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________.

Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal. [img]https://cdn.artofproblemsolving.com/attachments/7/1/77cf4958931df1c852c347158ff1e2bbcf45fd.png[/img]

Let m be a positive interger, let $p$ be a prime, let $a_1=8p^m$, and let $a_n=(n+1)^{\frac{a_{n-1}}{n}}$, $n=2,3...$. Determine the primes $p$ for which the products $a_n(1-\frac{1}{a_1})(1-\frac{1}{a_2})...(1-\frac{1}{a_n})$, $n=1,2,3...$ are all integral.

Let $I$ be the incenter of a non-equilateral triangle $ABC$ and $T_1$, $T_2$, and $T_3$ be the tangency points of the incircle with the sides $BC$, $CA$ and $AB$, respectively. Prove that the orthocenter of triangle $T_1T_2T_3$ lies on the line $OI$, where $O$ is the circumcenter of triangle $ABC$. [i]Proposed by Georgi Ganchev[/i]

The bisector of angle $A$ of triangle $ABC$ ($AB > AC$) meets its circumcircle at point $P$. The perpendicular to $AC$ from $C$ meets the bisector of angle $A$ at point $K$. A cừcle with center $P$ and radius $PK$ meets the minor arc $PA$ of the circumcircle at point $D$. Prove that the quadrilateral $ABDC$ is circumscribed.

The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97$, then the shorter base is: $ \textbf{(A)}\ 94 \qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 89 $

Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?