Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.

Let $a_1,a_2,a_3,\ldots$ be a sequence of all composite positive integers in increasing order. A sequence $b_1,b_2,b_3,\ldots$ is given for all positive integers $i$ by equation $$b_i=ia_1^2+(i-1)a_2^2+\ldots+2a_{i-1}^2+a_i^2$$ What is the maximum amount of consecutive elements of sequence $b_1,b_2,b_3,\ldots$ which can be divisible by $3$? [i]M. Shutro[/i]

A $2 \times 4 \times 8$ rectangular prism and a cube have the same volume. What is the difference between their surface areas?

([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.

The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.

We say a number is irie if it can be written in the form $1+\dfrac{1}{k}$ for some positive integer $k$. Prove that every integer $n \geq 2$ can be written as the product of $r$ distinct irie numbers for every integer $r \geq n-1$.

A rectangular grazing area is to be fenced off on three sides using part of a $ 100$ meter rock wall as the fourth side. Fence posts are to be placed every $ 12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $ 36$ m by $ 60$ m? \[ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16 \]

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$. Let $X$ and $Y$ be variable points on $AB$ and $AC$, respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$. Let $Z$ be the reflection of $A$ across $XY$. Prove that the circumcircle of $XYZ$ passes through a fixed point. (Dominik Burek, Poland)

In triangle $ABC$, let $I$ be the incentre, $O$ be the circumcentre, and $H$ be the orthocentre. It is given that $IO = IH$. Show that one of the angles of triangle $ABC$ must be equal to $60$ degrees.

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$

Let $0 = a_1 < a_2 < a_3 < \cdots < a_n < 1$ and $0 = b_1 < b_2 < b_3 \cdots < b_m < 1$ be real numbers such that for no $a_j$ and $b_k$ the relation $a_j + b_k = 1$ is satisfied. Prove that if the $mn$ numbers ${\ a_j + b_k : 1 \leq j \leq n , 1 \leq k \leq m \}}$ are reduced modulo $1$, then at least $m+n -1$ residues will be distinct.

a) Do there exist 5 circles in the plane such that each circle passes through exactly 3 centers of other circles? b) Do there exist 6 circles in the plane such that each circle passes through exactly 3 centers of other circles?

Let $n >3$ be an odd positive integer and $n=\prod_{i=1}^k p_i^{\alpha_i}$ where $p_i$ are primes and $\alpha_i$ are positive integers. We know that \[m=n(1-\frac{1}{p_1})(1-\frac{1}{p_2})(1-\frac{1}{p_3}) \cdots (1-\frac{1}{p_n}).\] Prove that there exists a prime $P$ such that $P|2^m -1$ but $P \nmid n.$

Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$. [i]Proposed by Isabella Grabski[/i]

Sam and Jonathan play a game where they take turns flipping a weighted coin, and the game ends when one of them wins. The coin has a $\frac89$ chance of landing heads and a $\frac19$ chance of landing tails. Sam wins when he flips heads, and Jonathan wins when he flips tails. Find the probability that Samwins, given that he takes the first turn. [i]Proposed by Samuel Tsui[/i]

[b]8.1[/b] Find all primes $p,q$ and $r$ such that $$pqr= 5(p + q + r).$$ [b]8.2 [/b] Prove that if $\overline{ab}/\overline{bc} = a/c$, then $$\overline{abb...bb}/\overline{bb...bbc} = a/c$$ (each number has $n$ digits). [b]8.3 / 9.1[/b] Construct a triangle with perimeter, altitude and angle at the base. [b]8.4. / 9.4[/b] Prove that the square of the sum of N distinct non-zero squares of integers is also the sum of $N $squares of non-zero integers. [b]8.5.[/b] In the quadrilateral $ABCD$ the diagonals $AC$ and $BD$ are drawn. Prove that if the circles inscribed in $ABC$ and $ ADC$ touch each other each other, then the circles inscribed in $BAD$ and in $BCD$ also touch each other. [b]8.6 / 9.6[/b] If the numbers $A$ and $n$ are coprime, then there are integers $X$ and $Y$ such that $ |X| <\sqrt{n}$, $|Y| <\sqrt{n} $ and $AX-Y$ is divided by $n$. Prove it. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.

Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy: \[ \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2,\\ f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases}\] where $b = \frac{1+c}{2+c}$, $c > 0$. Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$

In a non isosceles triangle $ ABC$ let $ w$ be the angle bisector of the exterior angle at $ C$. Let $ D$ be the point of intersection of $ w$ with the extension of $ AB$. Let $ k_A$ be the circumcircle of the triangle $ ADC$ and analogy $ k_B$ the circumcircle of the triangle $ BDC$. Let $ t_A$ be the tangent line to $ k_A$ in A and $ t_B$ the tangent line to $ k_B$ in B. Let $ P$ be the point of intersection of $ t_A$ and $ t_B$. Given are the points $ A$ and $ B$. Determine the set of points $ P\equal{}P(C )$ over all points $ C$, so that $ ABC$ is a non isosceles, acute-angled triangle.

One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.

Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$. Let $E$ and $F$ lie on a line passing through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, and $E$ and $F$ are different from $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.

Prove that there exist two subsets $D, K$ of $\mathbb{N}^2$ such that for any $4$-element sets $A_1, A_2$ we have $|A_1 \cap A_2|=1$ if and only if there exist $4$-element sets $A_3, A_4, \ldots$, such that $A_i \cap A_j=\emptyset$ for all $(i, j) \in D$ and $|A_i \cap A_j|=2$ for all $(i, j) \in K$.

A circle inscribed in triangle $ABC$ touches $BC$ at point $K$, $M$ is the midpoint of the altitude drawn on $BC$. The straight line $KM$ intersects the circle inscribed in $ABC$ for the second time at point $P$. Prove that the circle passing through $B$, $C$ and $P$ touches the circle inscribed in triangle $ABC$.