Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob’s win? [i]Maxim Didin, Russia[/i]

$ABCDE$ is a convex pentagon and: \[BC=CD, \;\;\; DE=EA, \;\;\; \angle BCD=\angle DEA=90^{\circ}\] Prove, that it is possible to build a triangle from segments $AC$, $CE$, $EB$. Find the value of its angles if $\angle ACE=\alpha$ and $\angle BEC=\beta$.

Let $ABC$ be a scalene triangle, $AM$ be the median through $A$, and $\omega$ be the incircle. Let $\omega$ touch $BC$ at point $T$ and segment $AT$ meet $\omega$ for the second time at point $S$. Let $\delta$ be the triangle formed by lines $AM$ and $BC$ and the tangent to $\omega$ at $S$. Prove that the incircle of triangle $\delta$ is tangent to the circumcircle of triangle $ABC$.

In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2.$ Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$? [asy] size(250); defaultpen(fontsize(10pt)); pair A =origin; pair B = (4.75,0); pair E1=(0,3); pair F = (4.75,3); pair G = (5.95,4.2); pair C = (5.95,1.2); pair D = (1.2,1.2); pair H= (1.2,4.2); pair M = ((4.75+5.95)/2,3.6); draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H); draw(B--C); draw(F--G); draw(A--D--H--C--D,dashed); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,W); label("$E$",E1,W); label("$F$",F,SW); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,N); dot(A); dot(B); dot(E1); dot(F); dot(G); dot(C); dot(D); dot(H); dot(M); label("3",A/2+B/2,S); label("2",C/2+G/2,E); label("1",C/2+B/2,SE);[/asy] $\textbf{(A) } 1 \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{5}{3} \qquad \textbf{(E) } 2$

Let $n$ be a positive integer. Determine the size of the largest subset of $\{ -n, -n+1, \dots, n-1, n\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$.

Let $n \geq 2$ be a positive integer. Let $\mathcal{R}$ be a connected set of unit squares on a grid. A [i]bar[/i] is a rectangle of length or width $1$ which is fully contained in $\mathcal{R}$. A bar is [i]special[/i] if it is not fully contained within any larger bar. Given that $\mathcal{R}$ contains special bars of sizes $1 \times 2,1 \times 3,\ldots,1 \times n$, find the smallest possible number of unit squares in $\mathcal{R}$. [i]Srinivas Arun[/i]

Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.

Given is a number $a$ with 0 $\le \alpha \le \pi$. A sequence $c_0,c_1, c_2,...$ is defined as $$c_0=\cos \alpha$$ $$C_{n+1}=\sqrt{\frac{1+c_n}{2}} \,\, for \,\,\, n=0,1,2,...$$ Calculate $\lim_{n\to \infty}2^{2n+1}(1-c_n)$

Positive integers $a,b,c,d$ satisfy $a+c=10$ and \[\displaystyle S=\frac{a}{b} + \frac{c}{d} <1.\] Find the maximum value of $S$.

Find the length of the longer side of the rectangle on the picture, if the shorter side has length $1$ and the circles touch each other and the sides of the rectangle as shown. [img]https://cdn.artofproblemsolving.com/attachments/b/8/3986683247293bd089d8e83911309308ce0c3a.png[/img]

A circle of diameter $AB$ is given. There are points $C$ and $ D$ on this circle, on different sides of the diameter such that holds $AC <BC$ or $AC<AD$. The point $P$ lies on the segment $BC$ and $\angle CAP = \angle ABC$. The perpendicular from the point $C$ to the line $AB$ intersects the direction $BD$ at the point $Q$. Lines $PQ$ and $AD$ intersect at point $R$, and the lines $PQ$ and $CD$ intersect at point $T$. If $AR=RQ$, prove that the lines $AT$ and $PQ$ are perpendicular.

$\boxed{A1}$ Find all ordered triplets of $(x,y,z)$ real numbers that satisfy the following system of equation $x^3=\frac{z}{y}-\frac {2y}{z}$ $y^3=\frac{x}{z}-\frac{2z}{x}$ $z^3=\frac{y}{x}-\frac{2x}{y}$

The set $A$ has exactly $n>4$ elements. Ann chooses $n+1$ distinct subsets of $A$, such that every subset has exactly $3$ elements. Prove that there exist two subsets chosen by Ann which have exactly one common element.

Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.

Which of these has the smallest maxima on positive real numbers? $\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$

Define $f(n)$ to be the sum of positive integers $k$ less than or equal to $n$ such that $\gcd(n, k)$ is prime. Find $f(2024)$.

For a real-valued function $f(x,y)$ of two positive real variables $x$ and $y$, define $f$ to be [i]linearly bounded[/i] if and only if there exists a positive number $K$ such that $|f(x,y)| < K(x+y)$ for all positive $x$ and $y.$ Find necessary and sufficient conditions on the real numbers $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is linearly bounded.

Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be $ \text{(A)}\ 10\qquad\text{(B)}\ 11\qquad\text{(C)}\ 19\qquad\text{(D)}\ 20\qquad\text{(E)}\ 25 $

Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.

We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$

The circumcentre $O$ of a given cyclic quadrilateral $ABCD$ lies inside the quadrilateral but not on the diagonal $AC$. The diagonals of the quadrilateral intersect at $I$. The circumcircle of the triangle $AOI$ meets the sides $AD$ and $AB$ at points $P$ and $Q$, respectively; the circumcircle of the triangle $COI$ meets the sides $CB$ and $CD$ at points $R$ and $S$, respectively. Prove that $PQRS$ is a parallelogram.

If $r$ is a rational number, let $f(r) = \left( \frac{1-r^2}{1+r^2} , \frac{2r}{1+r^2} \right)$. Then the images of $f$ forms a curve in the $xy$ plane. If $f(1/3) = p_1$ and $f(2) = p_2$, what is the distance along the curve between $p_1$ and $p_2$?

Let real angles $\theta_1, \theta_2, \theta_3, \theta_4$ satisfy \begin{align*} \sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4 &= 0, \\ \cos\theta_1+\cos\theta_2+\cos\theta_3+\cos\theta_4 &= 0. \end{align*} If the maximum possible value of the sum \[\sum_{i<j}\sqrt{1-\sin\theta_i\sin\theta_j-\cos\theta_i\cos\theta_j}\] for $i, j \in \{1, 2, 3, 4\}$ can be expressed as $a+b\sqrt{c}$, where $c$ is square-free and $a,b,c$ are positive integers, find $a+b+c$ [i]Proposed by Alex Li[/i]

Find all polynomials $P(x)$ with integer coefficients, such that for any positive integer $n$ number $P(n)$ is a positive integer and a divisor of $n!$. [i]Proposed by Mykyta Kharin[/i]

Let $ ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal. [i](Karl Czakler)[/i]