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Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.
[i]Proposed by Oleksii Masalitin[/i] | 1. We start by evaluating the given expression for specific values of \(m\) and \(n\). We note that:
\[
253^2 = 64009 \quad \text{and} \quad 40^3 = 64000
\]
Therefore, the difference is:
\[
|253^2 - 40^3| = |64009 - 64000| = 9
\]
This shows that a difference of 9 is possible.
2. Next, we need t... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$. For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$? | 1. Given the condition that the arithmetic mean of numbers \(a, b, c\) is twice smaller than the arithmetic mean of numbers \(a, b, c, d\), we can write this as:
\[
\frac{a + b + c}{3} = \frac{1}{2} \left( \frac{a + b + c + d}{4} \right)
\]
2. Simplify the equation:
\[
\frac{a + b + c}{3} = \frac{1}{2} ... | 10 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ for which one can select $n$ distinct real numbers such that each of them is equal to the sum of some two other selected numbers.
[i]Proposed by Anton Trygub[/i] | To find the smallest positive integer \( n \) for which one can select \( n \) distinct real numbers such that each of them is equal to the sum of some two other selected numbers, we will proceed as follows:
1. **Understanding the Problem:**
We need to find the smallest \( n \) such that there exist \( n \) distinc... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$. | To solve this problem, we need to find the minimum number of colors \( k \) such that the subgraph constructed by the edges colored with these \( k \) colors is connected, regardless of the initial coloring of the edges in the complete graph \( K_{2024} \).
1. **Understanding the Problem**:
- We have a complete gra... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ ... | 1. We start by setting up the equation for the probability of getting three heads and five tails, and the probability of getting five heads and three tails. The probability of getting exactly \( k \) heads in \( n \) flips of a biased coin (with probability \( p \) of heads) is given by the binomial distribution formul... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Ear... | 1. **Understanding the Problem:**
We need to find the number of possible seating arrangements for a committee of 15 members (5 Martians, 5 Venusians, and 5 Earthlings) around a round table with specific constraints:
- A Martian must occupy chair 1.
- An Earthling must occupy chair 15.
- No Earthling can sit... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For $ k>0$, let $ I_k\equal{}10\ldots 064$, where there are $ k$ zeros between the $ 1$ and the $ 6$. Let $ N(k)$ be the number of factors of $ 2$ in the prime factorization of $ I_k$. What is the maximum value of $ N(k)$?
$ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \t... | 1. We start by expressing \( I_k \) in a more manageable form. Given \( I_k = 10\ldots064 \) with \( k \) zeros between the 1 and the 6, we can write:
\[
I_k = 10^{k+2} + 64
\]
This is because \( 10^{k+2} \) represents the number 1 followed by \( k+2 \) zeros, and adding 64 gives us the desired number.
2. ... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends. | 1. **Initial Setup and Assumptions:**
- We are given a group of 2009 people where every two people have exactly one common friend.
- We need to find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends.
2. **Choosing a Person wi... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For the give functions in $\mathbb{N}$:
[b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);
[b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.
solve the equation $\phi(\sigma(2^x))=2^x$. | 1. **Understanding the Functions:**
- Euler's $\phi$ function, $\phi(n)$, counts the number of integers up to $n$ that are coprime with $n$.
- The $\sigma$ function, $\sigma(n)$, is the sum of the divisors of $n$.
2. **Given Equation:**
We need to solve the equation $\phi(\sigma(2^x)) = 2^x$.
3. **Simplifyin... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In convex pentagon $ ABCDE$, denote by
$ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command.
= B',CF%Error. "capDB" is a bad command.
= C',DG\cap EC = D',EH\cap AD = E'.$
Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdo... | To prove the given equation, we will use the concept of areas of triangles and the properties of cyclic products. Let's denote the area of triangle $\triangle XYZ$ by $S_{XYZ}$.
1. **Expressing the ratios in terms of areas:**
\[
\frac{EA'}{A'B} \cdot \frac{DA_1}{A_1C} = \frac{S_{EIA'}}{S_{BIA'}} \cdot \frac{S_{D... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \... | 1. **Understanding the Problem:**
- We are given an integer \( m > 1 \) and an odd number \( n \) such that \( 3 \le n < 2m \).
- We have a matrix \( a_{i,j} \) with \( 1 \le i \le m \) and \( 1 \le j \le n \).
- The matrix satisfies two conditions:
1. For any \( 1 \le j \le n \), the elements \( a_{1,j},... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following:
[list](i) $1\le a_1<a_2<\cdots < a_n\le 50$
(ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that \[mb_i=c_i^{a_i}\qquad\text... | 1. **Understanding the Problem:**
We are given an \( n \)-tuple \((a_1, a_2, \ldots, a_n)\) of integers such that \(1 \le a_1 < a_2 < \cdots < a_n \le 50\). For any \( n \)-tuple \((b_1, b_2, \ldots, b_n)\) of positive integers, there exists a positive integer \( m \) and an \( n \)-tuple \((c_1, c_2, \ldots, c_n)\)... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that
$$ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $$
If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that... | 1. **Identify the midpoints:**
Given that \( D \) and \( E \) are points on \( AB \) and \( AC \) respectively, and the condition:
\[
\overrightarrow{DA} + \overrightarrow{DB} + \overrightarrow{EA} + \overrightarrow{EC} = \overrightarrow{O}
\]
Since \( \overrightarrow{DA} = -\overrightarrow{AD} \) and \(... | -1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.
How ... | 1. **Understanding the problem**: We need to find how many different values are taken by \( a_j \) if all the numbers \( a_j \) (for \( 1 \leq j \leq n \)) and \( P \) are prime. Here, \( P \) is the sum of all \( p_k \) with odd values of \( k \) less than or equal to \( n \), where \( p_k \) is the sum of the product... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$. | 1. Given that the greatest common divisor (GCD) of the polynomials \(x^2 + ax + b\) and \(x^2 + bx + c\) is \(x + 1\), we can write:
\[
\gcd(x^2 + ax + b, x^2 + bx + c) = x + 1
\]
This implies that both polynomials share \(x + 1\) as a factor.
2. Given that the least common multiple (LCM) of the polynomial... | -6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) =x^4+14x^3+52x^2+56x+16.$ Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_az_b+z_cz_d|$ where $\{a,b,c,d\}=\{1,2,3,4\}$. | 1. **Show that all four roots are real and negative:**
We start with the polynomial:
\[
f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16
\]
We can factorize it as follows:
\[
x^4 + 14x^3 + 52x^2 + 56x + 16 = (x^2 + 7x + 4)^2 - 5x^2
\]
This can be rewritten as:
\[
(x^2 + (7 - \sqrt{5})x + 4)(x^2 +... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In a book with page numbers from $ 1$ to $ 100$ some pages are torn off. The sum of the numbers on the remaining pages is $ 4949$. How many pages are torn off? | 1. First, calculate the sum of all page numbers from 1 to 100. This can be done using the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
where \( n = 100 \). Thus,
\[
S = \frac{100 \cdot 101}{2} = 5050
\]
2. Given that the sum of the numbers on the remaining pages is 4949,... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group. | **
- We need to show that 11 rounds are sufficient.
- Number the 2008 programmers using 11-digit binary numbers from 0 to 2047 (adding leading zeros if necessary).
- In each round \( i \), divide the programmers into teams based on the \( i \)-th digit of their binary number.
- This ensures that in each rou... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$. | 1. **Understanding the problem**: We need to find the smallest odd integer \( k \) such that for every cubic polynomial \( f \) with integer coefficients, if there exist \( k \) integers \( n \) such that \( |f(n)| \) is a prime number, then \( f \) is irreducible in \( \mathbb{Z}[n] \).
2. **Initial observation**: We... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number. | 1. **Define the invariant:**
Let \( I \) be the remainder when the sum of all the integers on the board is divided by 13. Initially, the numbers from 1 to 2009 are written on the board. The sum of these numbers is:
\[
S = \sum_{k=1}^{2009} k = \frac{2009 \times 2010}{2} = 2009 \times 1005
\]
We need to f... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s).
Find all possible initial position of the black stone, to make all stones black by finite actions. | 1. **Restate the Problem:**
We have 2008 white stones and 1 black stone in a row. An 'action' means selecting one black stone and changing the color of neighboring stone(s). We need to find all possible initial positions of the black stone to make all stones black by finite actions.
2. **Generalize the Problem:**
... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0 \\
ax + 23y + cz &= 0 \\
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$. What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c -... | 1. We start with the given system of equations:
\[
\begin{align*}
13x + by + cz &= 0 \quad \text{(1)} \\
ax + 23y + cz &= 0 \quad \text{(2)} \\
ax + by + 42z &= 0 \quad \text{(3)}
\end{align*}
\]
2. Subtract equation (1) from equation (2):
\[
(ax + 23y + cz) - (13x + by + cz) = 0 - 0
\]
... | -2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The three roots of the cubic $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$ are distinct real numbers between $ 0$ and $ 1$. For every nonnegative integer $ n$, let $ s_n$ be the sum of the $ n$th powers of these three roots. What is the value of the infinite series
\[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{... | 1. Let the roots of the cubic polynomial \( P(x) = 30x^3 - 50x^2 + 22x - 1 \) be \( a, b, \) and \( c \). We are given that these roots are distinct real numbers between \( 0 \) and \( 1 \).
2. We need to find the value of the infinite series \( s_0 + s_1 + s_2 + s_3 + \dots \), where \( s_n \) is the sum of the \( n ... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
When the integer $ {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103}$ is divided by 9, what is the remainder? | 1. Let \( a_n = (\sqrt{3} + 5)^n + (\sqrt{3} - 5)^n \). Notice that \( a_n \) is an integer because the irrational parts cancel out.
2. We need to find the remainder when \( (\sqrt{3} + 5)^{103} - (\sqrt{3} - 5)^{103} \) is divided by 9. Define \( b_n = (\sqrt{3} + 5)^n - (\sqrt{3} - 5)^n \). We are interested in \( b_... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$ 1 \le n \le 455$ and $ n^3 \equiv 1 \pmod {455}$. The number of solutions is ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$ | 1. We start by noting that \( 455 = 5 \times 7 \times 13 \). We need to solve \( n^3 \equiv 1 \pmod{455} \). This can be broken down into solving the congruences modulo 5, 7, and 13 separately.
2. First, consider the congruence modulo 5:
\[
n^3 \equiv 1 \pmod{5}
\]
The possible values of \( n \) modulo 5 a... | 9 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$ | Given the sequence \( (a_n)_{n=0}^\infty \) defined by \( a_{n+1} = a_n^3 + a_n^2 \), we need to determine the maximum number of distinct residues of \( a_i \) modulo 11.
1. **List all possible residues modulo 11:**
\[
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \pmod{11}
\]
2. **Calculate the squares of these residues... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For every $ 0 \le i \le 17$, $ a_i \equal{} \{ \minus{} 1, 0, 1\}$.
How many $ (a_0,a_1, \dots , a_{17})$ $ 18 \minus{}$tuples are there satisfying :
$ a_0 \plus{} 2a_1 \plus{} 2^2a_2 \plus{} \cdots \plus{} 2^{17}a_{17} \equal{} 2^{10}$
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)... | 1. We need to find the number of 18-tuples \((a_0, a_1, \dots, a_{17})\) where each \(a_i \in \{-1, 0, 1\}\) and they satisfy the equation:
\[
a_0 + 2a_1 + 2^2a_2 + \cdots + 2^{17}a_{17} = 2^{10}
\]
2. Notice that \(2^{10}\) is a power of 2, and we need to express it as a sum of terms of the form \(2^i a_i\) ... | 8 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
We divide entire $ Z$ into $ n$ subsets such that difference of any two elements in a subset will not be a prime number. $ n$ is at least ?
$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$ | 1. **Assume division into two subsets \( A \) and \( B \):**
- Let \( a \in A \). Then \( a + 2 \) must be in \( B \) because the difference \( 2 \) is a prime number.
- Similarly, \( a + 5 \) must be in neither \( A \) nor \( B \) because the difference \( 5 \) is a prime number.
- This leads to a contradicti... | 4 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many of
$ 11^2 \plus{} 13^2 \plus{} 17^2$, $ 24^2 \plus{} 25^2 \plus{} 26^2$, $ 12^2 \plus{} 24^2 \plus{} 36^2$, $ 11^2 \plus{} 12^2 \plus{} 132^2$ are perfect square ?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 d)1 \qquad\textbf{(E)}\ 0$ | 1. **First Term: \( 11^2 + 13^2 + 17^2 \)**
We will check if this sum is a perfect square by considering it modulo 4:
\[
11^2 \equiv 1 \pmod{4}, \quad 13^2 \equiv 1 \pmod{4}, \quad 17^2 \equiv 1 \pmod{4}
\]
Therefore,
\[
11^2 + 13^2 + 17^2 \equiv 1 + 1 + 1 \equiv 3 \pmod{4}
\]
Since 3 modulo... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
There are $ n$ sets having $ 4$ elements each. The difference set of any two of the sets is equal to one of the $ n$ sets. $ n$ can be at most ? (A difference set of $A$ and $B$ is $ (A\setminus B)\cup(B\setminus A) $)
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 15 \qquad\textb... | 1. **Understanding the Problem:**
We are given \( n \) sets, each containing 4 elements. The difference set of any two of these sets is equal to one of the \( n \) sets. We need to determine the maximum possible value of \( n \).
2. **Difference Set Definition:**
The difference set of two sets \( A \) and \( B \... | 7 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
On a faded piece of paper it is possible to read the following:
\[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\]
Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore ... | 1. Given the equation:
\[
(x^2 + x + a)(x^{15} - \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90,
\]
we need to determine the constant term \(a\).
2. To find \(a\), we can substitute specific values for \(x\) and analyze the resulting equations.
3. **Setting \(x = 0\):**
\[
(0^2 + 0 + a)(0^{15} -... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009 \\
& x^3 + 5x^2-49x-245 \\
& x^3 + 39x^2 - 117x - 1435\end{align*} | 1. **Identify the polynomials:**
\[
\begin{align*}
P(x) &= x^3 + 41x^2 - 49x - 2009, \\
Q(x) &= x^3 + 5x^2 - 49x - 245, \\
R(x) &= x^3 + 39x^2 - 117x - 1435.
\end{align*}
\]
2. **Factorize the constant terms:**
\[
\begin{align*}
2009 &= 7^2 \cdot 41, \\
245 &= 5 \cdot 7^2, \\
1435 &... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs of integers $x$ and $y$ such that $x^2 + xy + y^2 = 28$. | To find the number of pairs of integers \( (x, y) \) such that \( x^2 + xy + y^2 = 28 \), we start by analyzing the given equation.
1. **Rewrite the equation**:
\[
x^2 + xy + y^2 = 28
\]
2. **Express \( x \) in terms of \( y \)**:
\[
x^2 + xy + y^2 - 28 = 0
\]
This is a quadratic equation in \( x... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose you are given that for some positive integer $n$, $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$. | 1. We start by examining the sum of factorials for small values of \( n \) to determine if the sum is a perfect square. We will check \( n = 1, 2, 3 \) and then analyze the behavior for \( n \geq 4 \).
2. For \( n = 1 \):
\[
1! = 1
\]
Since \( 1 \) is a perfect square (\( 1^2 = 1 \)), \( n = 1 \) works.
3... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition? | To solve this problem, we need to determine the minimum number of weights, \( k \), such that any integer mass from 1 to 2009 can be measured using these weights on a mechanical balance. The key insight is that the weights can be placed on either side of the balance, allowing us to use the concept of ternary (base-3) r... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \[a^2 + b^2 = ab(a+b).\] | To find the number of ordered pairs \((a, b)\) of positive integers that satisfy the equation \(a^2 + b^2 = ab(a + b)\), we will follow these steps:
1. **Rearrange the given equation:**
\[
a^2 + b^2 = ab(a + b)
\]
We can rewrite this equation as:
\[
a^2 + b^2 = a^2b + ab^2
\]
2. **Move all terms ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$. | 1. **Substitute \( k = 1 \) into the given functional equation:**
\[
f(h+1) + f(h) = f(h)f(1) + 1 \quad \text{(Equation 1)}
\]
2. **Substitute \( h = k = 0 \) into the given functional equation:**
\[
f(0+0) + f(0) = f(0)f(0) + 1 \implies 2f(0) = f(0)^2 + 1
\]
Solving the quadratic equation \( f(0)... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $n$ players in a round-robin ping-pong tournament (i.e. every two persons will play exactly one game). After some matches have been played, it is known that the total number of matches that have been played among any $n-2$ people is equal to $3^k$ (where $k$ is a fixed integer). Find the sum of all possible v... | 1. We start with the given information that the total number of matches played among any \( n-2 \) people is equal to \( 3^k \), where \( k \) is a fixed integer. The number of matches played among \( n-2 \) people can be represented by the binomial coefficient \( \binom{n-2}{2} \).
2. The binomial coefficient \( \bin... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$. | 1. **Initial Setup and Sequence Definition:**
Let $(x_n)$ be a sequence of positive integers where $x_1$ is a fixed six-digit number. For any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$.
2. **Bounding the Sequence:**
Since $x_1$ is a six-digit number, we have $x_1 \leq 999999$. Therefore, $x_1 + 1 \leq... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p_1 = 2, p_2 = 3, p_3 = 5 ...$ be the sequence of prime numbers. Find the least positive even integer $n$ so that $p_1 + p_2 + p_3 + ... + p_n$ is not prime. | 1. We need to find the least positive even integer \( n \) such that the sum of the first \( n \) prime numbers is not a prime number.
2. Let's start by calculating the sums for even values of \( n \) until we find a composite number.
- For \( n = 2 \):
\[
p_1 + p_2 = 2 + 3 = 5 \quad (\text{prime})
\... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ such that for every prime number $p, p^2 + n$ is never prime. | 1. **Understanding the problem**: We need to find the smallest positive integer \( n \) such that for every prime number \( p \), the expression \( p^2 + n \) is never a prime number.
2. **Initial consideration**: Let's consider the nature of \( p^2 \). For any prime \( p \), \( p^2 \) is always an odd number except w... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\
\cos4 & \cos5 & \cos 6 \\
\cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ i... | 1. Consider the \( n \times n \) matrix \( A_n \) whose entries are \( \cos 1, \cos 2, \ldots, \cos n^2 \). We need to evaluate the limit of the determinant of this matrix as \( n \) approaches infinity, i.e., \( \lim_{n \to \infty} d_n \).
2. To understand the behavior of \( d_n \), we need to analyze the structure o... | 0 | Calculus | other | Yes | Yes | aops_forum | false |
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ be nine points in space such that $ABCDE$, $ABFGH$, and $GFCDI$ are each regular pentagons with side length $1$. Determine the lengths of the sides of triangle $EHI$. | 1. **Understanding the Problem:**
We are given three regular pentagons \(ABCDE\), \(ABFGH\), and \(GFCDI\) with side length 1. We need to determine the lengths of the sides of triangle \(EHI\).
2. **Analyzing the Geometry:**
- Since \(ABCDE\) is a regular pentagon, all its sides are equal to 1.
- Similarly, \... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $ | 1. **Define the function and the domain:**
We need to find the infimum of the function \( f(z) = |z^2 - az + a| \) where \( z \in \mathbb{C} \) and \( |z| \leq 1 \).
2. **Consider the polynomial \( P(z) = z^2 - az + a \):**
We need to analyze the behavior of \( P(z) \) on the boundary of the unit disk \( |z| = ... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares. | 1. **Initial Setup and Definitions:**
We need to find natural numbers \( n \ge 2 \) such that the ring of integers modulo \( n \), denoted \( \mathbb{Z}/n\mathbb{Z} \), has exactly one element that is not a sum of two squares.
2. **Multiplicative Closure of Sums of Two Squares:**
The set of elements which, modul... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$.
Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$.
Find the mini... | 1. **Determine the form of \( g(x) \):**
Given \( g(x) = x - \int_0^1 f(t) \, dt \), we assume \( g(x) \) is linear and of the form:
\[
g(x) = x + \alpha
\]
2. **Substitute \( g(x) \) into the equation for \( f(x) \):**
\[
f(x) = \frac{x^3}{2} + 1 - x \int_0^x g(t) \, dt
\]
Substituting \( g(t)... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ .
Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$. | Given the function \( f(x) = x^2 + 3 \) and the line \( y = g(x) \) with slope \( a \) passing through the point \( (0, f(0)) \), we need to find the maximum and minimum values of \( I(a) = 3 \int_{-1}^1 |f(x) - g(x)| \, dx \).
First, note that \( f(0) = 3 \), so the line \( g(x) \) passing through \( (0, 3) \) with s... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2.
When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$. | 1. Given the polynomial \( f(x) = ax^2 + bx + c \) with the constraints \( f(0) = 0 \) and \( f(2) = 2 \), we need to find the minimum value of \( S = \int_0^2 |f'(x)| \, dx \).
2. From the constraint \( f(0) = 0 \), we have:
\[
f(0) = a(0)^2 + b(0) + c = 0 \implies c = 0
\]
Thus, the polynomial simplifies... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$. | To find the length of the curve given by the polar equation \( r = 1 + \cos \theta \) for \( 0 \leq \theta \leq \pi \), we use the formula for the length of a curve in polar coordinates:
\[
L = \int_{\alpha}^{\beta} \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 } \, d\theta
\]
1. **Calculate \(\frac{dr}{d\theta}\)... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$.
(1) Find $ I_1,\ I_2$.
(2) Find $ \lim_{n\to\infty} I_n$. | 1. **Finding \( I_1 \) and \( I_2 \):**
For \( I_1 \):
\[
I_1 = \int_0^{\sqrt{3}} \frac{1}{1 + x} \, dx
\]
This is a standard integral:
\[
I_1 = \left[ \ln(1 + x) \right]_0^{\sqrt{3}} = \ln(1 + \sqrt{3}) - \ln(1 + 0) = \ln(1 + \sqrt{3}) - \ln(1) = \ln(1 + \sqrt{3})
\]
For \( I_2 \):
\[
... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For $ n \equal{} 1,\ 2,\ 3,\ \cdots$, let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$.
(1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$.
(2) Find ... | 1. To find the point of intersection \((p_n, q_n)\) of the curves \(y = \ln(nx)\) and \((x - \frac{1}{n})^2 + y^2 = 1\), we start by setting \(y = q_n\) and \(x = p_n\). Thus, we have:
\[
q_n = \ln(np_n) \implies p_n = \frac{1}{n} e^{q_n}
\]
Substituting \(p_n\) into the circle equation:
\[
\left(\fra... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$.
(1)... | 1. To express \(A_n\) and \(B_n\) in terms of \(n\) and \(g(n)\), we need to analyze the areas described in the problem.
**Step 1: Express \(A_n\) in terms of \(n\) and \(g(n)\)**
The area \(A_n\) is the area enclosed by the line passing through the points \((n, \ln n)\) and \((n+1, \ln (n+1))\), and the curve ... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}... | 1. **Identify the region \( K \) and the plane \( \alpha \):**
- The cuboid \( K \) is defined by the inequalities \( |x| \leq \sqrt{3} \), \( |y| \leq \sqrt{3} \), and \( 0 \leq z \leq 2 \).
- The plane \( \alpha \) is given by \( z = 2 \).
2. **Determine the condition \( \overline{OP} \leq \overline{PH} \):**
... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$.
(1) Find the local minimum value of $ f(x)$.
(2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$. | ### Part 1: Finding the Local Minimum Value of \( f(x) \)
1. **Define the function:**
\[
f(x) = \frac{1}{\sin x \sqrt{1 - \cos x}}
\]
2. **Simplify the expression:**
Using the identity \(1 - \cos x = 2 \sin^2 \left(\frac{x}{2}\right)\), we get:
\[
f(x) = \frac{1}{\sin x \sqrt{2 \sin^2 \left(\frac{x}... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Each of $10$ identical jars contains some milk, up to $10$ percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other $9$ jars, the same amount in each case. Prove that we can have the same amount of mi... | 1. **Initial Setup**: Let the amount of milk in the jars be denoted by \(a_1, a_2, \ldots, a_{10}\), where \(0 \leq a_i \leq 10\) for all \(i\). Assume without loss of generality that the jars are sorted in ascending order of milk volume, i.e., \(a_1 \leq a_2 \leq \cdots \leq a_{10}\).
2. **Move Description**: In each... | 9 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts?
[i](5 points for Juniors and 4 points for Seniors)[/i] | To determine if we can open the safe in less than 7 attempts, we need to ensure that each attempt has at least one digit in the correct position. We will use a strategy to cover all possible digits in each position over multiple attempts.
1. **Understanding the Problem:**
- The password consists of 7 different digi... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$ | To solve the problem, we need to find the smallest possible value of \( n \) such that for every integer \( N \), there exists an integer \( k_i \) from the set \( \{k_1, k_2, \ldots, k_n\} \) that divides \( N - a_i \) for some \( 1 \leq i \leq n \).
1. **Initial Assumptions and Setup:**
- Given \( 1 < k_1 < k_2 ... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$?
[i]Proposed by S. Berlov[/i] | 1. **Define the problem and setup the geometry:**
- We are given a cyclic quadrilateral \(ABCD\) with a point \(X\) on side \(AB\).
- Diagonal \(BD\) bisects \(CX\) and diagonal \(AC\) bisects \(DX\).
- We need to find the minimum possible value of \(\frac{AB}{CD}\).
2. **Introduce parallel lines and intersec... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$, $ P(2)$, $ P(3)$, $ \dots?$
[i]Proposed by A. Golovanov[/i] | 1. Let \( P(x) = ax^2 + bx + c \) be a quadratic polynomial. We need to find the maximum number of integers \( n \ge 2 \) such that \( P(n+1) = P(n) + P(n-1) \).
2. First, express \( P(n+1) \), \( P(n) \), and \( P(n-1) \):
\[
P(n+1) = a(n+1)^2 + b(n+1) + c = a(n^2 + 2n + 1) + b(n + 1) + c = an^2 + (2a + b)n + (... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $ \{x_n\}$ is defined by \[ \left\{ \begin{array}{l}x_1 \equal{} \frac{1}{2} \\x_n \equal{} \frac{{\sqrt {x_{n \minus{} 1} ^2 \plus{} 4x_{n \minus{} 1} } \plus{} x_{n \minus{} 1} }}{2} \\\end{array} \right.\]
Prove that the sequence $ \{y_n\}$, where $ y_n\equal{}\sum_{i\equal{}1}^{n}\frac{1}{{{x}_{i}... | 1. **Define the sequence and initial conditions:**
The sequence $\{x_n\}$ is defined by:
\[
\begin{cases}
x_1 = \frac{1}{2} \\
x_n = \frac{\sqrt{x_{n-1}^2 + 4x_{n-1}} + x_{n-1}}{2} \quad \text{for } n \geq 2
\end{cases}
\]
2. **Simplify the recursive formula:**
Let's simplify the expression for... | 6 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In right triangle $ ABC$ with right angle at $ C$, $ \angle BAC < 45$ degrees and $ AB \equal{} 4$. Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$. The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$, where $ p,q,r$ are positive integers an... | 1. **Define the angles and segments:**
Let $\angle ACP = \theta$ and $\angle APC = 2\theta$. Also, let $AP = x$ and $PB = 4 - x$ because $AB = 4$. Let $AC = y$.
2. **Apply the Law of Sines in $\triangle ACP$:**
\[
\frac{\sin \theta}{x} = \frac{\sin 2\theta}{y}
\]
Since $\sin 2\theta = 2 \sin \theta \cos... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ ABCDEF$ be a regular hexagon. Let $ G$, $ H$, $ I$, $ J$, $ K$, and $ L$ be the midpoints of sides $ AB$, $ BC$, $ CD$, $ DE$, $ EF$, and $ AF$, respectively. The segments $ AH$, $ BI$, $ CJ$, $ DK$, $ EL$, and $ FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of ... | 1. **Define the problem and setup:**
Let \( ABCDEF \) be a regular hexagon with side length \( s \). Let \( G, H, I, J, K, \) and \( L \) be the midpoints of sides \( AB, BC, CD, DE, EF, \) and \( FA \), respectively. The segments \( AH, BI, CJ, DK, EL, \) and \( FG \) bound a smaller regular hexagon inside \( ABCDE... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.
P.S. for 10 grade gives same p... | 1. **Define the problem and variables:**
- We have 24 pencils, divided into 4 different colors: \( R_1, R_2, R_3, R_4 \).
- Each color has 6 pencils.
- There are 6 children, each receiving 4 pencils.
2. **Consider the worst-case distribution:**
- We need to find the minimum number of children to guarantee ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two t... | 1. **Lemma.** Let \( m, n \ge 2 \) be two positive integers. A graph \( G(V, E) \) is \( mn \)-colorable if and only if there exists an edge partition \( E = E_1 \cup E_2 \) such that \( G_1 = G(V, E_1) \) and \( G_2 = G(V, E_2) \) are \( m \)- and \( n \)-colorable, respectively.
2. **Proof of Lemma.**
- **If dir... | 4 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$ | 1. **Identify the problem**: We need to determine the minimum number of socks to pull from a drawer containing red, green, blue, and white socks to guarantee a matching pair.
2. **Consider the worst-case scenario**: In the worst-case scenario, we would pull out one sock of each color without getting a matching pair. S... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
In $ \triangle ABC, \ \cos(2A \minus{} B) \plus{} \sin(A\plus{}B) \equal{} 2$ and $ AB\equal{}4.$ What is $ BC?$
$ \textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$ | 1. Given the equation \(\cos(2A - B) + \sin(A + B) = 2\), we know that the maximum values of \(\cos x\) and \(\sin y\) are both 1. Therefore, for the sum to be 2, both \(\cos(2A - B)\) and \(\sin(A + B)\) must be equal to 1.
2. Since \(\cos(2A - B) = 1\), it implies that \(2A - B = 360^\circ k\) for some integer \(... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$? | 1. **Understanding the Problem:**
We need to find the smallest amusing odd integer greater than 1. A positive integer \( n \) is said to be amusing if there exists a positive integer \( k \) such that \( d(k) = s(k) = n \), where \( d(k) \) is the number of divisors of \( k \) and \( s(k) \) is the digit sum of \( k... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=cx(x-1)$, where $c$ is a positive real number. We use $f^n(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^n(x)$ are real. What is the smallest possible value of $c$? | To solve the problem, we need to determine the smallest value of \( c \) such that all the roots of \( f^n(x) \) are real for every positive integer \( n \). The function given is \( f(x) = cx(x-1) \).
1. **Find the fixed points of \( f(x) \):**
The fixed points of \( f(x) \) are the solutions to the equation \( f(... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$. | 1. **Define the sequence and the limit to be computed:**
The sequence \( \{x_n\} \) is defined by:
\[
x_1 = \sqrt{5}, \quad x_{n+1} = x_n^2 - 2 \quad \text{for} \quad n \geq 1.
\]
We need to compute:
\[
\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot \ldots \cdot x_n}{x_{n+1}}.
\]
2. **... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer. | 1. Let \(a, b, c\) be three distinct positive integers such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer. Without loss of generality (WLOG), assume \(a > b > c\).
2. We claim that \(a = 6\), \(b = 3\), and \(c = 2\) is a solution. First, ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find maximal numbers of planes, such there are $6$ points and
1) $4$ or more points lies on every plane.
2) No one line passes through $4$ points. | To solve the problem, we need to find the maximum number of planes such that:
1. Each plane contains at least 4 points.
2. No line passes through 4 points.
Let's analyze the problem step by step.
1. **Case 1: All 6 points lie on a single plane.**
- If all 6 points lie on a single plane, then we can only have one p... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\]
(a) prove that $ ABC$ is a right-angled triangle, and
(b)... | ### Part (a): Prove that \( \triangle ABC \) is a right-angled triangle
1. Given that \( \angle ABP = \angle PBQ = \angle QBC \), we need to show that \( \triangle ABC \) is a right-angled triangle.
2. Note that \( P \) and \( Q \) are points on the angle bisectors of \( \angle BAC \) and \( \angle ABC \) respectively... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $p$ is a prime number, what is the product of elements like $g$ such that $1\le g\le p^2$ and $g$ is a primitive root modulo $p$ but it's not a primitive root modulo $p^2$, modulo $p^2$?($\frac{100}{6}$ points) | 1. **Identify the primitive roots modulo \( p \):**
Let \( p \) be a prime number. A primitive root modulo \( p \) is an integer \( g \) such that the smallest positive integer \( k \) for which \( g^k \equiv 1 \pmod{p} \) is \( k = p-1 \). This means \( g \) generates all the non-zero residues modulo \( p \).
2. *... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Bob has picked positive integer $1<N<100$. Alice tells him some integer, and Bob replies with the remainder of division of this integer by $N$. What is the smallest number of integers which Alice should tell Bob to determine $N$ for sure? | To determine the smallest number of integers Alice should tell Bob to determine \( N \) for sure, we need to consider the information obtained from each guess and how it narrows down the possible values of \( N \).
1. **Initial Consideration**:
- Alice needs to determine \( N \) where \( 1 < N < 100 \).
- Each g... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers. | 1. We start by defining the problem. We need to find the smallest integer \( k \geq 3 \) such that there exist two numbers \( a \) and \( b \) from the set \(\{1, 2, \ldots, k\}\) whose product equals the sum of the remaining \( k-2 \) numbers. Mathematically, this can be expressed as:
\[
ab = \sum_{i=1}^k i - a ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12. | 1. Given that triangles \(OAB\), \(OBC\), and \(OCD\) are isosceles right triangles with \(\angle OAB = \angle OBC = \angle OCD = 90^\circ\), we can infer that these triangles are 45-45-90 triangles. In a 45-45-90 triangle, the legs are equal, and the hypotenuse is \(\sqrt{2}\) times the length of each leg.
2. Let \(O... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$, compute the value of the expression
\[
\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
\left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
\left(
\frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^... | 1. **Simplify the first term:**
\[
\frac{b^2}{a^2} + \frac{a^2}{b^2} - 2
\]
We can rewrite this as:
\[
\frac{b^4 + a^4 - 2a^2b^2}{a^2b^2} = \frac{(a^2 - b^2)^2}{a^2b^2}
\]
2. **Simplify the second term:**
\[
\frac{a + b}{b - a} + \frac{b - a}{a + b}
\]
Using the common denominator, we ... | -8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of:
$ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$ | 1. Given \( x_1, x_2, \ldots, x_n \) are positive real numbers with sum \( 1 \), we need to find the integer part of:
\[
E = x_1 + \frac{x_2}{\sqrt{1 - x_1^2}} + \frac{x_3}{\sqrt{1 - (x_1 + x_2)^2}} + \cdots + \frac{x_n}{\sqrt{1 - (x_1 + x_2 + \cdots + x_{n-1})^2}}
\]
2. Since \( \sqrt{1 - (x_1 + x_2 + \cdots... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
What is the coefficient for $\text{O}_2$ when the following reaction
$\_\text{As}_2\text{S}_3+\_\text{O}_2 \rightarrow \_\text{As}_2\text{O}_3+\_\text{SO}_2$
is correctly balanced with the smallest integer coefficients?
$ \textbf{(A)} 5 \qquad\textbf{(B)} 6 \qquad\textbf{(C)} 8 \qquad\textbf{(D)} 9 \qquad $ | 1. Write the unbalanced chemical equation:
\[
\text{As}_2\text{S}_3 + \text{O}_2 \rightarrow \text{As}_2\text{O}_3 + \text{SO}_2
\]
2. Balance the sulfur atoms. There are 3 sulfur atoms in \(\text{As}_2\text{S}_3\), so we need 3 \(\text{SO}_2\) molecules on the right-hand side:
\[
\text{As}_2\text{S}_3 ... | 9 | Other | MCQ | Yes | Yes | aops_forum | false |
How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None}
$ | 1. **Understanding the problem**: We need to find the number of positive integers less than 2010 such that the sum of the factorials of its digits equals the number itself.
2. **Constraints on digits**: Since $7! = 5040 > 2010$, no digit in the number can be greater than 6. Therefore, the digits of the number can only... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Two players are playing a turn based game on a $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a m... | 1. **Understanding the Game Mechanics**:
- The game is played on an \( n \times n \) chessboard.
- The piece starts at the bottom-left corner \((1,1)\).
- Players take turns moving the piece to either the square just above, the square just right, or the diagonal square just right-top.
- The player who canno... | 4 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, what is the smallest distance that the grasshoper can be away from the point $2010$?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 18
\qquad\textbf{(D)}\ 34
\qquad\textbf{(E)}\ 164
$ | To solve this problem, we need to determine the smallest distance the grasshopper can be from the point \(2010\) on the real number line, given that it can jump either \(364\) or \(715\) units at a time.
1. **Understanding the Problem:**
The grasshopper starts at \(0\) and can jump either \(364\) or \(715\) units. ... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For how many integers $1\leq n \leq 2010$, $2010$ divides $1^2-2^2+3^2-4^2+\dots+(2n-1)^2-(2n)^2$?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 5
$ | 1. We start by simplifying the given expression:
\[
1^2 - 2^2 + 3^2 - 4^2 + \dots + (2n-1)^2 - (2n)^2
\]
We can pair the terms as follows:
\[
(1^2 - 2^2) + (3^2 - 4^2) + \dots + ((2n-1)^2 - (2n)^2)
\]
Each pair can be simplified using the difference of squares formula:
\[
a^2 - b^2 = (a - ... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None}
$ | 1. Let the polynomial be \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where each coefficient \( a_i \) is either \( 1 \) or \( -1 \).
2. By Vieta's formulas, the sum of the roots of \( P(x) \) is given by:
\[
-\frac{a_{n-1}}{a_n} = \pm 1
\]
This implies that the sum of the roots is either... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many integers $n$ with $0\leq n < 840$ are there such that $840$ divides $n^8-n^4+n-1$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 8
$ | 1. First, we note that \( 840 = 3 \cdot 5 \cdot 7 \cdot 8 \). We need to find the integers \( n \) such that \( 0 \leq n < 840 \) and \( 840 \) divides \( n^8 - n^4 + n - 1 \). This means we need \( n^8 - n^4 + n - 1 \equiv 0 \pmod{840} \).
2. We will consider the congruences modulo \( 3 \), \( 5 \), \( 7 \), and \( 8... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Which one does not divide the numbers of $500$-subset of a set with $1000$ elements?
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ 17
$ | To determine which number does not divide the number of $500$-subsets of a set with $1000$ elements, we need to evaluate the binomial coefficient $\binom{1000}{500}$.
1. **Express the binomial coefficient:**
\[
\binom{1000}{500} = \frac{1000!}{500! \cdot 500!}
\]
2. **Prime factorization of factorials:**
... | 11 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For which value of $m$, there is no triple of integer $(x,y,z)$ such that $3x^2+4y^2-5z^2=m$?
$ \textbf{(A)}\ 16
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ 8
$ | To determine the value of \( m \) for which there is no triple of integers \((x, y, z)\) such that \( 3x^2 + 4y^2 - 5z^2 = m \), we need to analyze the possible values of \( 3x^2 + 4y^2 - 5z^2 \).
1. **Check the given values:**
- For \( m = 8 \):
\[
(x, y, z) = (2, 2, 2) \implies 3(2^2) + 4(2^2) - 5(2^2) ... | 10 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$. | 1. **Express \(a_k\) in a different form:**
\[
a_k = k^2 + k + 1 = \frac{k^3 - 1}{k - 1} \quad \text{for} \quad k \geq 2
\]
This is because:
\[
k^3 - 1 = (k - 1)(k^2 + k + 1)
\]
Therefore:
\[
a_k = \frac{k^3 - 1}{k - 1}
\]
2. **Rewrite the product \(a_1 a_2 \ldots a_{p-1}\):**
\[
... | 3 | Number Theory | proof | Yes | Yes | aops_forum | false |
On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$. | 1. **Understanding the Problem:**
- We have a circle with diameter \(CD\).
- Points \(A\) and \(B\) are on opposite arcs of the diameter \(CD\).
- Line segments \(CE\) and \(DF\) are perpendicular to \(AB\) such that \(A, E, F, B\) are collinear in this order.
- Given \(AE = 1\), we need to find the length ... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a circle with centre at $O$ and diameter $AB$, two chords $BD$ and $AC$ intersect at $E$. $F$ is a point on $AB$ such that $EF \perp AB$. $FC$ intersects $BD$ in $G$. If $DE = 5$ and $EG =3$, determine $BG$. | 1. **Identify the given elements and relationships:**
- Circle with center \( O \) and diameter \( AB \).
- Chords \( BD \) and \( AC \) intersect at \( E \).
- \( F \) is a point on \( AB \) such that \( EF \perp AB \).
- \( FC \) intersects \( BD \) at \( G \).
- Given \( DE = 5 \) and \( EG = 3 \).
2... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$. | 1. We start with the equation \(4^x = x^4\). To find the solutions, we can consider both graphical and algebraic approaches.
2. First, let's identify the integer solutions. We test some integer values:
- For \(x = 2\):
\[
4^2 = 16 \quad \text{and} \quad 2^4 = 16 \quad \Rightarrow \quad 4^2 = 2^4
\]
... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball? | To solve this problem, we need to ensure that any light ray emanating from the point source intersects at least one of the solid balls. This means we need to cover the entire surface of a sphere centered at the light source with these solid balls.
1. **Understanding the Problem**:
- A point source of light emits ra... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that $\sigma(n) = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$. | 1. **Understanding the problem**: We need to find the largest positive integer \( n \) such that the sum of its divisors, denoted by \( \sigma(n) \), equals 28. The function \( \sigma(n) \) is defined as the sum of all positive divisors of \( n \), including \( n \) itself.
2. **Exploring the properties of \( \sigma(n... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square. ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$) | To find the largest positive integer \( n \) such that \( n\varphi(n) \) is a perfect square, we need to analyze the properties of \( n \) and its Euler's totient function \( \varphi(n) \).
1. **Express \( n \) in terms of its prime factorization:**
Let \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), where \( p_i \... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$. | 1. Let the roots of the polynomial \(x^3 + 3x + 1\) be \(\alpha, \beta, \gamma\). By Vieta's formulas, we have:
\[
\alpha + \beta + \gamma = 0,
\]
\[
\alpha\beta + \beta\gamma + \gamma\alpha = 3,
\]
\[
\alpha\beta\gamma = -1.
\]
2. We need to find the value of \(\frac{m}{n}\) where \(m\) and... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$ | To find the smallest possible value of \( AB \) given that \( A, B, \) and \( C \) are noncollinear points with integer coordinates and the distances \( AB, AC, \) and \( BC \) are integers, we will analyze the problem step-by-step.
1. **Upper Bound Consideration**:
Notice that a triangle with side lengths 3, 4, an... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit. | 1. **Lemma 1**: If, for some \( n \) we have \( x_n < \frac{1}{n} \), then for all \( m > n \) we have \( x_m < \frac{1}{n} \).
**Proof**:
- Assume \( x_n < \frac{1}{n} \).
- For \( m = n \), we have \( x_{n+1} = \left| x_n - \frac{1}{n} \right| \).
- Since \( 0 < x_n < \frac{1}{n} \), it follows that \( ... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of
\[AB^2 + 2AC^2 - 3AD^2.\] | 1. **Set up the coordinate system:**
- Place the origin at point \( E \), the foot of the altitude from \( A \).
- Let \( A = (0, a) \), \( B = (b, 0) \), and \( C = (3-b, 0) \).
- Point \( D \) is on \( BC \) such that \( BD = 2 \), so \( D = (2-b, 0) \).
2. **Calculate \( AB^2 \):**
\[
AB^2 = (0 - b)^... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$. | 1. **Problem Restatement**: We need to fill a \( (2n) \times (2n) \) table with entries of either \( 1 \) or \( -1 \) such that there are exactly \( 2n^2 \) entries of each kind. We are to determine the largest possible value of \( M \), where \( M \) is the minimum of the absolute values of all row sums and all column... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only? | 1. **Define the sequence and sum**:
Let \( x \) be the first number of the pucelana sequence. The sequence consists of 16 consecutive odd numbers starting from \( x \). Therefore, the sequence is:
\[
x, x+2, x+4, \ldots, x+30
\]
The sum of these 16 numbers can be written as:
\[
S = x + (x+2) + (x+4... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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