Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.

Consider a subset $A$ of $84$ elements of the set $\{1,\,2,\,\dots,\,169\}$ such that no two elements in the set add up to $169$. Show that $A$ contains a perfect square.

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

Determine the number of ordered triples $(a, b, c)$, with $0 \le a, b, c \le 10$ for which there exists $(x, y)$ such that $ax^2 + by^2 \equiv c$ (mod $11$)

Determine all $n$ positive integers such that exists an $n\times n$ where we can write $n$ times each of the numbers from $1$ to $n$ (one number in each cell), such that the $n$ sums of numbers in each line leave $n$ distinct remainders in the division by $n$, and the $n$ sums of numbers in each column leave $n$ distinct remainders in the division by $n$.

[b]p1.[/b] There are $4$ mirrors facing the inside of a $5\times 7$ rectangle as shown in the figure. A ray of light comes into the inside of a rectangle through $A$ with an angle of $45^o$. When it hits the sides of the rectangle, it bounces off at the same angle, as shown in the diagram. How many times will the ray of light bounce before it reaches any one of the corners $A$, $B$, $C$, $D$? A bounce is a time when the ray hit a mirror and reflects off it. [img]https://cdn.artofproblemsolving.com/attachments/1/e/d6ea83941cdb4b2dab187d09a0c45782af1691.png[/img] [b]p2.[/b] Jerry cuts $4$ unit squares out from the corners of a $45\times 45$ square and folds it into a $43\times 43\times 1$ tray. He then divides the bottom of the tray into a $43\times 43$ grid and drops a unit cube, which lands in precisely one of the squares on the grid with uniform probability. Suppose that the average number of sides of the cube that are in contact with the tray is given by $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p3.[/b] Compute $2021^4 - 4 \cdot 2023^4 + 6 \cdot 2025^4 - 4 \cdot 2027^4 + 2029^4$. [b]p4.[/b] Find the number of distinct subsets $S \subseteq \{1, 2,..., 20\}$, such that the sum of elements in $S$ leaves a remainder of $10$ when divided by $32$. [b]p5.[/b] Some $k$ consecutive integers have the sum $45$. What is the maximum value of $k$? [b]p6.[/b] Jerry picks $4$ distinct diagonals from a regular nonagon (a regular polygon with $9$-sides). A diagonal is a segment connecting two vertices of the nonagon that is not a side. Let the probability that no two of these diagonals are parallel be $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p7.[/b] The Olympic logo is made of $5$ circles of radius $1$, as shown in the figure [img]https://cdn.artofproblemsolving.com/attachments/1/7/9dafe6b72aa8471234afbaf4c51e3e97c49ee5.png[/img] Suppose that the total area covered by these $5$ circles is $a+b\pi$ where $a, b$ are rational numbers. Find $10a + 20b$. [b]p8.[/b] Let $P(x)$ be an integer polynomial (polynomial with integer coefficients) with $P(-5) = 3$ and $P(5) = 23$. Find the minimum possible value of $|P(-2) + P(2)|$. [b]p9. [/b]There exists a unique tuple of rational numbers $(a, b, c)$ such that the equation $$a \log 10 + b \log 12 + c \log 90 = \log 2025.$$ What is the value of $a + b + c$? [b]p10.[/b] Each grid of a board $7\times 7$ is filled with a natural number smaller than $7$ such that the number in the grid at the $i$th row and $j$th column is congruent to $i + j$ modulo $7$. Now, we can choose any two different columns or two different rows, and swap them. How many different boards can we obtain from a finite number of swaps? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares. [i](Proposed by Orestis Lignos, Greece)[/i]

A drawer has $5$ pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$? [i]Author: Ray Li[/i]

Show that $ 2005^{2005}$ is a sum of two perfect squares, but not a sum of two perfect cubes.

For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\]

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$. (1) $ $ $x\mid y^2-3, y\mid x^2-2$ (2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $

For prime number $p$, prove that there are integers $a$, $b$, $c$, $d$ such that for every integer $n$, the expression $n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)$ is a multiple of $p$.

For how many ordered pairs of positive integers $ (x,y)$ is $ x \plus{} 2y \equal{} 100$? $ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 49 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 99 \qquad \textbf{(E)}\ 100$

For each positive integer $n$, denote by $O(n)$ its greatest odd divisor. Given any positive integers $x_1 = a$ and $x_2 = b$, construct an innite sequence of positive integers as follows: $x_n = O(x_{n-1} + x_{n-2})$, where $n = 3,4,...$ (a) Prove that starting from some place, all terms of the sequence are equal to the same integer. (b) Express this integer in terms of $a$ and $b$.

Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Find all non-negative integers $a, b, c, d$ such that $7^a = 4^b + 5^c + 6^d$

The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows: $\circ$ The first member $s_1$ is 2. $\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation. $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation. $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation. The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$.\\ Does this sequence contain a) 2001, b) 2006?

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

Find all triples $ (a,b,c)$ of positive integers such that $ a^2\plus{}2^{b\plus{}1}\equal{}3^c$.

Let $n$ and $k$ be two positive integers. Prove that if $n$ is relatively prime with $30$, then there exist two integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2 - b^2 + k}{n}$ is an integer. Malik Talbi

Determine all positive integers $n\geq 2$ for which there exist integers $x_1,x_2,\ldots ,x_{n-1}$ satisfying the condition that if $0<i<n,0<j<n, i\neq j$ and $n$ divides $2i+j$, then $x_i<x_j$.

Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$