Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.

The bisector of the exterior angle at vertex $C$ of the triangle $ABC$ intersects the bisector of the interior angle at vertex $B$ in point $K$. Consider the diameter of the circumcircle of the triangle $BCK$ whose one endpoint is $K$. Prove that $A$ lies on this diameter.

In the night, stars in the sky are seen in different time intervals. Suppose for every $k$ stars ($k>1$), at least $2$ of them can be seen in one moment. Prove that we can photograph $k-1$ pictures from the sky such that each of the mentioned stars is seen in at least one of the pictures. (The number of stars is finite. Define the moments that the $n^{th}$ star is seen as $[a_n,b_n]$ that $a_n<b_n$.)

Points $A$ and $B$ move on circles centered at $O_A$ and $O_B$ such that $O_AA$ and $O_BB$ rotate at the same speed. Prove that vertex $C$ of the equilateral triangle $ABC$ moves along a certain circle at the same angular velocity. (The vertices of $ABC$ are oriented clockwise.)

There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?

We say that a triangle of sides $a,b,c$ is [i] virtual[/i] if such measures satisfy $$\begin{cases} a^{2024}+b^{2024}> c^{2024},\\ b^{2024}+c^{2024}> a^{2024},\\ c^{2024}+a^{2024}> b^{2024} \end{cases}$$ Find the number of ordered triples $(a,b,c)$ such that $a,b,c$ are integers between $1$ and $2024$ (inclusive) and $a,b,c$ are the sides of a [i]virtual [/i] triangle.

Given $2012$ stones divided into several groups, a [i]legal move[/i] is to merge two of the groups into one, as long as the size of the new group is less than or equal to $51$. Two players, $A$ and $B$, take turns making legal moves, starting with $A$. Initially, each stone is in a separate group. The player who cannot make a legal move on their turn loses. Determine which of the two players has a winning strategy and provide that strategy.

Every member of a certain parliament has not more than $3$ enemies. Prove that it is possible to divide it onto two subparliaments so, that everyone will have not more than one enemy in his subparliament. ($A$ is the enemy of $B$ if and only if $B$ is the enemy of $A$.)

Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is [asy] // note: diagram deliberately not to scale -- azjps void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); } size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3); real min = -0.6, step = 0.5; pair[] A, B; D(unitcircle); for(int i = 0; i < 3; ++i) { A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]); D(D(A[i])--D(B[i])); } MP("10",(A[0]+B[0])/2,N); MP("\sqrt{a}",(A[1]+B[1])/2,N); MP("14",(A[2]+B[2])/2,N); htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy] $\textbf{(A)}\ 144 \qquad \textbf{(B)}\ 156 \qquad \textbf{(C)}\ 168 \qquad \textbf{(D)}\ 176 \qquad \textbf{(E)}\ 184$

A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?

Charlie noticed his golden ticket was golden in two ways! In addition to being gold, it was a rectangle whose side lengths had ratio the golden ratio $\varphi = \tfrac{1+\sqrt{5}}{2}$. He then folds the ticket so that two opposite corners (vertices connected by a diagonal) coincide and makes a sharp crease (the ticket folds just as any regular piece of paper would). The area of the resulting shape can be expressed as $a + b \varphi$. What is $\tfrac{b}{a}$?

If in a convex quadrilateral $ ABCD, E$ and $ F$ are the midpoints of the sides $ BC$ and $ DA$ respectively. Show that the sum of the areas of the triangles $ EDA$ and $ FBC$ is equal to the area of the quadrangle.

A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour? (NN Konstantinov, Moscow)

Consider $n$ weights, $n \ge 2$, of masses $m_1, m_2, ..., m_n$, where $m_k$ are positive integers such that $1 \le m_ k \le k$ for all $k \in \{1,2,...,n\} $: Prove that we can place the weights on the two pans of a balance such that the pans stay in equilibrium if and only if the number $m_1 + m_2 + ...+ m_n$ is even. Estonian Olympiad

What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping? (A) $Mg$ (B) $\mu_k Mg$ (C) $\mu_k Mg/\mu_s$ (D) $(\mu_s + \mu_k)Mg$ (E) $Mg/\mu_s$

Let $a, b$, and $c$ be integers simultaneously satisfying the equations $4abc + a + b + c = 2018$ and $ab + bc + ca = -507$. Find $|a| + |b|+ |c|$.

On one of the sides of the $60$ degree angle with vertex $O$ a fixed point $F$ is marked. On the other side of the angle a point $A$ is chosen, and on the ray $OF$, but not the segment $OF$, a point $B$ such that $OA=FB$. On the segment $AB$ equilateral triangle $ABC$ and $ABD$ are built such that points $O$ and $C$ lie in the same half-plane with respect to $AB$, and $D$ in the other. a) Prove that the point $C$ does not depend on $A$. b) Prove that all points $D$ lie on a line.

Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.

$x$ and $y$ are real numbers such that $6 -x$, $3 + y^2$, $11 + x$, $14 - y^2$ are greater than zero. Find the maximum of the function $$f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}.$$

Suppose that $p>3$ is prime. Prove that the products of the primitive roots of $p$ between $1$ and $p-1$ is congruent to $1$ modulo $p$.

Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)

Ants, named Anna and Bob, are located at vertices \(A\) and \(B\) respectively of a cube \(ABCD A_1 B_1 C_1 D_1\), with a sugar cube placed at vertex \(C_1\). It is known that Bob can move at a speed of $20$ meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the sugar cube at \(C_1\) before Bob. [i]Proposed by Tamar Turashvili, Georgia [/i]

A square house is partitioned into an $n \times n$ grid, where each cell is a room. All neighboring rooms have a door connecting them, and each door can either be normalor inversive. If USJL walks over an inversive door, he would become inverted-USJL,and vice versa. USJL must choose a room to begin and walk pass each room exactly once. If it is inverted-USJL showing up after finishing, then he would be trapped for all eternity. Prove that USJL could always escape.

Let $(a_n)_{n=1}^{\infty}$ be a sequence with $a_n \in \{0,1\}$ for every $n$. Let $F:(-1,1) \to \mathbb{R}$ be defined by \[F(x)=\sum_{n=1}^{\infty} a_nx^n\] and assume that $F\left(\frac{1}{2}\right)$ is rational. Show that $F$ is the quotient of two polynomials with integer coefficients.

In $\triangle ABC$, $\angle C=90^{\circ},\angle B=30^{\circ}, AC=2$. $M$ is the midpoint of $AB$. Fold up $\triangle ACM$ along $CM$, satisfying that $|AB|=2\sqrt2$. The volume of triangular pyramid $A-BCM$ is________.