For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$8^k=x^3+y^3+z^3-3xyz$$

Alex has a piece of cheese. He chooses a positive number a and cuts the piece into several pieces one by one. Every time he choses a piece and cuts it in the same ratio $1 : a$. His goal is to divide the cheese into two piles of equal masses. Can he do it if $(a) a$ is irrational? $(b) a$ is rational, $a \neq 1?$

Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

Given an acute angle triangle $ABC$ with circumcircle $\Gamma$ and circumcenter $O$. A point $P$ is taken on the line $BC$ but not on $[BC]$. Let $K$ be the reflection of the second intersection of the line $AP$ and $\Gamma$ with respect to $OP$. If $M$ is the intersection of the lines $AK$ and $OP$, prove that $\angle OMB+\angle OMC=180^{\circ}$.

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

Let $ ABC$ be a triangle with $ \angle BAC\equal{}60^{\circ}$. The incircle of $ ABC$ is tangent to $ AB$ at $ D$. Construct a circle with radius $ DA$ and cut the incircle of $ ABC$ at $ E$. If $ AF$ is an altitude, prove that $ AE\ge AF$.

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$. [i]Proposed by Evan Chen[/i]

We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system \begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*} is divisible by 10.

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$, where $n$ is a given positive integer.

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Points $P$ and $Q$ lie on diagonals $AC$ and $BD$, respectively and $\angle APD = \angle BQC$. Prove that $\angle AQD = \angle BPC$.

A ball of mass $M$ and radius $R$ has a moment of inertia of $I=\frac{2}{5}MR^2$. The ball is released from rest and rolls down the ramp with no frictional loss of energy. The ball is projected vertically upward off a ramp as shown in the diagram, reaching a maximum height $y_{max}$ above the point where it leaves the ramp. Determine the maximum height of the projectile $y_{max}$ in terms of $h$. [asy] size(250); import roundedpath; path A=(0,0)--(5,-12)--(20,-12)--(20,-10); draw(roundedpath(A,1),linewidth(1.5)); draw((25,-10)--(12,-10),dashed+linewidth(0.5)); filldraw(circle((1.7,-1),1),lightgray); draw((25,-1)--(-1.5,-1),dashed+linewidth(0.5)); draw((23,-9.5)--(23,-1.5),Arrows(size=5)); label(scale(1.1)*"$h$",(23,-6.5),2*E); [/asy] (A) $h$ (B) $\frac{25}{49}h$ (C) $\frac{2}{5}h$ (D) $\frac{5}{7}h$ (E) $\frac{7}{5}h$

The sum of non-negative numbers $ x $, $ y $ and $ z $ is $3$. Prove the inequality $$ {1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1. $$

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. What is the maximum value of the tension in the rod? $\textbf{(A) } mg\\ \textbf{(B) } 2mg\\ \textbf{(C) } mL\theta_0/T_0^2\\ \textbf{(D) } mg \sin \theta_0\\ \textbf{(E) } mg(3 - 2 \cos \theta_0)$

Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.

$ABCDE$ is convex pentagon. $m(\widehat{B})=m(\widehat{D})=90^\circ$, $m(\widehat{C})=120^\circ$, $|AB|=2$, $|BC|=|CD|=\sqrt3$, and $|ED|=1$. $|AE|=?$ $ \textbf{(A)}\ \frac{3\sqrt3}{2} \qquad\textbf{(B)}\ \frac{2\sqrt3}{3} \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \sqrt3 - 1 \qquad\textbf{(E)}\ \sqrt3 $

Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number \[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \] is a perfect square.

Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and define $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and $$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$. (Salvatore Tringali)

Find the $2016$th smallest positive integer that is a solution to $x^x\equiv x\pmod{5}$.

suppose that $a=3^{100}$ and $b=5454$. how many $z$s in $[1,3^{99})$ exist such that for every $c$ that $gcd(c,3)=1$, two equations $x^z\equiv c$ and $x^b\equiv c$ (mod $a$) have the same number of answers?($\frac{100}{6}$ points)

Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people. P1. Let $f$ be a function satisfying $f(x + 1) + f(x - 1) = \sqrt{2} f(x)$, for all reals $x$. If $f(x - 1) = a$ and $f(x) = b$, determine the value of $f(x + 4)$. [hide=Remarks]We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.[/hide] P2. The sequence of "Sanga" numbers is formed by the following procedure. i. Pick a positive integer $n$. ii. The first term of the sequence $(U_1)$ is $9n$. iii. For $k \geq 2$, $U_k = U_{k-1} - 17$. Sanga$[r]$ is the "Sanga" sequence whose smallest positive term is $r$. As an example, for $n = 3$, the "Sanga" sequence which is formed is $27, 10, -7, -24, -41, \ldots.$ Since the smallest positive term of such sequence is $10$, for $n = 3$, the sequence formed is called Sanga$[10]$. For $n \leq 100$, determine the sum of all $n$ which makes the sequence Sanga$[4]$. P3. The cube $ABCD.EFGH$ has an edge length of 6 cm. Point $R$ is on the extension of line (segment) $EH$ with $EH : ER = 1 : 2$, such that triangle $AFR$ cuts edge $GH$ at point $P$ and cuts edge $DH$ at $Q$. Determine the area of the region bounded by the quadrilateral $AFPQ$. [url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4[/url]. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed. P5. After pressing the start button, a game machine works according to the following procedure. i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen. ii. It shows the product of the seven chosen numbes on screen. iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even. iv. Shows the seven chosen numbers and their sum and products. v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.) Kiki says that this game is really easy since the probability of winning is greater than $90$%. Explain, whether you agree with Kiki.

Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.

Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB=11$ and $CD=19$. Point $P$ is on segment $AB$ with $AP=6$, and $Q$ is on segment $CD$ with $CQ=7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ=27$, find $XY$.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]