1\) from the first 11 terms:
\(-5, -4, 1, 2, 7, 8, 13, 14, 19, 20, 26\) (sum of a... | To solve this problem, we need to find the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101. Let's break down the solution step-by-step:
1. **Understanding the Problem:**
- We need to find a sequence of integers where the sum of any 11 consecutive in... |
Jesse has ten squares, which are labeled $1, 2, \dots, 10$ . In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$ , if $i$ divides $j$ , then the $i$ -th and $j$ -th squares have different colors? | Okay, let me try to figure out how to solve this problem. Jesse has ten squares labeled 1 to 10, and he wants to color each square either red, green, yellow, or blue. The condition is that for any two squares i and j where i divides j (and i < j), the colors of the i-th and j-th squares must be different. I need to fin... | Jesse has ten squares labeled 1 to 10, each of which he can color red, green, yellow, or blue. The constraint is that if a square \(i\) divides \(j\) (where \(i < j\)), then the colors of squares \(i\) and \(j\) must be different. The goal is to determine the number of valid colorings under this constraint.
1. **Key I... | To solve this problem, we need to ensure that for all \(1 \le i < j \le 10\), if \(i\) divides \(j\), then the \(i\)-th and \(j\)-th squares have different colors. We will use a step-by-step approach to count the number of valid colorings.
1. **Color the 1st square:**
- There are 4 possible colors for the 1st squar... |
Determine all integers $n$ such that both of the numbers: $$ |n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8| $$ are both prime numbers. | Okay, so I need to find all integers n where both |n³ - 4n² + 3n - 35| and |n² + 4n + 8| are prime numbers. Hmm, let's start by understanding the problem step by step.
First, let's break down the two expressions:
1. The first expression is the absolute value of a cubic polynomial: |n³ - 4n² + 3n - 35|. Let's call thi... | To determine all integers \( n \) such that both \( |n^3 - 4n^2 + 3n - 35| \) and \( |n^2 + 4n + 8| \) are prime numbers, we analyze both expressions step by step.
1. **Expression \( |n^2 + 4n + 8| \)**:
- This quadratic expression is always positive since its discriminant is negative.
- We check for integer val... | To determine all integers \( n \) such that both \( |n^3 - 4n^2 + 3n - 35| \) and \( |n^2 + 4n + 8| \) are prime numbers, we will analyze the given expressions and check for integer solutions.
1. **Analyze the expression \( |n^2 + 4n + 8| \):**
- We need \( n^2 + 4n + 8 \) to be a prime number.
- Let's consider ... |
Triangle $ ABC$ has side lengths $ AB \equal{} 5$ , $ BC \equal{} 6$ , and $ AC \equal{} 7$ . Two bugs start simultaneously from $ A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $ D$ . What is $ BD$ ?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad ... | Okay, let's try to solve this problem step by step. So, we have triangle ABC with sides AB=5, BC=6, and AC=7. Two bugs start from point A at the same time and crawl in opposite directions around the triangle. They meet at point D, and we need to find the length BD. The answer choices are from 1 to 5, so BD is an intege... | Triangle \( ABC \) has side lengths \( AB = 5 \), \( BC = 6 \), and \( AC = 7 \). Two bugs start simultaneously from \( A \) and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point \( D \). We need to find the length \( BD \).
First, we calculate the perimeter of triangle... | 1. Let's denote the points where the bugs meet as \( D \). Since the bugs start from \( A \) and crawl along the sides of the triangle \( ABC \) in opposite directions at the same speed, they will meet at a point \( D \) such that the distances they have crawled are equal.
2. Let the bug starting from \( A \) and craw... |
Find the number of solutions to the given congruence $$ x^{2}+y^{2}+z^{2} \equiv 2 a x y z \pmod p $$ where $p$ is an odd prime and $x,y,z \in \mathbb{Z}$ .
*Proposed by math_and_me* | Okay, so I need to find the number of solutions to the congruence equation \( x^2 + y^2 + z^2 \equiv 2axyz \pmod{p} \), where \( p \) is an odd prime and \( x, y, z \) are integers modulo \( p \). The problem is proposed by math_and_me, so I guess it's a problem related to number theory and modular equations. Let me st... | To determine the number of solutions to the congruence \( x^2 + y^2 + z^2 \equiv 2axyz \pmod{p} \) where \( p \) is an odd prime and \( x, y, z \in \mathbb{Z} \), we analyze the equation through various approaches and cases.
1. **Case Analysis for Zero Variables**:
- **At least one variable is zero**:
- If \(... | 1. **Lemma 1**: Let \( S_n = \sum_{x=0}^{p-1} x^n \). Then \( S_n \equiv -1 \pmod{p} \) if \( p-1 \mid n \) and \( S_n \equiv 0 \pmod{p} \) otherwise.
2. **Lemma 2**: Let \( a, b, c \in \mathbb{Z} \) and \( p \nmid a \). Then
\[
\sum_{x=0}^{p-1} \left( \frac{ax^2 + bx + c}{p} \right) =
\begin{cases}
- \... |
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$ .
Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$ .
*(I. Gorodnin)* | Okay, let's try to figure out this problem. So, we have several real numbers in the interval [0,1], and their total sum doesn't exceed S. We need to find the maximum S such that no matter what these numbers are, we can always partition them into two groups. One group should have a sum A ≤ 8 and the other B ≤ 4. Wait, b... | To solve the problem of finding the maximum value \( S \) such that any set of real numbers from \([0, 1]\) with a sum not exceeding \( S \) can be partitioned into two groups with sums \( A \leq 8 \) and \( B \leq 4 \), we need to consider the constraints and possible configurations of the numbers.
1. **Understanding... | 1. **Initial Consideration**:
- We need to find the maximum value of \( S \) such that any set of real numbers from \([0,1]\) with a sum not exceeding \( S \) can be partitioned into two groups with sums \( A \leq 8 \) and \( B \leq 4 \).
- Clearly, if \( S > 12 \), it is impossible to partition the numbers as re... |
Let $A$ be an $n\times n$ square matrix with integer entries. Suppose that $p^2A^{p^2}=q^2A^{q^2}+r^2I_n$ for some positive integers $p,q,r$ where $r$ is odd and $p^2=q^2+r^2$ . Prove that $|\det A|=1$ . (Here $I_n$ means the $n\times n$ identity matrix.) | Okay, so I need to prove that the absolute value of the determinant of matrix A is 1. The given conditions are that A is an n×n matrix with integer entries, and there's this equation involving p, q, r which are positive integers. Also, r is odd and p² = q² + r². The equation given is p²A^{p²} = q²A^{q²} + r²I_n.
Firs... | Given an \( n \times n \) square matrix \( A \) with integer entries, we need to prove that \( |\det A| = 1 \) under the conditions \( p^2A^{p^2} = q^2A^{q^2} + r^2I_n \) for positive integers \( p, q, r \) where \( r \) is odd and \( p^2 = q^2 + r^2 \).
1. **Eigenvalue Analysis**:
- Consider an eigenvalue \( \lamb... | 1. **Define the polynomial $\Pi(x)$:**
\[
\Pi(x) = p^2 x^{p^2} - q^2 x^{q^2} - r^2
\]
Given that \( p^2 A^{p^2} = q^2 A^{q^2} + r^2 I_n \), we have:
\[
\Pi(A) = p^2 A^{p^2} - q^2 A^{q^2} - r^2 I_n = 0
\]
This implies that \( \Pi(A) = 0 \).
2. **Evaluate $\Pi(\lambda)$ for eigenvalues $\lambda$:... |
Compute $e^{\pi}+\pi^e$ . If your answer is $A$ and the correct answer is $C$ , then your score on this problem will be $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$ (note that the closer you are to the right answer, the higher your score is).
*2017 CCA Math Bonanza Lightning Round #5.2* | Okay, so I need to compute e^π + π^e. Hmm, that seems straightforward, but I remember that sometimes these exponent problems can be tricky. Let me think step by step.
First, let me recall what e and π are approximately. e is approximately 2.71828, and π is approximately 3.14159. So, e^π means e raised to the power of ... | To compute \( e^{\pi} + \pi^e \), we start by approximating the values of \( e^{\pi} \) and \( \pi^e \).
1. **Calculating \( e^{\pi} \)**:
- \( e \approx 2.71828 \) and \( \pi \approx 3.14159 \).
- Using the Taylor series expansion for \( e^x \) around \( x = 0 \), we approximate \( e^{0.141592653589793} \):
... | 1. **Estimate \( e^{\pi} \):**
- We know that \( e \approx 2.718 \) and \( \pi \approx 3.14159 \).
- To estimate \( e^{\pi} \), we can use the approximation \( e \approx 2.7 \) and \( \pi \approx 3.1 \).
- Therefore, \( e^{\pi} \approx 2.7^{3.1} \).
2. **Estimate \( 2.7^{3.1} \):**
- We can approximate \( ... |
There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$ . Suppose that at some point in time, all three flies... | Okay, so I have this problem about three flies starting at the same position on a circular track with a circumference of 1000 meters. They fly clockwise at speeds of 2, 6, and k meters per second, where k is an integer between 7 and 2013. The question is asking how many possible values of k there are such that all thre... | To solve the problem of determining how many possible values of \( k \) exist such that all three flies meet at a location different from their starting point, we need to analyze their meeting conditions on a circular track with circumference 1000 meters.
1. **Position Equations**:
- The positions of the flies at t... | 1. We need to find the values of \( k \) such that the three flies meet at a location different from their starting point. The flies travel at speeds of 2, 6, and \( k \) meters per second, respectively, on a circular track with a circumference of 1000 meters.
2. Let \( t \) be the time in seconds when the flies meet ... |
Given a convex pentagon $ABCDE$ , with $\angle BAC = \angle ABE = \angle DEA - 90^o$ , $\angle BCA = \angle ADE$ and also $BC = ED$ . Prove that $BCDE$ is parallelogram. | Okay, let's try to tackle this geometry problem step by step. So, we have a convex pentagon ABCDE with several angle conditions and a side length condition. The goal is to prove that BCDE is a parallelogram. Hmm, okay. Let's start by recalling what defines a parallelogram. A quadrilateral is a parallelogram if both pai... | Given a convex pentagon \(ABCDE\) with the following conditions:
1. \(\angle BAC = \angle ABE = \angle DEA - 90^\circ\)
2. \(\angle BCA = \angle ADE\)
3. \(BC = ED\)
We need to prove that \(BCDE\) is a parallelogram.
**Key Steps:**
1. **Assign Coordinates:**
- Place \(A\) at \((0, 0)\) and \(B\) at \((1, 0)\).
... | 1. **Denote the angles and point:**
Let $\angle ABE = \alpha$, $\angle BCA = \beta$, and let $P$ be the intersection point of $AC$ with the line parallel to $AB$ from $E$.
2. **Establish parallel lines and angles:**
Since $EP \parallel AB$, we have $\angle PAB = \angle CAB = \angle ABE = \alpha$. Therefore, $\an... |
For a constant $\alpha>0$ , take the points $O(0,\ 0),\ A(\alpha,\ \ln\ (\alpha +1)))$ on the curve $C: y=\ln\ (x+1)$ .
Consider a moving point $P(t,\ \ln\ (t+1))$ with $0<t<\alpha$ on the curve $C$ . Denote by $S_1$ the area of the part
bounded by the curve $C$ and the line $OP$ , and by $S_2$ that of ... | Okay, let's try to tackle this problem step by step. It has three parts, and each part seems to build on the previous one. Let me start by understanding what each part is asking and how they connect.
First, part (1) asks to show that for x > 0, the inequality x/(x+1) < ln(x+1) < x holds. That seems like an inequality ... | ### Summary
#### (1) Show that for \( x > 0 \), \( \frac{x}{x+1} < \ln(x+1) < x \) holds.
To show the inequality:
- **Upper Bound**: For \( x > 0 \), using the inequality \( \ln(1+x) \leq x \), we have \( \ln(x+1) < x \).
- **Lower Bound**: Define \( f(x) = \ln(x+1) - \frac{x}{x+1} \). Compute the derivative \( f'(x)... | **(1)** Show that for \( x > 0 \), \(\frac{x}{x+1} < \ln(x+1) < x\) holds.
1. Define the functions \( f(x) = x - \ln(x+1) \) and \( g(x) = \ln(x+1) - \frac{x}{x+1} \) for \( x > 0 \).
2. Evaluate \( f(0) \) and \( g(0) \):
\[
f(0) = 0 - \ln(1) = 0
\]
\[
g(0) = \ln(1) - \frac{0}{1} = 0
\]
3. Compute... |
Let $X$ be a set with $n$ elements and $0 \le k \le n$ . Let $a_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at least $k$ common components (where a common component of $f$ and g is an $x \in X$ such that $f(x) = g(x)$ ). Let $b_{n,k}$ be the maximum num... | Okay, let's try to tackle part (a) first. The problem states that for a set X with n elements, and integers 0 ≤ k ≤ n, we need to show that the product of a_{n,k} and b_{n,k-1} is at most n!. Here, a_{n,k} is the maximum number of permutations where every two permutations have at least k common components, and b_{n,k-1... | To solve part (a), we need to show that \(a_{n,k} \cdot b_{n,k-1} \le n!\). We use a contradiction argument based on the pigeonhole principle. Suppose \(|A| \cdot |B| > n!\). Then, there exist two distinct pairs \((f_1, g_1)\) and \((f_2, g_2)\) such that \(f_1^{-1} g_1 = f_2^{-1} g_2\). This implies \(f_2^{-1} f_1 = g... | ### Part (a)
1. **Define the sets and their properties:**
Let \( f_k(n) \) denote a set of permutations of \(\{1, 2, \cdots, n\}\) such that any two permutations share at least \( k \) elements. Let \( g_k(n) \) denote a set of permutations of \(\{1, 2, \cdots, n\}\) such that no two permutations share more than \( ... |
For any integer $n\ge 2$ and two $n\times n$ matrices with real
entries $A,\; B$ that satisfy the equation $$ A^{-1}+B^{-1}=(A+B)^{-1}\; $$ prove that $\det (A)=\det(B)$ .
Does the same conclusion follow for matrices with complex entries?
(Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)
| Alright, so I need to prove that for any integer n ≥ 2 and two n×n real matrices A and B that satisfy A⁻¹ + B⁻¹ = (A + B)⁻¹, the determinants of A and B are equal. Then, I also need to check whether the same conclusion holds for complex matrices. Hmm, okay. Let me start by understanding the given equation and seeing wh... | To prove that \(\det(A) = \det(B)\) for real matrices \(A\) and \(B\) satisfying \(A^{-1} + B^{-1} = (A + B)^{-1}\), we proceed as follows:
1. **Manipulate the Given Equation**:
Start by left-multiplying both sides by \(A + B\):
\[
(A + B)(A^{-1} + B^{-1}) = I.
\]
Expanding the left-hand side:
\[
... | 1. Given the equation \( A^{-1} + B^{-1} = (A + B)^{-1} \), we start by multiplying both sides by \( A \) from the left and \( B \) from the right:
\[
A(A^{-1} + B^{-1})B = A(A + B)^{-1}B
\]
2. Simplifying the left-hand side:
\[
A A^{-1} B + A B^{-1} B = A (A + B)^{-1} B
\]
\[
B + A = A (A + B)^... |
Given that six-digit positive integer $\overline{ABCDEF}$ has distinct digits $A,$ $B,$ $C,$ $D,$ $E,$ $F$ between $1$ and $8$ , inclusive, and that it is divisible by $99$ , find the maximum possible value of $\overline{ABCDEF}$ .
*Proposed by Andrew Milas* | Okay, so I need to find the largest six-digit number made up of distinct digits from 1 to 8 that's divisible by 99. Let me start by recalling that a number divisible by 99 must be divisible by both 9 and 11. Since 9 and 11 are co-prime, their least common multiple is 99. That means the number has to satisfy the divisib... | To find the maximum six-digit number $\overline{ABCDEF}$ with distinct digits from 1 to 8 that is divisible by 99, we need to satisfy both the divisibility rules for 9 and 11.
1. **Divisibility by 9**: The sum of the digits must be 27 (the only multiple of 9 within the possible range of sums).
2. **Divisibility by 11... | To solve the problem, we need to find the maximum six-digit number $\overline{ABCDEF}$ with distinct digits between 1 and 8 that is divisible by 99. A number is divisible by 99 if and only if it is divisible by both 9 and 11.
1. **Divisibility by 9**:
A number is divisible by 9 if the sum of its digits is divisible... |
Find the minimal positive integer $m$ , so that there exist positive integers $n>k>1$ , which satisfy $11...1=11...1.m$ , where the first number has $n$ digits $1$ , and the second has $k$ digits $1$ . | Alright, let's try to figure out this problem. We need to find the minimal positive integer m such that there exist positive integers n > k > 1 where a number consisting of n ones is equal to a number consisting of k ones multiplied by m. So, in other words, we have a repunit (a number made up entirely of 1s) with n di... | To find the minimal positive integer \( m \) such that there exist positive integers \( n > k > 1 \) where a number consisting of \( n \) ones (denoted as \( R_n \)) is equal to a number consisting of \( k \) ones (denoted as \( R_k \)) multiplied by \( m \), we need to explore the properties of repunits.
Repunits ca... | 1. We start with the given equation \( 11\ldots1 = 11\ldots1 \cdot m \), where the first number has \( n \) digits of 1, and the second has \( k \) digits of 1. This can be written as:
\[
\frac{10^n - 1}{9} = \frac{10^k - 1}{9} \cdot m
\]
Simplifying, we get:
\[
m = \frac{10^n - 1}{10^k - 1}
\]
2.... |
The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$ th page of this question, you write the fraction $i/p$ to denote that this is the $i$ th page out of $p$ for this question. When you turned in your submissions f... | Okay, so I need to solve this USAMO problem where the total sum of fractions from all pages turned in equals 2017, and I need to find the total number of pages. Let me try to break this down step by step.
First, let me understand the problem again. For each question, you submit a positive integer number of pages, p. O... | For each question, if you submit \( p \) pages, the sum of the fractions written on each page of that question is \(\frac{1}{p} + \frac{2}{p} + \cdots + \frac{p}{p}\). This sum simplifies to \(\frac{1 + 2 + \cdots + p}{p} = \frac{p(p+1)/2}{p} = \frac{p+1}{2}\).
Given that there are six questions, the total sum of the... | 1. Let \( a_k \) be the number of pages submitted for the \( k \)-th question, where \( k = 1, 2, \ldots, 6 \).
2. For each question, the sum of the fractions written on the pages is given by:
\[
\sum_{i=1}^{a_k} \frac{i}{a_k}
\]
3. We can simplify this sum as follows:
\[
\sum_{i=1}^{a_k} \frac{i}{a_k} =... |
Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$ , for which the following statement is true:
If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function
and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that
$f(x_{0})=f(x_{1})=\ldots =f(x_{n}... | Okay, so I need to find all sequences \( a_0, a_1, \ldots, a_n \) of real numbers where \( a_n \neq 0 \), such that if we have any \( n \)-times differentiable function \( f: \mathbb{R} \to \mathbb{R} \) with \( n+1 \) zeros at points \( x_0 < x_1 < \ldots < x_n \), then there exists some \( h \in (x_0, x_n) \) where t... | To solve the problem, we need to find all sequences \(a_0, a_1, \ldots, a_n\) of real numbers such that \(a_n \neq 0\), and for any \(n\)-times differentiable function \(f\) with \(n+1\) zeros, there exists some \(h \in (x_0, x_n)\) where the linear combination \(a_0 f(h) + a_1 f'(h) + \ldots + a_n f^{(n)}(h) = 0\).
1... | 1. **Understanding the problem**: We need to find sequences \(a_0, a_1, \ldots, a_n\) of real numbers such that \(a_n \neq 0\) and for any \(n\)-times differentiable function \(f\) with \(f(x_0) = f(x_1) = \ldots = f(x_n) = 0\), there exists \(h \in (x_0, x_n)\) such that
\[
a_0 f(h) + a_1 f'(h) + \ldots + a_n f... |
The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$ . Determine the value of $k$ . | Okay, let's see. I need to find the value of k in the quartic equation x⁴ - 18x³ + kx² + 200x - 1984 = 0, given that the product of two of its roots is -32. Hmm, quartic equations have four roots, right? So maybe I can use Vieta's formulas here. Vieta's relates the coefficients of a polynomial to sums and products of i... | Given the quartic equation \( x^4 - 18x^3 + kx^2 + 200x - 1984 = 0 \) with the product of two of its roots being \(-32\), we need to determine the value of \( k \).
Using Vieta's formulas, we know:
- The sum of the roots \( r_1 + r_2 + r_3 + r_4 = 18 \).
- The product of all roots \( r_1 r_2 r_3 r_4 = -1984 \).
- The ... | 1. Let \( a, b, c, \) and \( d \) be the roots of the quartic equation \( x^4 - 18x^3 + kx^2 + 200x - 1984 = 0 \).
2. By Vieta's formulas, we have the following relationships:
\[
a + b + c + d = 18
\]
\[
ab + ac + ad + bc + bd + cd = k
\]
\[
abc + abd + acd + bcd = -200
\]
\[
abcd = -1... |
What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6? | Okay, so I need to figure out the maximum number of positive integers less than or equal to 2016 such that no two of them differ by 1, 2, or 6. Hmm, that sounds like a problem about selecting numbers with certain differences forbidden. Let me try to think through this step by step.
First, the problem is asking for the... | To determine the greatest number of positive integers less than or equal to 2016 such that no two of them differ by 1, 2, or 6, we can use a systematic approach involving modular arithmetic and block analysis.
1. **Forbidden Differences Analysis**:
- The problem requires selecting numbers such that no two numbers d... | 1. **Constructing the Sequence:**
- We need to construct a sequence of integers such that no two integers differ by 1, 2, or 6.
- Consider the sequence: \(1, 4, 8, 11, 15, 18, 22, 25, \ldots, 2013\).
- The difference between consecutive integers alternates between 3 and 4.
- The \(2n\)-th integer in this se... |
Let $n$ be a positive integer with $k\ge22$ divisors $1=d_{1}< d_{2}< \cdots < d_{k}=n$ , all different. Determine all $n$ such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\] | Okay, let me try to figure out this problem step by step. The problem says that for a positive integer n with at least 22 divisors, ordered as 1 = d₁ < d₂ < ... < d_k = n, we need to find all n such that d₇² + d₁₀² = (n / d₂₂)². Hmm, okay. Let me start by understanding what each part means.
First, n has k ≥ 22 divisor... | To determine all positive integers \( n \) with at least 22 divisors such that \( d_7^2 + d_{10}^2 = \left( \frac{n}{d_{22}} \right)^2 \), we analyze the structure of the divisors and the given equation.
1. **Divisor Pairing**: For any divisor \( d_i \) of \( n \), \( \frac{n}{d_i} \) is also a divisor. Thus, \( d_{2... | 1. Let \( n \) be a positive integer with \( k \ge 22 \) divisors \( 1 = d_1 < d_2 < \cdots < d_k = n \), all different. We need to determine all \( n \) such that
\[
d_7^2 + d_10^2 = \left( \frac{n}{d_{22}} \right)^2.
\]
2. Note that \( d_{22} = d_{k+1-22} \). Since \( k \ge 22 \), we have \( d_{22} = d_{... |
For $f_n(x)=x^n$ and $a\neq 1$ consider
I. $(f_{11}(a)f_{13}(a))^{14}$ II. $f_{11}(a)f_{13}(a)f_{14}(a)$ III. $(f_{11}(f_{13}(a)))^{14}$ IV. $f_{11}(f_{13}(f_{14}(a)))$ Which of these equal $f_{2002}(a)$ ? $\textbf{(A) }\text{I and II only}\qquad\textbf{(B) }\text{II and III only}\qquad\textbf{(C) }\text{III... | Okay, so I have this problem here where I need to figure out which of the four given expressions equal f_{2002}(a), where f_n(x) = x^n. The options are I, II, III, IV, and the answer choices are combinations of these. Let me try to break this down step by step.
First, let me recall that f_n(a) is just a^n. So, f_{2002... | To determine which expressions equal \( f_{2002}(a) \) where \( f_n(x) = x^n \) and \( a \neq 1 \), we analyze each expression step by step:
1. **Expression I: \( (f_{11}(a) f_{13}(a))^{14} \)**
- \( f_{11}(a) = a^{11} \)
- \( f_{13}(a) = a^{13} \)
- Multiplying these gives \( a^{11} \cdot a^{13} = a^{24} \)
... | To determine which of the given expressions equal \( f_{2002}(a) \), we need to simplify each expression using the definition \( f_n(x) = x^n \).
1. **Expression I: \( (f_{11}(a)f_{13}(a))^{14} \)**
\[
f_{11}(a) = a^{11} \quad \text{and} \quad f_{13}(a) = a^{13}
\]
\[
f_{11}(a)f_{13}(a) = a^{11} \cdot a... |
Decipher the equality $(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S} $ where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from $0$ . | Okay, let's tackle this problem step by step. So, we have this equation: (LARN - ACA) : (CYP + RUS) = C^{Y^P} * R^{U^S}. Each letter represents a different digit, and none of them can be zero. Hmm, that's quite a complex equation with both arithmetic and exponentiation involved. Let me break it down.
First, let's pars... | To solve the given equation \((\overline{LARN} - \overline{ACA}) : (\overline{CYP} + \overline{RUS}) = C^{Y^P} \cdot R^{U^S}\) where different symbols correspond to different digits and all digits are non-zero, we need to consider the following steps:
1. **Identify the structure of the numbers**:
- \(LARN\) is a 4-... | To decipher the given equality, we need to analyze the constraints and the possible values for the digits. The equality is:
\[
(\overline{LARN} - \overline{ACA}) : (\overline{CYP} + \overline{RUS}) = C^{Y^P} \cdot R^{U^S}
\]
where different symbols correspond to different digits, and equal symbols correspond to equal... |
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$ , which are not less than $k$ , there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$ , such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots. | Okay, so I have this problem where I need to find the smallest positive real number k such that for any four different real numbers a, b, c, d that are all at least k, there exists a permutation (p, q, r, s) of (a, b, c, d) where the equation (x² + px + q)(x² + rx + s) = 0 has four different real roots. Hmm, interestin... | To find the smallest positive real number \( k \) such that for any four different real numbers \( a, b, c, d \geq k \), there exists a permutation \( (p, q, r, s) \) of \( (a, b, c, d) \) such that the equation \( (x^2 + px + q)(x^2 + rx + s) = 0 \) has four different real roots, we need to ensure that each quadratic ... | To find the smallest positive real \( k \) such that for any four different real numbers \( a, b, c, d \) which are not less than \( k \), there exists a permutation \((p, q, r, s)\) of \((a, b, c, d)\) such that the equation \((x^2 + px + q)(x^2 + rx + s) = 0\) has four different real roots, we need to ensure that bot... |
If $a$ and $b$ are positive integers such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$ , what is the greatest possible value of $a+b$ ? | Okay, so I need to find the greatest possible value of a + b where a and b are positive integers that satisfy the equation 1/a + 1/b = 1/9. Hmm, let me start by understanding what this equation means.
First, I remember that when you have equations with reciprocals like this, it might be helpful to combine the fractio... | Given the equation \(\frac{1}{a} + \frac{1}{b} = \frac{1}{9}\) where \(a\) and \(b\) are positive integers, we start by combining the fractions:
\[
\frac{a + b}{ab} = \frac{1}{9}
\]
Cross-multiplying gives:
\[
9(a + b) = ab
\]
Rearranging terms, we get:
\[
ab - 9a - 9b = 0
\]
Using Simon's Favorite Factoring Tric... | 1. Start with the given equation:
\[
\frac{1}{a} + \frac{1}{b} = \frac{1}{9}
\]
2. Combine the fractions on the left-hand side:
\[
\frac{a + b}{ab} = \frac{1}{9}
\]
3. Cross-multiply to eliminate the fractions:
\[
9(a + b) = ab
\]
4. Rearrange the equation to set it to zero:
\[
ab - ... |
A $6\text{-year stock}$ that goes up $30\%$ in the first year, down $30\%$ in the second, up $30\%$ in the third, down $30\%$ in the fourth, up $30\%$ in the fifth, and down $30\%$ in the sixth is equivalent to a $3\text{-year stock}$ that loses $x\%$ in each of its three years. Compute $x$ .
| Okay, so I need to figure out what the equivalent three-year loss percentage x would be for a stock that has these up and down movements over six years compared to another stock that loses x% each year for three years. Let me try to break this down step by step.
First, let me understand the problem. The six-year stock... | To determine the equivalent annual loss percentage \( x \) for a 3-year stock that results in the same final value as a 6-year stock with alternating 30% gains and 30% losses, we need to compare the compounded returns of both scenarios.
1. **Calculate the final value of the 6-year stock:**
- The stock alternates be... | 1. Let the initial price of the stock be \( a \).
2. Calculate the price of the stock at the end of each year:
- After the first year, the stock goes up by 30%:
\[
a \times 1.3
\]
- After the second year, the stock goes down by 30%:
\[
a \times 1.3 \times 0.7 = a \times 0.91
\]
-... |
A circle is inscribed in a triangle $ABC$ with sides $a,b,c$ . Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$ . In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in ... | Okay, so I need to find the sum of the areas of four inscribed circles in a specific geometric configuration. Let me try to visualize and understand the problem first.
We have a triangle ABC with sides a, b, c. A circle is inscribed in this triangle—that's the incircle, which touches each side of the triangle. Then, ... | To find the sum of the areas of all four inscribed circles (the original incircle and three smaller incircles) in terms of the sides \(a, b, c\) of triangle \(ABC\), we proceed as follows:
1. **Inradius of the original triangle**:
- The semi-perimeter \(s = \frac{a + b + c}{2}\).
- The area \(A\) of the triangle... | 1. Let \((I, r)\) be the inscribed circle of \(\triangle ABC\) with radius \(r\).
2. The tangent parallel to the side \(BC\) intersects the sides \(AB\) and \(AC\) at the points \(B_1\) and \(C_1\), forming the triangle \(T_1 = \triangle AB_1C_1\).
3. Since \(B_1C_1 \parallel BC\), we have \(\triangle AB_1C_1 \sim \tri... |
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$ s that may occur among the $100$ numbers. | Okay, so I need to figure out the minimum number of 1s among 100 positive integers where their sum equals their product. Hmm, let's start by understanding the problem better.
First, we have 100 positive integers. Let's denote them as \( a_1, a_2, \ldots, a_{100} \). The condition given is that the sum of these numbers... | Given 100 positive integers whose sum equals their product, we need to determine the minimum number of 1s among these integers.
We denote the integers as \(a_1, a_2, \ldots, a_{100}\) and require that their sum equals their product:
\[
a_1 + a_2 + \cdots + a_{100} = a_1 \times a_2 \times \cdots \times a_{100}
\]
To ... | To solve the problem, we need to find the minimum number of 1s among 100 positive integers such that their sum equals their product. Let's denote these integers as \(a_1, a_2, \ldots, a_{100}\).
1. **Initial Setup:**
We are given that the sum of these integers equals their product:
\[
a_1 + a_2 + \cdots + a_{... |
Compute the remainder when $99989796\ldots 121110090807 \ldots 01$ is divided by $010203 \ldots 091011 \ldots 9798$ (note that the first one starts at $99$ , and the second one ends at $98$ ).
*2020 CCA Math Bonanza Team Round #7* | Okay, so I have this problem where I need to compute the remainder when a very large number is divided by another large number. The numbers are constructed by concatenating sequences of digits. Let me try to parse the problem correctly.
The first number is described as "99989796…121110090807…01". It starts at 99 and g... | To solve the problem of finding the remainder when the number \( N = 99989796\ldots121110090807\ldots01 \) is divided by \( M = 010203\ldots091011\ldots9798 \), we need to consider the structure of these numbers and use modular arithmetic properties.
1. **Understanding the Numbers**:
- \( N \) is a 198-digit number... | 1. **Define the numbers:**
Let \( N \) be the second number, which is the concatenation of numbers from 01 to 98. We need to compute the remainder when the first number, which is the concatenation of numbers from 99 to 01, is divided by \( N \).
2. **Express \( N \) in a mathematical form:**
\[
N = 010203 \ld... |
Let $ABC$ be a triangle in a plane $\pi$ . Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$ , $BCP$ , $CAP$ have the same radii. | Okay, so I have this geometry problem here: Given a triangle ABC in a plane π, find all points P (not equal to A, B, or C) such that the circumcircles of triangles ABP, BCP, and CAP all have the same radii. Hmm, interesting. Let me try to figure this out step by step.
First, I need to recall that the circumradius of a... | To find the set of all points \( P \) (distinct from \( A, B, C \)) in the plane \( \pi \) such that the circumcircles of triangles \( ABP \), \( BCP \), and \( CAP \) have the same radii, we start by considering the circumradius formula for a triangle. The circumradius \( R \) of a triangle with sides \( a, b, c \) an... | To solve the problem, we need to find all points \( P \) in the plane \( \pi \) such that the circumcircles of triangles \( ABP \), \( BCP \), and \( CAP \) have the same radii. We will denote the angles \( \angle BPC \), \( \angle CPA \), and \( \angle APB \) as \( \alpha \), \( \beta \), and \( \gamma \), respectivel... |
Determine the smallest positive prime $p$ which satisfies the congruence \[p+p^{-1}\equiv 25\pmod{143}.\] Here, $p^{-1}$ as usual denotes multiplicative inverse. | Okay, so I need to find the smallest positive prime number p such that when you add p and its multiplicative inverse modulo 143, the result is congruent to 25 modulo 143. Hmm, let's start by understanding what exactly this means.
First, let's recall that a multiplicative inverse of p modulo 143 is a number p⁻¹ such t... | To determine the smallest positive prime \( p \) which satisfies the congruence \( p + p^{-1} \equiv 25 \pmod{143} \), we start by transforming the given congruence into a quadratic form. Multiplying both sides by \( p \), we get:
\[
p^2 - 25p + 1 \equiv 0 \pmod{143}
\]
We solve this quadratic congruence modulo 11 an... | 1. Given the congruence \( p + p^{-1} \equiv 25 \pmod{143} \), we start by noting that \( 143 = 11 \times 13 \). We can use the Chinese Remainder Theorem to break this down into two separate congruences:
\[
p + p^{-1} \equiv 25 \pmod{11} \quad \text{and} \quad p + p^{-1} \equiv 25 \pmod{13}
\]
2. Simplify eac... |
Let $ABCDEF$ be a regular hexagon with side length two. Extend $FE$ and $BD$ to meet at $G$ . Compute the area of $ABGF$ . | Alright, so I have this problem: Given a regular hexagon ABCDEF with side length 2, I need to extend FE and BD to meet at point G, and then compute the area of quadrilateral ABGF. Hmm, okay. Let me try to visualize this first.
First, let me recall that in a regular hexagon, all sides are equal, and all internal angles... | To solve the problem of finding the area of quadrilateral \(ABGF\) in a regular hexagon \(ABCDEF\) with side length 2, we start by assigning coordinates to the vertices of the hexagon. The coordinates are as follows:
- \(A(2, 0)\)
- \(B(1, \sqrt{3})\)
- \(C(-1, \sqrt{3})\)
- \(D(-2, 0)\)
- \(E(-1, -\sqrt{3})\)
- \(F(1... | 1. **Identify the Geometry and Angles:**
- Given a regular hexagon $ABCDEF$ with side length 2.
- Extend $FE$ and $BD$ to meet at $G$.
- In a regular hexagon, each internal angle is $120^\circ$.
- $\angle FED = 120^\circ$ and $\angle DEG = 60^\circ$ (since $FE$ and $BD$ are extended lines).
2. **Determine ... |
Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap? | Okay, so I need to find the circumference of the region where two overlapping disks of radius 1 intersect. Each disk's circumference passes through the other's center. Let me visualize this first. Both disks have the same radius, and each center is on the circumference of the other. That means the distance between the ... | To find the circumference of the region where two overlapping disks of radius 1 intersect, we start by noting that the distance between their centers is 1 unit. Each disk's circumference passes through the other's center, forming an equilateral triangle with the centers and one of the intersection points.
1. **Equilat... | 1. **Understanding the Geometry**:
- Each disk has a radius of 1.
- The circumference of each disk passes through the center of the other disk.
- This implies that the distance between the centers of the two disks is equal to the radius, which is 1.
2. **Forming the Equilateral Triangle**:
- The distance b... |
Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$ . Prove that the sum of the numbers in the four corners of the square is also $k$ .
| Okay, so I need to prove that in a 4x4 magic square where each row, column, and main diagonal sums to k, the sum of the four corner numbers is also k. Hmm, let's start by recalling what a magic square is. A magic square is a grid where the numbers in each row, each column, and both main diagonals all add up to the same... | To prove that the sum of the numbers in the four corners of a 4x4 magic square is equal to the magic constant \( k \), we start by denoting the four corners as \( a, b, c, d \). We use the properties of the magic square, where each row, column, and main diagonal sums to \( k \).
1. **Sum of the main diagonals**:
- ... | 1. **Define the Magic Square and its Properties:**
A magic square of side \(4\) is a \(4 \times 4\) grid where the sum of the numbers in each row, column, and main diagonal is equal to \(k\). Let the elements of the magic square be denoted as \(a_{ij}\) where \(i, j \in \{1, 2, 3, 4\}\).
2. **Label the Corners:**
... |
$ABC$ is a triangle
$E, F$ are points in $AB$ , such that $AE = EF = FB$
$D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$
$AD$ is perpendicular to $CF$ .
The angle CFA is the triple of angle BDF. ( $3\angle BDF = \angle CFA$ )
Determine the ratio $\frac{DB}{DC}$ .
%E... | Okay, so I need to find the ratio DB/DC in triangle ABC where points E and F divide AB into three equal parts (AE = EF = FB). D is a point on BC such that ED is perpendicular to BC, and AD is perpendicular to CF. Also, the angle CFA is three times the angle BDF. Hmm, this seems a bit complex, but let's break it down st... | Given triangle \(ABC\) with points \(E\) and \(F\) on \(AB\) such that \(AE = EF = FB\), and point \(D\) on \(BC\) such that \(ED\) is perpendicular to \(BC\) and \(AD\) is perpendicular to \(CF\). Additionally, \(\angle CFA = 3\angle BDF\). We need to determine the ratio \(\frac{DB}{DC}\).
1. **Coordinate System Setu... | 1. **Given Information and Initial Setup:**
- Triangle \(ABC\) with points \(E\) and \(F\) on \(AB\) such that \(AE = EF = FB\).
- Point \(D\) on line \(BC\) such that \(ED \perp BC\).
- \(AD \perp CF\).
- \(\angle CFA = 3 \angle BDF\).
2. **Angle Relationships:**
- Let \(\angle BDF = x\). Then \(\angle... |
Clarabelle wants to travel from $(0,0)$ to $(6,2)$ in the coordinate plane. She is able to move in one-unit steps up, down, or right, must stay between $y=0$ and $y=2$ (inclusive), and is not allowed to visit the same point twice. How many paths can she take?
*Proposed by Connor Gordon* | Okay, let's try to figure out how many paths Clarabelle can take from (0,0) to (6,2) with the given constraints. Hmm, so she can move up, down, or right, but she can't go below y=0 or above y=2. Also, she can't visit the same point more than once. That means her path has to be a self-avoiding walk within those y-bounda... | Clarabelle wants to travel from \((0,0)\) to \((6,2)\) in the coordinate plane, moving in one-unit steps up, down, or right, while staying between \(y=0\) and \(y=2\) and not revisiting any point. The key steps to solve this problem are as follows:
1. **Movement Constraints**: Clarabelle can move right, up, or down, b... | 1. Clarabelle needs to travel from \((0,0)\) to \((6,2)\) in the coordinate plane. She can move up, down, or right, but must stay within the bounds \(0 \leq y \leq 2\) and cannot visit the same point twice.
2. For each \(i \in \{0, 1, \ldots, 5\}\), Clarabelle has 3 choices for moving from \(x = i\) to \(x = i+1\). Th... |
Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$ .
\[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\]
If necessary, you may use $ \ln 3 \equal{} 1.10$ . | Okay, let me try to figure out the range of k for which the inequality holds for all x between 0 and 1. So, the problem is:
For 0 ≤ x ≤ 1, find the range of k such that
∫₀^x [1 / √(3 + t²)³] dt ≥ k ∫₀^x [1 / √(3 + t²)] dt.
Hmm. Alright, so we need to compare two integrals from 0 to x. The left integral is 1 over (3 ... | To find the range of \( k \) for which the inequality
\[
\int_0^x \frac{dt}{\sqrt{(3 + t^2)^3}} \geq k \int_0^x \frac{dt}{\sqrt{3 + t^2}}
\]
holds for \( 0 \leq x \leq 1 \), we need to compare the two integrals and determine the minimum value of their ratio over the interval \([0, 1]\).
First, we compute the integrals... | To find the range of \( k \) for which the inequality
\[ \int_0^x \frac{dt}{\sqrt{(3 + t^2)^3}} \geq k \int_0^x \frac{dt}{\sqrt{3 + t^2}} \]
holds for \( 0 \leq x \leq 1 \), we start by analyzing the integrands.
1. **Simplify the ratio of integrands:**
\[
\frac{(3 + x^2)^{-3/2}}{(3 + x^2)^{-1/2}} = \frac{1}{3 + ... |
In triangle $ABC$ , $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$ . What is the area of triangle $ABC$ ?
*Proposed by Evan Chen* | Okay, so I need to find the area of triangle ABC given that the product of its sides AB·BC·CA is 1000 and the product of the sines of its angles sin A sin B sin C is 1/1000. Hmm, that seems interesting. Let me try to figure this out step by step.
First, I remember that in a triangle, there are relationships between th... | Given triangle \(ABC\) with \(\sin A \sin B \sin C = \frac{1}{1000}\) and \(AB \cdot BC \cdot CA = 1000\), we need to find the area of the triangle.
1. **Using the Law of Sines**:
The sides \(a\), \(b\), and \(c\) of the triangle can be expressed as \(a = 2R \sin A\), \(b = 2R \sin B\), and \(c = 2R \sin C\), where... | 1. **Given Information:**
- \(\sin A \sin B \sin C = \frac{1}{1000}\)
- \(AB \cdot BC \cdot CA = 1000\)
2. **Using the Extended Law of Sines:**
The extended law of sines states that for any triangle \(ABC\) with circumradius \(R\),
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
\]
... |
Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification. | Alright, let's tackle this series problem. The series given is the sum from n=2 to infinity of log(n³ +1) minus log(n³ -1). The first thing I notice is that this is a telescoping series because it's a difference of logarithms. Maybe I can rewrite it as a single logarithm to see if terms cancel out when summed.
So, log... | The given series is \(\sum_{n=2}^\infty \left[ \log(n^3 + 1) - \log(n^3 - 1) \right]\). To evaluate this, we first combine the logarithms:
\[
\log\left(\frac{n^3 + 1}{n^3 - 1}\right).
\]
Factoring the numerator and denominator using sum and difference of cubes:
\[
\frac{n^3 + 1}{n^3 - 1} = \frac{(n + 1)(n^2 - n + 1)... | 1. We start with the given series:
\[
\sum_{n=2}^\infty \left( \log(n^3 + 1) - \log(n^3 - 1) \right)
\]
Using the properties of logarithms, we can combine the terms inside the logarithm:
\[
\log(n^3 + 1) - \log(n^3 - 1) = \log \left( \frac{n^3 + 1}{n^3 - 1} \right)
\]
Thus, the series becomes:
... |
Given a triangle on the plane, construct inside the triangle the point $P$ for which the centroid of the triangle formed by the three projections of $P$ onto the sides of the triangle happens to be $P$ . | Alright, so I need to figure out how to construct a point P inside a triangle such that the centroid of the triangle formed by the three projections of P onto the sides of the original triangle is P itself. Hmm, that sounds a bit complex, but let me break it down step by step.
First, let me recall what a centroid is. ... | To construct the point \( P \) inside a triangle such that the centroid of the triangle formed by the projections of \( P \) onto the sides is \( P \) itself, follow these steps:
1. **Understanding Projections**: For each side of the triangle, the projection of \( P \) is the foot of the perpendicular from \( P \) to ... | 1. **Define the Problem and Setup:**
Let \( \triangle ABC \) be a given triangle. We need to find a point \( P \) inside the triangle such that the centroid of the triangle formed by the projections of \( P \) onto the sides of \( \triangle ABC \) is \( P \) itself.
2. **Projections and Midpoints:**
Let \( P_A, ... |
Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$ , $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$ . | Okay, let's try to tackle this problem. So, the question is asking for the maximum value of the sum of the squares of the sines of 2007 variables, each non-negative, and their total sum is π. Then, once we find that maximum value expressed as a reduced fraction a/b, we need to find a + b. Alright, let's start breaking ... | To find the maximum value of \(\sin^2 x_1 + \sin^2 x_2 + \cdots + \sin^2 x_{2007}\) where \(x_1 + x_2 + \cdots + x_{2007} = \pi\) and each \(x_i\) is a nonnegative real number, we need to consider the optimal distribution of the angles \(x_i\) to maximize the sum of their squares of sines.
1. **Key Insight**: The func... | To find the maximum possible value of \(\sin^2 x_1 + \sin^2 x_2 + \sin^2 x_3 + \cdots + \sin^2 x_{2007}\) given that \(x_1 + x_2 + x_3 + \cdots + x_{2007} = \pi\), we will use the properties of the sine function and some optimization techniques.
1. **Initial Assumptions and Simplifications**:
- We are given that \(... |
Let $ABC$ be a triangle with $\angle A=\tfrac{135}{2}^\circ$ and $\overline{BC}=15$ . Square $WXYZ$ is drawn inside $ABC$ such that $W$ is on $AB$ , $X$ is on $AC$ , $Z$ is on $BC$ , and triangle $ZBW$ is similar to triangle $ABC$ , but $WZ$ is not parallel to $AC$ . Over all possible triangles ... | Okay, so I need to find the maximum area of square WXYZ inside triangle ABC with angle A being 135/2 degrees, which is 67.5 degrees, and BC = 15. The square is positioned such that point W is on AB, X is on AC, Z is on BC, and triangle ZBW is similar to triangle ABC. Also, WZ isn't parallel to AC. Hmm, this seems a bit... | Given triangle \(ABC\) with \(\angle A = \frac{135}{2}^\circ = 67.5^\circ\) and \(\overline{BC} = 15\), we need to find the maximum area of the square \(WXYZ\) inscribed inside \(ABC\) such that \(W\) is on \(AB\), \(X\) is on \(AC\), \(Z\) is on \(BC\), and \(\triangle ZBW \sim \triangle ABC\) with \(WZ\) not parallel... | 1. **Given Information and Initial Setup:**
- We are given a triangle \(ABC\) with \(\angle A = \frac{135}{2}^\circ\) and \(\overline{BC} = 15\).
- A square \(WXYZ\) is drawn inside \(ABC\) such that \(W\) is on \(AB\), \(X\) is on \(AC\), \(Z\) is on \(BC\), and triangle \(ZBW\) is similar to triangle \(ABC\).
... |
Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones? | Okay, so I have this problem here: I need to figure out the maximum number of checkers I can place on an 8x8 chessboard, with each row and each column containing twice as many white checkers as black ones. Hmm, let's start breaking this down.
First, let's parse the problem statement again to make sure I understand it ... | To determine the maximum number of checkers that can be placed on an 8x8 chessboard such that each row and each column contains twice as many white checkers as black ones, we start by noting the ratio constraint: the number of white checkers (W) must be twice the number of black checkers (B) in each row and column. Thi... | 1. **Understanding the Problem:**
We need to place checkers on an $8 \times 8$ chessboard such that each row and each column contains twice as many white checkers as black ones.
2. **Constraints:**
- Each row and each column can have at most 6 checkers.
- The ratio of white to black checkers in each row and ... |
Set $P$ as a polynomial function by $p_n(x)=\sum_{k=0}^{n-1} x^k$ .
a) Prove that for $m,n\in N$ , when dividing $p_n(x)$ by $p_m(x)$ , the remainder is $$ p_i(x),\forall i=0,1,...,m-1. $$ b) Find all the positive integers $i,j,k$ that make $$ p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x). $$ *(square1zoa)* | Okay, so I have this problem about polynomial functions, specifically about these polynomials defined as p_n(x) = sum_{k=0}^{n-1} x^k. There are two parts: part a asks to prove that when dividing p_n(x) by p_m(x), the remainder is p_i(x) for some i between 0 and m-1. Part b is to find all positive integers i, j, k such... | a) To prove that when dividing \( p_n(x) \) by \( p_m(x) \), the remainder is \( p_i(x) \) for some \( i \in \{0, 1, \ldots, m-1\} \):
1. Recall that \( p_n(x) = \sum_{k=0}^{n-1} x^k \) and \( p_m(x) = \sum_{k=0}^{m-1} x^k \).
2. These polynomials can be expressed as geometric series: \( p_n(x) = \frac{x^n - 1}{x - 1}... | ### Part (a)
1. **Define the Polynomial**:
The polynomial \( p_n(x) \) is given by:
\[
p_n(x) = \sum_{k=0}^{n-1} x^k = 1 + x + x^2 + \cdots + x^{n-1}
\]
2. **Division of Polynomials**:
We need to divide \( p_n(x) \) by \( p_m(x) \) and find the remainder. Let's denote the quotient by \( q(x) \) and the... |
A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$ | Okay, so I need to prove that in this cyclic quadrilateral BCDE, where there's a circle centered on side ED that touches the other three sides, EB + CD equals ED. Hmm, cyclic quadrilaterals... properties... tangent circles... Let me start by drawing a diagram to visualize this.
First, cyclic quadrilateral BCDE means a... | To prove that \( EB + CD = ED \) in the cyclic quadrilateral \( BCDE \) with a circle centered on \( ED \) and tangent to the other three sides, we proceed as follows:
**Step 1: Identify Tangent Segments**
- Let the circle touch \( EB \) at \( P \), \( BC \) at \( Q \), and \( CD \) at \( R \).
- By the tangent segmen... | 1. Let \( M \) be the center of the incircle of the cyclic quadrilateral \( BCDE \). This circle touches the sides \( BC \), \( CD \), and \( EB \) at points \( P \), \( Q \), and \( R \) respectively.
2. Since the circle is tangent to \( BC \), \( CD \), and \( EB \), we have the following equal tangents from a point ... |
Problem:
Find all polynomials satisfying the equation $ f(x^2) = (f(x))^2 $ for all real numbers x.
I'm not exactly sure where to start though it doesn't look too difficult. Thanks! | Okay, let's try to figure out this problem: find all polynomials satisfying the equation \( f(x^2) = (f(x))^2 \) for all real numbers x. Hmm, the user mentioned they're not sure where to start, so maybe I should start by recalling some basic properties of polynomials and functional equations.
First, let's think about ... | To find all polynomials satisfying the equation \( f(x^2) = (f(x))^2 \) for all real numbers \( x \), we start by analyzing the degrees and leading coefficients of the polynomials.
1. **Degree Analysis**:
- If \( f \) is a polynomial of degree \( n \), then \( f(x^2) \) is a polynomial of degree \( 2n \).
- The... | 1. Let the polynomial be \( f(x) = a_m x^{k_m} + a_{m-1} x^{k_{m-1}} + \cdots + a_1 x^{k_1} \) where all of the \( a_i \) are nonzero and \( k_1 < k_2 < \cdots < k_m \).
2. We need to satisfy the equation \( f(x^2) = (f(x))^2 \) for all real numbers \( x \).
3. Compute \( f(x^2) \):
\[
f(x^2) = a_m (x^2)^{k_m} ... |
For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$ ? | Okay, let me try to tackle this problem. The question is asking for which integers n ≥ 2, we can arrange the numbers 1 to 16 in a 4x4 grid such that the 8 sums of the rows and columns are all different and each divisible by n.
First, let me make sure I understand the problem correctly. We need to place numbers 1 thro... | The integers from 1 to 16 can be arranged in a 4×4 grid such that all row and column sums are distinct and divisible by **n = 2**. Here's the systematic reasoning:
1. **Total Sum Analysis**:
The total sum of numbers 1 to 16 is 136. For all row and column sums to be divisible by **n**, **n** must divide 136 (since... | 1. **Define the sums of rows and columns:**
For \( k \in \{1,2,3,4\} = S \), let \( R_k \) and \( C_k \) be the sum of the four numbers in row \( k \) and column \( k \) respectively. According to the given information, there exist 8 distinct positive integers \( x_1, x_2, \ldots, x_8 \) such that \( R_k = nx_k \) a... |
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
\[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$ . | Okay, so I need to find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the inequality:
\[
f(a^2) - f(b^2) \leq (f(a) + b)(a - f(b))
\]
for all real numbers \( a \) and \( b \). Hmm, functional equations can be tricky, especially inequalities. Let me try to break this down step by step.
First, maybe I c... | To determine all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the inequality
\[
f(a^2) - f(b^2) \leq (f(a) + b)(a - f(b))
\]
for all \( a, b \in \mathbb{R} \), we proceed as follows:
1. **Case \( a = 0 \)**:
Substituting \( a = 0 \) gives:
\[
f(0) - f(b^2) \leq -f(0)f(b) - b f(b)
\]
Rearrang... | 1. **Define the assertion**: Let \( P(a, b) \) be the given assertion:
\[
P(a, b): f(a^2) - f(b^2) \leq (f(a) + b)(a - f(b))
\]
2. **Evaluate \( P(0, 0) \)**:
\[
P(0, 0): f(0^2) - f(0^2) \leq (f(0) + 0)(0 - f(0)) \implies 0 \leq 0
\]
This is always true, but it doesn't provide new information.
3.... |
Given integers $a_0,a_1, ... , a_{100}$ , satisfying $a_1>a_0$ , $a_1>0$ , and $a_{r+2}=3 a_{r+1}-2a_r$ for $r=0, 1, ... , 98$ . Prove $a_{100}>299$ | Okay, let's see. I need to prove that \( a_{100} > 2^{99} \) given the initial conditions \( a_1 > a_0 \), \( a_1 > 0 \), and the recurrence relation \( a_{r+2} = 3a_{r+1} - 2a_r \) for \( r = 0, 1, \ldots, 98 \). Hmm. Alright, let's start by understanding the problem step by step.
First, the recurrence relation is li... | Given the recurrence relation \(a_{r+2} = 3a_{r+1} - 2a_r\) with initial conditions \(a_1 > a_0\) and \(a_1 > 0\), we first solve the recurrence. The characteristic equation \(r^2 - 3r + 2 = 0\) has roots \(r = 1\) and \(r = 2\), leading to the general solution:
\[
a_n = A + B \cdot 2^n
\]
Using the initial condition... | 1. **Define the sequence and initial conditions:**
Given the sequence \(a_0, a_1, \ldots, a_{100}\) with the recurrence relation:
\[
a_{r+2} = 3a_{r+1} - 2a_r \quad \text{for} \quad r = 0, 1, \ldots, 98
\]
and initial conditions:
\[
a_1 > a_0, \quad a_1 > 0
\]
2. **Propose a general form for \(... |
What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?
| Okay, so I need to find all prime numbers p where 7^p - 6^p + 2 is divisible by 43. That means 7^p - 6^p + 2 ≡ 0 mod 43. So, equivalently, 7^p - 6^p ≡ -2 mod 43. My goal is to find all primes p for which this congruence holds.
First, I remember that when dealing with exponents modulo a prime number (and 43 is a prime)... | To solve the problem of finding all primes \( p \) such that \( 7^p - 6^p + 2 \) is divisible by 43, we need to check when the expression \( 7^p - 6^p + 2 \equiv 0 \pmod{43} \). This simplifies to finding primes \( p \) for which \( 7^p - 6^p \equiv -2 \pmod{43} \).
Using Fermat's Little Theorem, we know that for any ... | 1. **Determine the order of 7 modulo 43:**
The order of an integer \( a \) modulo \( n \) is the smallest positive integer \( d \) such that \( a^d \equiv 1 \pmod{n} \). For \( 7 \mod 43 \), we need to find the smallest \( d \) such that \( 7^d \equiv 1 \pmod{43} \).
By Fermat's Little Theorem, \( 7^{42} \equiv ... |
On $\triangle ABC$ let points $D,E$ on sides $AB,BC$ respectivily such that $AD=DE=EC$ and $AE \ne DC$ . Let $P$ the intersection of lines $AE, DC$ , show that $\angle ABC=60$ if $AP=CP$ . | Okay, so I need to show that angle ABC is 60 degrees given the conditions AD = DE = EC and AP = CP, where P is the intersection of AE and DC. Hmm, let me try to visualize the problem first.
First, let's draw triangle ABC. Points D and E are on sides AB and BC respectively. AD = DE = EC. So, AD is a segment on AB, DE i... | Given triangle \( \triangle ABC \) with points \( D \) and \( E \) on sides \( AB \) and \( BC \) respectively such that \( AD = DE = EC \) and \( AE \neq DC \). Let \( P \) be the intersection of lines \( AE \) and \( DC \). We need to show that \( \angle ABC = 60^\circ \) if \( AP = CP \).
1. **Coordinate Setup and ... | 1. Let $\angle DAE = \angle DEA = a$ and $\angle CAE = \angle PCA = b$.
2. Since $AD = DE = EC$, $\triangle ADE$ and $\triangle DEC$ are isosceles triangles.
3. We know that $\angle APC = 180^\circ - 2b$ because $AP = CP$ implies $\triangle APC$ is isosceles with $\angle APC$ as the vertex angle.
4. Therefore, $\angle... |
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$ | Okay, so I need to find all positive integers n for which there exist positive integers a₁, a₂, ..., aₙ satisfying two conditions:
1. The sum a₁ + 2a₂ + 3a₃ + ... + naₙ equals 6n.
2. The sum (1/a₁) + (2/a₂) + (3/a₃) + ... + (n/aₙ) equals 2 + 1/n.
Hmm, this looks like a problem involving some sort of optimization or b... | To determine all positive integers \( n \) for which there exist positive integers \( a_1, a_2, \ldots, a_n \) satisfying the conditions:
1. \( a_1 + 2a_2 + 3a_3 + \cdots + na_n = 6n \)
2. \( \frac{1}{a_1} + \frac{2}{a_2} + \frac{3}{a_3} + \cdots + \frac{n}{a_n} = 2 + \frac{1}{n} \)
we start by examining small values... | 1. **Applying the Cauchy-Schwarz Inequality:**
We start by applying the Cauchy-Schwarz inequality to the given sums. The Cauchy-Schwarz inequality states that for any sequences of real numbers \(x_i\) and \(y_i\), we have:
\[
\left( \sum_{i=1}^n x_i y_i \right)^2 \leq \left( \sum_{i=1}^n x_i^2 \right) \left( ... |
Theseus starts at the point $(0, 0)$ in the plane. If Theseus is standing at the point $(x, y)$ in the plane, he can step one unit to the north to point $(x, y+1)$ , one unit to the west to point $(x-1, y)$ , one unit to the south to point $(x, y-1)$ , or one unit to the east to point $(x+1, y)$ . After a seque... | Okay, let me try to figure out how to approach this problem. So Theseus starts at (0, 0), takes a step south to (0, -1), and then continues moving in steps of north, south, east, or west. After more than two moves (so the total path is at least three steps, starting from (0,0) to (0,-1) and then some more steps), he en... | To prove that \( |X - Y| = 1 \) for Theseus's path, we analyze the sequence of moves and their contributions to the path's orientation. Here's a step-by-step explanation:
1. **Path Properties**:
- Theseus's path is a self-avoiding closed loop starting and ending at \((0,0)\), with the first step being south to \((0... | To prove that \( |X - Y| = 1 \), we will use induction on the area \( A \) enclosed by the path Theseus takes. We will also consider the direction of the path (clockwise or counterclockwise) to establish the relationship between \( X \) and \( Y \).
1. **Base Case: \( A = 1 \)**
- If the area enclosed by the path i... |
For an integer $n$ , let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$ . Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$ . Find the smallest possible value of $m$ . | Okay, let's try to figure out this problem. So, we need to find the smallest possible value of m such that for all n > m, the function f_9(n) can be expressed as a linear combination of the previous m values of f_9(n - j), with coefficients b_j. Hmm, that sounds like a linear recurrence relation. But f_9(n) counts the ... | To solve the problem, we need to find the smallest possible value of \( m \) such that the function \( f_9(n) \), which counts the number of divisors of \( n \) up to 9, can be expressed as a linear combination of its previous \( m \) values for all \( n > m \).
1. **Understanding \( f_9(n) \)**:
- \( f_9(n) \) is ... | 1. **Define the function \( f_9(n) \)**:
- \( f_9(n) \) is the number of positive integers \( d \leq 9 \) that divide \( n \).
- The divisors of \( n \) that are less than or equal to 9 are \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \).
2. **Generating function for \( f_9(n) \)**:
- We define the generating function \( F(x... |
Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies:
**I.** $a_0 = 1, a_1 = 337$ ;
**II.** $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$ ;
**III.** $\frac{1}{6}(a_n + 1)(2a_n + ... | Okay, so I need to prove that there exists a natural number m such that there's an integer sequence {a_n} satisfying three conditions. Let me break down the problem step by step.
First, the given conditions are:
I. The sequence starts with a_0 = 1 and a_1 = 337.
II. For all n ≥ 1, the recurrence relation is (a_{n+1}... | To prove that there exists \( m \in \mathbb{N} \) such that there exists an integral sequence \( \{a_n\} \) satisfying the given conditions, we analyze the problem step by step:
1. **Condition II Analysis**:
- The recurrence relation is given by:
\[
(a_{n+1} a_{n-1} - a_n^2) + \frac{3}{4}(a_{n+1} + a_{n-1... | 1. **Define a new sequence**: Let \( b_n = 4a_n + 3 \). This transformation helps simplify the given conditions.
2. **Initial conditions**: From the given initial conditions \( a_0 = 1 \) and \( a_1 = 337 \), we can find the initial values for \( b_n \):
\[
b_0 = 4a_0 + 3 = 4 \cdot 1 + 3 = 7
\]
\[
b_1 ... |
Consider the set $M=\{1,2,...,n\},n\in\mathbb N$ . Find the smallest positive integer $k$ with the following property: In every $k$ -element subset $S$ of $M$ there exist two elements, one of which divides the other one.
| Okay, so I need to figure out the smallest positive integer k such that any k-element subset of M = {1, 2, ..., n} will necessarily contain two elements where one divides the other. Hmm, interesting problem. Let me start by understanding the problem better.
First, the set M is just the natural numbers from 1 up to som... | To determine the smallest positive integer \( k \) such that every \( k \)-element subset \( S \) of \( M = \{1, 2, \ldots, n\} \) contains two elements where one divides the other, we need to find the largest subset of \( M \) where no element divides another (an antichain) and then add 1 to its size.
1. **Understand... | 1. **Define the problem and the set \( M \):**
We are given the set \( M = \{1, 2, \ldots, n\} \) where \( n \in \mathbb{N} \). We need to find the smallest positive integer \( k \) such that in every \( k \)-element subset \( S \) of \( M \), there exist two elements \( a \) and \( b \) such that \( a \) divides \(... |
**Determine all four digit numbers** $\bar{a}\bar{b}\bar{c}\bar{d}$ **such that** $$ a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d} $$ | Okay, let's see. I need to find all four-digit numbers \(\overline{a}\overline{b}\overline{c}\overline{d}\) such that:
\[
a(a + b + c + d)(a^2 + b^2 + c^2 + d^2)(a^6 + 2b^6 + 3c^6 + 4d^6) = \overline{a}\overline{b}\overline{c}\overline{d}
\]
First, let's parse the problem. The left side of the equation is a product o... | To determine all four-digit numbers \(\overline{a}\overline{b}\overline{c}\overline{d}\) such that:
\[
a(a + b + c + d)(a^2 + b^2 + c^2 + d^2)(a^6 + 2b^6 + 3c^6 + 4d^6) = \overline{a}\overline{b}\overline{c}\overline{d}
\]
we start by analyzing the equation and considering the constraints for four-digit numbers. The ... | To determine all four-digit numbers $\overline{abcd}$ such that
\[ a(a+b+c+d)(a^2+b^2+c^2+d^2)(a^6+2b^6+3c^6+4d^6) = \overline{abcd}, \]
we need to analyze the constraints and possible values for the digits \(a, b, c,\) and \(d\).
1. **Analyzing the constraint \(a^{10} < 1000(a+1)\):**
- Since \(a\) is a digit, \(... |
Of all positive integral solutions $(x,y,z)$ to the equation \[x^3+y^3+z^3-3xyz=607,\] compute the minimum possible value of $x+2y+3z.$ *Individual #7* | Okay, so I need to find all positive integer solutions (x, y, z) to the equation x³ + y³ + z³ - 3xyz = 607, and then compute the minimum value of x + 2y + 3z. Hmm, let me start by understanding the problem better.
First, the equation given is x³ + y³ + z³ - 3xyz = 607. I remember there's a formula related to this expr... | Given the equation \(x^3 + y^3 + z^3 - 3xyz = 607\), we use the identity \(x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)\). Since 607 is a prime number, the factors must be 1 and 607. Given \(x, y, z\) are positive integers, the only feasible factorization is \(x + y + z = 607\) and \(x^2 + y^2 +... | 1. We start with the given equation:
\[
x^3 + y^3 + z^3 - 3xyz = 607
\]
This can be factored using the identity for the sum of cubes:
\[
x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx)
\]
Therefore, we have:
\[
(x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx) = 607
\]
2. Since ... |
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$ . | Okay, so I need to find all real numbers a for which there exists a function f that's defined on all real numbers, takes all real numbers as its values exactly once (so it's bijective, right?), and satisfies the equation f(f(x)) = x²f(x) + a x² for all real x. Hmm, let me try to figure this out step by step.
First, le... | To find all real numbers \( a \) for which there exists a bijective function \( f \) defined on all real numbers such that \( f(f(x)) = x^2 f(x) + ax^2 \) for all real \( x \), we proceed as follows:
1. **Bijectivity and Polynomial Assumption**:
- We initially considered if \( f \) could be a polynomial. Assuming \... | 1. **Define the assertion**: Let \( P(x) \) be the assertion \( f(f(x)) = x^2 f(x) + a x^2 \).
2. **Existence of \( u \)**: Since \( f \) is bijective, there exists a real number \( u \) such that \( f(u) = -a \).
3. **Evaluate at \( x = 0 \)**:
\[
P(0) \implies f(f(0)) = 0^2 f(0) + a \cdot 0^2 \implies f(f(0)... |
A positive integer is *detestable* if the sum of its digits is a multiple of $11$ . How many positive integers below $10000$ are detestable?
*Proposed by Giacomo Rizzo* | Okay, so I need to figure out how many positive integers below 10,000 are detestable. A detestable number is defined as one where the sum of its digits is a multiple of 11. Hmm, let's start by understanding the problem.
First, positive integers below 10,000 include all numbers from 1 up to 9999. That's 9999 numbers in... | To determine how many positive integers below 10,000 are detestable (i.e., their digit sum is a multiple of 11), we consider numbers from 0000 to 9999 (inclusive) and then subtract 1 to exclude 0000.
Using generating functions and roots of unity, we analyze the generating function for the digit sum of four-digit numb... | To determine how many positive integers below \(10000\) are detestable, we need to find the number of integers whose digits sum to a multiple of \(11\). We will break this down step-by-step.
1. **Understanding the Problem:**
A number is detestable if the sum of its digits is a multiple of \(11\). We need to count s... |
Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$ -axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than v... | Alright, so I need to figure out the smallest integer d such that there's a polynomial of degree d whose graph intersects all sides of a regular 2n-sided polygon. The polygon has one of its largest diagonals parallel to the x-axis. Hmm, okay. Let's break this down step by step.
First, understanding the problem: We hav... | To solve the problem of finding the smallest integer \( d \) such that there exists a polynomial \( P \) of degree \( d \) whose graph intersects all sides of a regular \( 2n \)-sided polygon (with one of its largest diagonals parallel to the \( x \)-axis), we need to consider the symmetry and structure of the polygon.... | 1. **Assume the Setup:**
Suppose we have a regular polygon with \(2n\) sides, and one of its largest diagonals lies on the \(x\)-axis. This means the polygon is symmetric about the \(x\)-axis.
2. **Intersection Points:**
Let \(x_1 < x_2 < \cdots < x_{2n}\) be the \(x\)-coordinates of the intersection points of t... |
Let us have $n$ ( $n>3$ ) balls with different rays. On each ball it is written an integer number. Determine the greatest natural number $d$ such that for any numbers written on the balls, we can always find at least 4 different ways to choose some balls with the sum of the numbers written on them divisible by $d... | Okay, let's try to figure out this problem. So, we have n balls, each with a different integer written on them, and n is greater than 3. We need to determine the greatest natural number d such that no matter what integers are on the balls, there are always at least 4 different ways to choose some balls (subsets) whose ... | To determine the greatest natural number \( d \) such that for any numbers written on \( n \) balls (\( n > 3 \)), there are always at least 4 different ways to choose some balls with the sum divisible by \( d \), we analyze the problem using combinatorial reasoning and modular arithmetic.
1. **Understanding the Probl... | 1. **Understanding the Problem:**
We need to determine the greatest natural number \( d \) such that for any set of \( n \) balls (where \( n > 3 \)) with integers written on them, there are at least 4 different ways to choose some balls such that the sum of the numbers on the chosen balls is divisible by \( d \).
... |
In a $k$ -player tournament for $k > 1$ , every player plays every other player exactly once. Find with proof the smallest value of $k$ such that it is possible that for any two players, there was a third player who beat both of them. | Okay, so I need to find the smallest number k of players in a tournament where for any two players, there's a third player who beat both of them. Let me try to understand the problem first.
A tournament means that every pair of players plays exactly once, and there are no ties. So for any two players A and B, either ... | To determine the smallest value of \( k \) such that in a \( k \)-player tournament, every pair of players has a third player who beats both, we need to ensure the tournament's structure allows for this property.
1. **Initial Considerations**:
- For \( k = 2 \), no third player exists.
- For \( k = 3 \), cyclic... | 1. **Establishing the requirement:**
We need to find the smallest value of \( k \) such that in a \( k \)-player tournament, for any two players, there exists a third player who beat both of them.
2. **Analyzing the minimum losses:**
Suppose \( k \) players are labeled as \( P_1, P_2, \ldots, P_k \). Each playe... |
The *equatorial algebra* is defined as the real numbers equipped with the three binary operations $\natural$ , $\sharp$ , $\flat$ such that for all $x, y\in \mathbb{R}$ , we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\]
An *equatorial expression*... | Okay, so I need to figure out how many distinct functions f: R^3 → R can be expressed as equatorial expressions with complexity at most 3. The operations allowed are natural (which is just addition), sharp (max), and flat (min). Each operation combines two expressions, and the complexity is 1 plus the sum of the comple... | To compute the number of distinct functions \( f: \mathbb{R}^3 \rightarrow \mathbb{R} \) that can be expressed as equatorial expressions of complexity at most 3, we need to consider the operations \(\natural\) (addition), \(\sharp\) (max), and \(\flat\) (min) applied recursively. The complexity of an expression is defi... | To solve the problem, we need to compute the number of distinct functions \( f: \mathbb{R}^3 \rightarrow \mathbb{R} \) that can be expressed as equatorial expressions of complexity at most 3. We will analyze the problem by considering the complexity of the expressions step by step.
1. **Complexity 0:**
- The expres... |
Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$ . Find the area of $S$ . | Okay, so I need to find the area of the set S, which is the locus of all points (x, y) in the first quadrant such that x/t + y/(1 - t) = 1 for some t between 0 and 1. Hmm, let me start by understanding what exactly this problem is asking.
First, the equation given is x/t + y/(1 - t) = 1, where t is a parameter between... | To find the area of the locus \( S \) of all points \((x, y)\) in the first quadrant such that \(\frac{x}{t} + \frac{y}{1 - t} = 1\) for some \( t \) with \( 0 < t < 1 \), we start by analyzing the given equation.
Each line \(\frac{x}{t} + \frac{y}{1 - t} = 1\) has intercepts \((t, 0)\) and \((0, 1 - t)\) as \( t \) ... | 1. We start with the given equation of the locus:
\[
\frac{x}{t} + \frac{y}{1-t} = 1 \quad \text{for some} \quad 0 < t < 1.
\]
Rearrange the equation to solve for \( y \):
\[
\frac{y}{1-t} = 1 - \frac{x}{t} \implies y = (1-t) \left(1 - \frac{x}{t}\right).
\]
Simplify the expression inside the pa... |
In how many ways can we fill the cells of a $m\times n$ board with $+1$ and $-1$ such that the product of numbers on each line and on each column are all equal to $-1$ ?
| Okay, so I need to figure out how many ways we can fill an m x n board with +1s and -1s such that the product of numbers in each row and each column is all equal to -1. Hmm, let's start by understanding what the problem is asking.
First, each cell in the m x n grid can be either +1 or -1. The condition is that when yo... | To determine the number of ways to fill an \( m \times n \) board with \( +1 \) and \( -1 \) such that the product of numbers in each row and each column is \( -1 \), we need to consider the parity of \( m \) and \( n \).
1. **Key Observations**:
- The product of all row products is \( (-1)^m \).
- The product o... | 1. **Consider the product of all \( mn \) numbers on the board:**
- Each cell in the \( m \times n \) board can be either \( +1 \) or \( -1 \).
- The product of all numbers in the board is the product of the products of each row and each column.
2. **Analyze the product of each row and column:**
- For the pro... |
If $X$ is a finite set, let $X$ denote the number of elements in $X$ . Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $ , and $ t > |S| $ for each $ t \in T $ . How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots... | Okay, so I need to figure out how many admissible ordered pairs (S, T) of subsets of {1, 2, ..., 10} there are. An admissible pair is defined such that for each s in S, s > |T|, and for each t in T, t > |S|. Hmm, let me try to unpack this.
First, let's restate the problem in my own words. We have two subsets S and T o... | To determine the number of admissible ordered pairs \((S, T)\) of subsets of \(\{1, 2, \ldots, 10\}\), we need to consider the constraints: each element \(s \in S\) must be greater than \(|T|\) and each element \(t \in T\) must be greater than \(|S|\).
1. **Define Variables**: Let \(|S| = k\) and \(|T| = m\). The con... | To solve the problem, we need to count the number of admissible ordered pairs \((S, T)\) of subsets of \(\{1, 2, \cdots, 10\}\) such that \(s > |T|\) for each \(s \in S\) and \(t > |S|\) for each \(t \in T\).
1. **Understanding the conditions**:
- For each element \(s \in S\), \(s > |T|\).
- For each element \(t... |
There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who... | Okay, so I need to figure out whether Geoff or Nazar has a winning strategy in this game where they take turns connecting blue and red points in 3D space. The first person to create a cycle loses? Wait, no, wait, the problem says "the one who first makes a cycle wins." Hmm, so actually creating a cycle is a winning mov... | The problem involves determining who has a winning strategy between Geoff (first player) and Nazar (second player) in a game where they take turns connecting blue and red points in 3D space, with no four points coplanar. The first player to create a cycle wins.
Key steps and reasoning:
1. **Cycle Formation**: A cycle ... | 1. **Translation to Graph Theory**:
- We can translate the problem into a graph theory problem where blue points and red points are vertices of two different sets in a bipartite graph. Geoff and Nazar take turns adding edges between these two sets.
- The goal is to determine who has a winning strategy, i.e., who ... |
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called *beautiful* if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of bea... | Okay, so I need to figure out how many 10-digit numbers are considered "beautiful" based on the given conditions. Let me start by understanding the problem statement carefully.
First, the arrangement is a permutation of the digits 0 through 9. That means each digit from 0 to 9 must appear exactly once. The term "beaut... | To determine the number of beautiful arrangements of the digits 0 through 9, we need to consider the following conditions:
1. The digits 0, 1, 2, 3, 4 must form an increasing sequence, and the digits 5, 6, 7, 8, 9 must form a decreasing sequence when read left to right.
2. The digit 0 must not be the leftmost digit.
F... | To determine the number of beautiful arrangements of the digits \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\), we need to follow the given conditions:
1. The digits \(0, 1, 2, 3, 4\) must form an increasing sequence.
2. The digits \(5, 6, 7, 8, 9\) must form a decreasing sequence.
3. The digit \(0\) cannot be the leftmost digit.
L... |
<span style="color:darkred">The altitude $[BH]$ dropped onto the hypotenuse of a triangle $ABC$ intersects the bisectors $[AD]$ and $[CE]$ at $Q$ and $P$ respectively. Prove that the line passing through the midpoints of the segments $[QD]$ and $[PE]$ is parallel to the line $AC$ .</span> | Okay, so I need to prove that the line passing through the midpoints of segments QD and PE is parallel to AC in triangle ABC where BH is the altitude to the hypotenuse, and it intersects the bisectors AD and CE at Q and P respectively. Hmm, let me start by drawing a diagram to visualize this.
First, triangle ABC with... | To prove that the line passing through the midpoints of the segments \([QD]\) and \([PE]\) is parallel to the line \(AC\) in triangle \(ABC\) with altitude \([BH]\) intersecting the angle bisectors \([AD]\) and \([CE]\) at \(Q\) and \(P\) respectively, we can use coordinate geometry and specific case analysis.
1. **Se... | 1. Consider the triangle \( \triangle ABC \) with \( BH \) as the altitude dropped onto the hypotenuse \( AC \). The bisectors \( AD \) and \( CE \) intersect \( BH \) at points \( Q \) and \( P \) respectively.
2. Let \( M \) and \( N \) be the midpoints of segments \( [PE] \) and \( [QD] \) respectively.
3. Since \( ... |
Let $ABC$ be an equilateral triangle with side length $2$ , and let $M$ be the midpoint of $\overline{BC}$ . Points $X$ and $Y$ are placed on $AB$ and $AC$ respectively such that $\triangle XMY$ is an isosceles right triangle with a right angle at $M$ . What is the length of $\overline{XY}$ ? | Okay, let me try to figure out how to solve this geometry problem. So, we have an equilateral triangle ABC with side length 2. M is the midpoint of BC, so BM = MC = 1. Points X and Y are on AB and AC respectively, such that triangle XMY is an isosceles right triangle with a right angle at M. We need to find the length ... | Given an equilateral triangle \(ABC\) with side length 2, and \(M\) as the midpoint of \(\overline{BC}\). Points \(X\) and \(Y\) are placed on \(AB\) and \(AC\) respectively such that \(\triangle XMY\) is an isosceles right triangle with a right angle at \(M\). We need to find the length of \(\overline{XY}\).
1. **Coo... | 1. **Identify the given information and set up the problem:**
- $ABC$ is an equilateral triangle with side length $2$.
- $M$ is the midpoint of $\overline{BC}$, so $BM = MC = 1$.
- $\triangle XMY$ is an isosceles right triangle with a right angle at $M$.
2. **Determine the coordinates of the points:**
- Pl... |
Find the number of solutions to the equation: $6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $ With {x} we denote the fractional part of the number x. | Okay, so I need to find the number of solutions to the equation \(6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018\). Hmm, let's start by recalling that \(\{x\}\) denotes the fractional part of \(x\), which means \(\{x\} = x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) is the greatest integer less than or equal to \(x\). S... | To find the number of solutions to the equation \(6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018\), where \(\{x\}\) denotes the fractional part of \(x\), we start by expressing \(x\) as \(n + f\), where \(n = \lfloor x \rfloor\) is an integer and \(f = \{x\} \in [0, 1)\). Substituting \(x = n + f\) into the equation, we get:
... | 1. **Understanding the problem**: We need to find the number of solutions to the equation \(6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018\), where \(\{x\}\) denotes the fractional part of \(x\). Recall that \(0 \leq \{x\} < 1\).
2. **Isolate the integer part**: Let \(x = [x] + \{x\}\), where \([x]\) is the integer part of \(... |
Three strictly increasing sequences
\[a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots\]
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$ , the following conditions hold:
(a) $c_{a_n}=b_n+1$ ;
(b) $a_{n... | Okay, let's try to tackle this problem step by step. It seems a bit complex at first glance, but maybe breaking it down will help. So, we have three strictly increasing sequences of positive integers: a_n, b_n, c_n. Every positive integer is in exactly one of these sequences. That means the union of all three sequences... | To solve the problem, we analyze the given conditions and deduce the structure of the sequences \(a_n\), \(b_n\), and \(c_n\). Each positive integer belongs to exactly one sequence, and the conditions must hold for all \(n\).
### Key Steps:
1. **Condition (a):** \(c_{a_n} = b_n + 1\)
2. **Condition (b):** \(a_{n+1} > ... | To solve this problem, we need to find the sequences \(a_n\), \(b_n\), and \(c_n\) that satisfy the given conditions. Let's analyze each condition step-by-step.
1. **Condition (a): \(c_{a_n} = b_n + 1\)**
This condition tells us that the \(a_n\)-th term of the sequence \(c\) is equal to the \(n\)-th term of the se... |
For each positive integer $n$ , define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$ (1) Find $H_1(x),\ H_2(x),\ H_3(x)$ .
(2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction.
(3) Let $a$ be real number. For $n\geq 3$... | Okay, let me try to work through these problems step by step. I need to start with part (1), finding H₁(x), H₂(x), H₃(x). The definition given is Hₙ(x) = (-1)ⁿ e^{x²} dⁿ/dxⁿ e^{-x²}.
First, let me recall that taking derivatives of e^{-x²} will involve the product of polynomials and e^{-x²}, and then multiplying by (-... | To solve the given problems, we proceed as follows:
**(1) Finding \( H_1(x), H_2(x), H_3(x) \):**
Using the definition \( H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \):
- For \( n = 1 \):
\[
H_1(x) = (-1)^1 e^{x^2} \frac{d}{dx} e^{-x^2} = -e^{x^2} (-2x e^{-x^2}) = 2x
\]
Thus, \( H_1(x) = 2x \).
- For... | ### Part (1)
We need to find \( H_1(x), H_2(x), H_3(x) \).
1. **For \( H_1(x) \):**
\[
H_1(x) = (-1)^1 e^{x^2} \frac{d}{dx} e^{-x^2}
\]
\[
\frac{d}{dx} e^{-x^2} = -2x e^{-x^2}
\]
\[
H_1(x) = -e^{x^2} \cdot (-2x e^{-x^2}) = 2x
\]
2. **For \( H_2(x) \):**
\[
H_2(x) = (-1)^2 e^{x^2} \fra... |
In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product. | Okay, so the problem is asking: In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? They consider m·n and n·m as the same product. Hmm, let's try to figure this out step by step.
First, I know that if I want to express a number as a product of two numbers, it's rel... | To determine the number of different ways 900 can be expressed as the product of two positive integers (regarding \(m \cdot n\) and \(n \cdot m\) as the same), we need to find the number of unique pairs \((m, n)\) such that \(m \leq n\) and \(m \cdot n = 900\).
First, we find the prime factorization of 900:
\[
900 = ... | 1. First, we need to find the prime factorization of 900. We have:
\[
900 = 2^2 \cdot 3^2 \cdot 5^2
\]
2. The number of divisors of 900 can be calculated using the formula for the number of divisors of a number \( n = p_1^{e_1} \cdot p_2^{e_2} \cdot \ldots \cdot p_k^{e_k} \), which is:
\[
(e_1 + 1)(e_2 +... |
Find all positive integers $m$ for which there exist three positive integers $a,b,c$ such that $abcm=1+a^2+b^2+c^2$ . | Okay, so I need to find all positive integers m such that there exist three positive positive integers a, b, c satisfying the equation abcm = 1 + a² + b² + c². Hmm, let's see.
First, let me understand the equation. It says that the product of a, b, c, and m equals 1 plus the sum of their squares. Since a, b, c, and m... | To find all positive integers \( m \) for which there exist three positive integers \( a, b, c \) such that \( abcm = 1 + a^2 + b^2 + c^2 \), we start by analyzing the equation and testing small values of \( m \).
1. **Testing \( m = 1 \)**:
- The equation becomes \( abc = 1 + a^2 + b^2 + c^2 \). Testing small valu... | 1. **Claim**: The only such \( m \) is \( m = 4 \), achieved by \((a, b, c) = (1, 1, 1)\).
2. **Proof**: Assume \( a, b, c \) are all positive integers such that \( abcm = 1 + a^2 + b^2 + c^2 \).
3. **Step 1**: Show that \( a, b, c \) are all odd.
- If all of them are even, then the LHS is even but the RHS is odd,... |
(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis.
(2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of... | Okay, let's tackle these two problems step by step. Both are about finding common tangents between a cubic curve and a parabola, and then using that information to calculate an area. Since these are part (1) and (2), I need to solve part (1) first to get the range of 'a', and then use that range in part (2) to find the... | To solve the given problems, we need to find the range of \( a \) for which there exist two common tangent lines to the curves \( y = \frac{8}{27}x^3 \) and \( y = (x + a)^2 \) other than the x-axis, and then determine the area bounded by these tangent lines and the parabola.
### Part (1): Finding the Range of \( a \)... | ### Part (1)
1. **Find the equation of the tangent line to the curve \( y = \frac{8}{27}x^3 \) at a point \( (t, \frac{8}{27}t^3) \):**
\[
\text{Slope of the tangent line} = \frac{d}{dx}\left(\frac{8}{27}x^3\right) = \frac{8}{9}x^2 \bigg|_{x=t} = \frac{8}{9}t^2
\]
The equation of the tangent line at \( (t, ... |
Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called *good* if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$ . Denote by $\mathcal{J}$ the set of polynomials of th... | Okay, so I have this problem about polynomials and symmetric functions. Let me try to parse it step by step.
First, the problem defines a "good" polynomial. A polynomial f in variables x₁, ..., xₙ is good if two conditions hold: first, f(0, ..., 0) = 0, which means that the constant term is zero. Second, the polynomia... | Given an integer \( n \geq 3 \), a polynomial \( f(x_1, \ldots, x_n) \) with integer coefficients is called *good* if \( f(0, \ldots, 0) = 0 \) and it is symmetric under any permutation of the variables. The set \( \mathcal{J} \) consists of polynomials that can be written as a sum of products of good polynomials and a... | 1. **Understanding the Problem:**
We need to find the smallest natural number \( D \) such that every monomial of degree \( D \) can be expressed as a sum of products of good polynomials and other polynomials with integer coefficients.
2. **Properties of Good Polynomials:**
A polynomial \( f(x_1, \ldots, x_n) \)... |
In a trapezoid $ABCD$ with $AB // CD, E$ is the midpoint of $BC$ . Prove that if the quadrilaterals $ABED$ and $AECD$ are tangent, then the sides $a = AB, b = BC, c =CD, d = DA$ of the trapezoid satisfy the equalities $a+c = \frac{b}{3} +d$ and $\frac1a +\frac1c = \frac3b$ . | Okay, so I need to prove that in a trapezoid ABCD with AB parallel to CD, and E being the midpoint of BC, if the quadrilaterals ABED and AECD are tangent, then the sides a = AB, b = BC, c = CD, d = DA satisfy a + c = (b/3) + d and 1/a + 1/c = 3/b. Hmm, let me start by understanding the problem step by step.
First, let... | To prove that in a trapezoid \(ABCD\) with \(AB \parallel CD\) and \(E\) as the midpoint of \(BC\), if quadrilaterals \(ABED\) and \(AECD\) are tangential, the sides \(a = AB\), \(b = BC\), \(c = CD\), \(d = DA\) satisfy \(a + c = \frac{b}{3} + d\) and \(\frac{1}{a} + \frac{1}{c} = \frac{3}{b}\), we proceed as follows:... | 1. **Given Conditions and Initial Setup:**
- We have a trapezoid \(ABCD\) with \(AB \parallel CD\).
- \(E\) is the midpoint of \(BC\).
- Quadrilaterals \(ABED\) and \(AECD\) are tangential.
2. **Tangential Quadrilateral Properties:**
- For a tangential quadrilateral, the sum of the lengths of opposite side... |
Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single $1\times1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w\times h$ ( $w$ columns and $h$ rows), then it costs $w^2$ dollars to add another row and ... | Alright, let's tackle this problem step by step. So, Nathan wants to build a chocolate bar from 1x1 up to 20x20, and each time he adds a row or a column, it costs w² or h² dollars respectively, depending on whether he's adding a row or a column. The goal is to find the minimum total cost to reach a 20x20 bar.
First, l... | Nathan wants to construct a chocolate bar from 1×1 to 20×20 with the minimum cost. The cost to add a row or column depends on the current dimensions \( w \times h \): adding a row costs \( w^2 \) dollars and adding a column costs \( h^2 \) dollars. The key is to balance the dimensions to minimize the total cost.
To so... | To find the minimum cost to construct a $20 \times 20$ chocolate bar, we need to consider the cost of adding rows and columns. The cost to add a row is $w^2$ dollars, and the cost to add a column is $h^2$ dollars, where $w$ is the current width and $h$ is the current height of the chocolate bar.
We will compare two st... |
The quadrilateral $ABCD$ has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$ . Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$ ? | Okay, let me try to work through this problem step by step. So, we have a quadrilateral ABCD with an incircle of diameter d. The incircle touches BC at K and DA at L. The question is asking whether the harmonic mean of AB and CD equals KL if and only if the geometric mean of AB and CD equals d. Hmm, that's a bit to unp... | The quadrilateral \(ABCD\) has an incircle of diameter \(d\) which touches \(BC\) at \(K\) and \(DA\) at \(L\). We need to determine if the harmonic mean of \(AB\) and \(CD\) is equal to \(KL\) if and only if the geometric mean of \(AB\) and \(CD\) is equal to \(d\).
1. **Tangential Quadrilateral Properties**:
- A ... | 1. **Define the harmonic mean and geometric mean:**
The harmonic mean \( h \) of \( AB \) and \( CD \) is given by:
\[
h = \frac{2}{\frac{1}{AB} + \frac{1}{CD}} = \frac{2AB \cdot CD}{AB + CD}
\]
The geometric mean \( g \) of \( AB \) and \( CD \) is given by:
\[
g = \sqrt{AB \cdot CD}
\]
2. **E... |
Let $a$ and $b$ be the roots of the quadratic $x^2-7x+c$ . Given that $a^2+b^2=17$ , compute $c$ . | Okay, so I need to find the value of c in the quadratic equation x² - 7x + c, given that a and b are the roots, and a² + b² equals 17. Hmm, let me start by recalling some properties of quadratic equations and their roots.
First, for a quadratic equation of the form x² + px + q = 0, the sum of the roots is -p, and the... | To determine the value of \( c \) in the quadratic equation \( x^2 - 7x + c \), given that the roots \( a \) and \( b \) satisfy \( a^2 + b^2 = 17 \):
1. **Sum and Product of Roots**:
- For the quadratic \( x^2 - 7x + c \), the sum of the roots \( a + b = 7 \).
- The product of the roots \( ab = c \).
2. **Rela... | 1. Let \( a \) and \( b \) be the roots of the quadratic equation \( x^2 - 7x + c = 0 \). By Vieta's formulas, we know:
\[
a + b = 7 \quad \text{and} \quad ab = c
\]
2. We are given that \( a^2 + b^2 = 17 \). We can use the identity for the sum of squares of the roots:
\[
a^2 + b^2 = (a + b)^2 - 2ab
... |
Let $ABC$ be a triangle in which $AB=AC$ . Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$ . | Okay, let's see. So, the problem is about an isosceles triangle ABC where AB equals AC. We need to find the ratio of AB to BC given that the orthocenter lies on the incircle. Hmm, orthocenter is the point where the three altitudes intersect, right? And the incircle is the circle inscribed inside the triangle, tangent t... | Given an isosceles triangle \(ABC\) with \(AB = AC\), we need to find the ratio \(\frac{AB}{BC}\) given that the orthocenter lies on the incircle.
1. **Setup and Variables**:
- Let \(AB = AC = x\) and \(BC = 2y\).
- The altitude from \(A\) to \(BC\) is \(AD = \sqrt{x^2 - y^2}\).
2. **Inradius Calculation**:
... | 1. **Identify Key Elements and Setup**:
- Given: \( \triangle ABC \) is isosceles with \( AB = AC \).
- Let \( D \) be the foot of the perpendicular from \( A \) to \( BC \).
- Denote \( H \) as the orthocenter and \( I \) as the incenter of \( \triangle ABC \).
- Since \( \triangle ABC \) is isosceles, \( ... |
Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$ , if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given). | Okay, so I need to find the roots of the equation (x - a)(x - b) = (x - c)(x - d). The given conditions are that a + d = b + c = 2015 and a ≠ c. Hmm, let's start by expanding both sides of the equation to see if I can simplify it.
First, expanding the left side: (x - a)(x - b) = x² - (a + b)x + ab. Similarly, the righ... | To find the roots of the equation \((x - a)(x - b) = (x - c)(x - d)\) given \(a + d = b + c = 2015\) and \(a \ne c\), we start by expanding both sides:
Left side: \((x - a)(x - b) = x^2 - (a + b)x + ab\)
Right side: \((x - c)(x - d) = x^2 - (c + d)x + cd\)
Subtracting the right side from the left side, we get:
\[
-(... | 1. Start with the given equation:
\[
(x-a)(x-b) = (x-c)(x-d)
\]
2. Expand both sides:
\[
x^2 - (a+b)x + ab = x^2 - (c+d)x + cd
\]
3. Subtract \(x^2\) from both sides:
\[
-(a+b)x + ab = -(c+d)x + cd
\]
4. Rearrange the equation to isolate \(x\):
\[
(c+d - a+b)x = ab - cd
\]
5. Giv... |
The polynomial $x^3 + px^2 + qx + r$ has three roots in the interval $(0,2)$ . Prove that $-2 <p + q + r < 0$ . | Okay, so I need to prove that for the polynomial \(x^3 + p x^2 + q x + r\) with all three roots in the interval \((0, 2)\), the sum \(p + q + r\) lies between \(-2\) and \(0\). Hmm, let's start by recalling some properties of polynomials and their roots.
First, since the polynomial is cubic and has three real roots i... | To prove that \(-2 < p + q + r < 0\) for the polynomial \(x^3 + px^2 + qx + r\) with all roots in \((0, 2)\), we analyze the polynomial's behavior at \(x = 1\).
1. **Factor Form and Coefficients**:
Let the roots be \(a, b, c \in (0, 2)\). The polynomial is \((x - a)(x - b)(x - c)\). Expanding this:
\[
x^3 -... | 1. Let \( a, b, c \) be the roots of the polynomial \( x^3 + px^2 + qx + r \). By Vieta's formulas, we know:
\[
a + b + c = -p,
\]
\[
ab + bc + ca = q,
\]
\[
abc = -r.
\]
2. Given that the roots \( a, b, c \) are in the interval \( (0, 2) \), we can write:
\[
0 < a, b, c < 2.
\]
3.... |
Let $n \geq 2$ . Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$ . Show that \[ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 . \] | Okay, so I need to show that the sum of these cyclic fractions a₁/a₂ + a₂/a₃ + ... + aₙ/a₁ is less than 2n - 1. The given conditions are that each a_k is less than 1, and the absolute difference between consecutive terms is less than 1. Hmm, let me start by understanding the problem step by step.
First, all the a_i ar... | Given \( n \geq 2 \) and real numbers \( a_1, a_2, \ldots, a_n \) all less than 1 with \( |a_k - a_{k+1}| < 1 \) for \( 1 \leq k \leq n-1 \), we need to show that:
\[
\dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2n - 1.
\]
### Key Steps and Reasoning:
1. **Understanding th... | To solve the problem, we need to show that the sum of the ratios of consecutive terms in the sequence is less than \(2n - 1\). Let's proceed step-by-step.
1. **Initial Setup:**
Given \(a_1, a_2, \ldots, a_n\) are real numbers such that \(a_i < 1\) for all \(i\) and \(|a_k - a_{k+1}| < 1\) for \(1 \leq k \leq n-1\).... |
Given that $\tan A=\frac{12}{5}$ , $\cos B=-\frac{3}{5}$ and that $A$ and $B$ are in the same quadrant, find the value of $\cos (A-B)$ . $ \textbf{(A) }-\frac{63}{65}\qquad\textbf{(B) }-\frac{64}{65}\qquad\textbf{(C) }\frac{63}{65}\qquad\textbf{(D) }\frac{64}{65}\qquad\textbf{(E) }\frac{65}{63} $ | Okay, let me try to solve this problem. So, we need to find the value of cos(A - B) given that tan A = 12/5 and cos B = -3/5, and both A and B are in the same quadrant. The answer choices are given, so after working through it, I can check which one matches. Alright, let's break this down step by step.
First, I rememb... | Given that \(\tan A = \frac{12}{5}\) and \(\cos B = -\frac{3}{5}\), and both \(A\) and \(B\) are in the same quadrant, we need to find the value of \(\cos(A - B)\).
1. **Determine the Quadrants**:
- \(\tan A = \frac{12}{5}\) is positive, so \(A\) must be in either quadrant I or III.
- \(\cos B = -\frac{3}{5}\) i... | 1. **Determine the values of $\sin A$ and $\cos A$:**
Given $\tan A = \frac{12}{5}$, we can use the identity $\tan A = \frac{\sin A}{\cos A}$.
Let $\sin A = 12k$ and $\cos A = 5k$ for some $k$. Using the Pythagorean identity $\sin^2 A + \cos^2 A = 1$:
\[
(12k)^2 + (5k)^2 = 1 \implies 144k^2 + 25k^2 = 1 \imp... |
There are $100$ cards with numbers from $1$ to $100$ on the table.Andriy and Nick took the same number of cards in a way such that the following condition holds:if Andriy has a card with a number $n$ then Nick has a card with a number $2n+2$ .What is the maximal number of cards that could be taken by the two g... | Okay, let's try to figure out this problem. So, we have 100 cards numbered from 1 to 100 on the table. Andriy and Nick each take the same number of cards, but there's a condition: if Andriy has a card with number n, then Nick must have a card with number 2n + 2. We need to find the maximal number of cards that both cou... | The maximal number of cards that Andriy and Nick could take together is **66**.
**Step-by-Step Explanation:**
1. **Understanding the Pairing Condition:**
- For each card \( n \) Andriy has, Nick must have \( 2n + 2 \).
- Andriy's cards (A) and Nick's cards (N) must be disjoint subsets of {1, 2, ..., 100}.
2. ... | 1. **Identify the relationship between the cards:**
If Andriy has a card with number \( n \), then Nick has a card with number \( 2n + 2 \). This relationship forms a sequence where each term is derived from the previous term using the formula \( a_{k+1} = 2a_k + 2 \).
2. **Form sequences (chains) based on the give... |
Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A *step* starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is the probability that the number 1 is face... | Okay, so I have this problem here: there's a fair coin and a fair six-sided die. The die starts with 1 facing up. A step involves tossing the coin. If it's heads, we roll the die; if tails, we do nothing. After 5 such steps, what's the probability that the number 1 is still face up on the die?
Hmm, let's break this do... | The probability that the number 1 is face up on the die after 5 steps is found by analyzing the process through recurrence relations and considering the effect of each coin toss and die roll.
**Key Steps:**
1. **Recurrence Relation:** Let \( P(n) \) be the probability the die shows 1 after \( n \) steps. The recurren... | 1. **Determine the probability of the coin landing tails all 5 times:**
- The probability of getting tails on a single coin flip is \( \frac{1}{2} \).
- The probability of getting tails 5 times in a row is \( \left( \frac{1}{2} \right)^5 = \frac{1}{32} \).
2. **Determine the probability of the die showing 1 if t... |
Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$ | Okay, let's tackle this problem step by step. The question is: What number do we need to subtract from the numerator and add to the denominator of the fraction \(\frac{\overline{28a3}}{7276}\) so that the resulting fraction equals \(\frac{2}{7}\)?
First, let me make sure I understand the problem correctly. The fractio... | To solve the problem, we need to find the number \( x \) such that subtracting \( x \) from the numerator and adding \( x \) to the denominator of the fraction \(\frac{\overline{28a3}}{7276}\) results in the fraction \(\frac{2}{7}\).
First, we express the four-digit number \(\overline{28a3}\) as \(2803 + 10a\). The eq... | 1. We start with the given equation:
\[
\frac{\overline{28a3} - x}{7276 + x} = \frac{2}{7}
\]
Here, $\overline{28a3}$ represents a four-digit number where $a$ is a digit.
2. Cross-multiplying to eliminate the fraction, we get:
\[
7(\overline{28a3} - x) = 2(7276 + x)
\]
3. Expanding both sides:
... |
(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$ . Call this $x_n$ .
(b) Find $\lim\limits_{n\rightarrow+\infty}x_n$ . | Okay, let's tackle part (a) first. The problem states that for every natural number n, there's exactly one positive real number x such that x^n + x^{n+1} = 1, and we need to call this x_n. Hmm, so I need to show existence and uniqueness for each n.
Starting with existence. Let's consider the function f(x) = x^n + x^{n... | (a) To show that for every \( n \in \mathbb{N} \), there is exactly one \( x \in \mathbb{R}^+ \) such that \( x^n + x^{n+1} = 1 \):
1. **Existence**: Consider the function \( f(x) = x^n + x^{n+1} - 1 \). As \( x \to 0 \), \( f(x) \to -1 \), and as \( x \to \infty \), \( f(x) \to \infty \). By the Intermediate Value Th... | ### Part (a)
1. Consider the function \( f(x) = x^n + x^{n+1} \) for \( x \in \mathbb{R}^+ \).
2. The function \( f(x) \) is continuous and strictly increasing on \( \mathbb{R}^+ \). This is because both \( x^n \) and \( x^{n+1} \) are continuous and strictly increasing functions for \( x > 0 \).
3. As \( x \) approach... |
For an integer $n\geq 0$ , let $a_n=\int_0^1 x^ne^x\ dx.$ (1) Find $a_1,\ a_2,\ a_3$ and $a_4$ .
(2) Show that $\frac 83<e<\frac{30}{11}.$ | Okay, so I need to solve this problem about integrals and then use those results to show that e is between 8/3 and 30/11. Let me start with part (1), finding a₁, a₂, a₃, and a₄ where each aₙ is the integral from 0 to 1 of xⁿeˣ dx.
First, I remember that integrating functions like xⁿeˣ can be done using integration by... | To solve the given problem, we start by finding the integrals \(a_n = \int_0^1 x^n e^x \, dx\) for \(n = 1, 2, 3, 4\) using integration by parts. We derive a recurrence relation and compute the values step-by-step.
### Part (1)
1. **Finding \(a_0\)**:
\[
a_0 = \int_0^1 e^x \, dx = e - 1
\]
2. **Using the re... | ### Part 1: Finding \(a_1, a_2, a_3, a_4\)
To find \(a_n = \int_0^1 x^n e^x \, dx\), we can use integration by parts. Let \(u = x^n\) and \(dv = e^x \, dx\). Then \(du = n x^{n-1} \, dx\) and \(v = e^x\).
Using integration by parts:
\[
\int u \, dv = uv - \int v \, du
\]
1. For \(a_1\):
\[
a_1 = \int_0^1 x e^x \, dx... |
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is... | Okay, so I need to prove that if a strictly increasing sequence of positive integers \( s_1, s_2, s_3, \ldots \) has the property that both \( s_{s_1}, s_{s_2}, s_{s_3}, \ldots \) and \( s_{s_1+1}, s_{s_2+1}, s_{s_3+1}, \ldots \) are arithmetic progressions, then the original sequence \( s_n \) must itself be an arithm... | Given a strictly increasing sequence of positive integers \( s_1, s_2, s_3, \ldots \) such that the sub-sequences \( s_{s_1}, s_{s_2}, s_{s_3}, \ldots \) and \( s_{s_1+1}, s_{s_2+1}, s_{s_3+1}, \ldots \) are both arithmetic progressions (APs), we need to prove that the original sequence \( s_1, s_2, s_3, \ldots \) is i... | 1. Let \( d_1 \) and \( d_2 \) denote the common differences of the sequences \( (s_{s_n})_{n \ge 1} \) and \( (s_{s_n+1})_{n \ge 1} \) respectively. We can write:
\[
s_{s_n} = a + nd_1 \quad \text{and} \quad s_{s_n+1} = b + nd_2
\]
where \( a \) and \( b \) are constants.
2. We claim that \( d_1 = d_2 \).... |
There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no... | Okay, let's try to tackle part (a) first. The problem states that there are n = 8 classes, each extracurricular group has exactly a = 4 classes participating, and any two groups share at most b = 1 class. We need to find the maximum possible number of groups m.
Hmm, this seems like a combinatorial problem. Maybe it's ... | To solve part (a) of the problem, we need to find the maximum number \( m \) of extracurricular groups when \( n = 8 \) classes, each group has exactly \( a = 4 \) classes, and any two groups share at most \( b = 1 \) class.
### Key Steps and Analysis:
1. **Problem Interpretation**: We need a family of 4-element subs... | ### Part (a)
Given:
- \( n = 8 \)
- \( a = 4 \)
- \( b = 1 \)
We need to find \( m \).
1. **Define the 3-tuple and calculate \( N \)**:
- Let the classes be \( c_1, c_2, \ldots, c_n \).
- Let the extracurricular groups be \( G_1, G_2, \ldots, G_m \).
- Define the 3-tuple \((G_i, G_j, c_k)\) if class \( c_k \... |
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