Let $p,q,r$ be distinct prime numbers and let \[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \] Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$. [i]Ioan Tomescu[/i]

A hexahedron $ABCDE$ is made of two regular congruent tetrahedra $ABCD$ and $ABCE.$ Prove that there exists only one isometry $\mathbf Z$ that maps points $A, B, C, D, E$ onto $B, C, A, E, D,$ respectively. Find all points $X$ on the surface of hexahedron whose distance from $\mathbf Z(X)$ is minimal.

Let $ABC$ be an acuted-angle triangle and let $H$ be it's orthocenter. Let $PQ$ be a segment through $H$ such that $P$ lies on $AB$ and $Q$ lies on $AC$ and such that $ \angle PHB= \angle CHQ$. Finally, in the circumcircle of $\triangle ABC$, consider $M$ such that $M$ is the mid point of the arc $BC$ that doesn't contain $A$. Prove that $MP=MQ$ Proposed by Eduardo Velasco/Marco Figueroa

Given a polygon with $n$ sides, we assign the numbers $0,1,...,n-1$ to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for $n = 5$. Notice that the numbers assigned to the sides are still in arithmetic progression. [img]https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png[/img] $\bullet$ Make the respective assignment for a $9$-sided polygon, and generalize for odd $n$. $\bullet$ Prove that this is not possible if $n$ is even.

Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.

There are \(n+1\) squares in a row, labelled from 0 to \(n\). Tony starts with \(k\) stones on square 0. On each move, he may choose a stone and advance the stone up to \(m\) squares where \(m\) is the number of stones on the same square (including itself) or behind it. Tony's goal is to get all stones to square \(n\). Show that Tony cannot achieve his goal in fewer than \(\frac{n}{1} + \frac{n}{2} + \cdots + \frac{n}{k}\) moves. [i]Proposed by Tony Wang[/i]

Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.

Max has $2015$ jars labeled with the numbers $1$ to $2015$ and an unlimited supply of coins. Consider the following starting configurations: (a) All jars are empty. (b) Jar $1$ contains $1$ coin, jar $2$ contains $2$ coins, and so on, up to jar $2015$ which contains $2015$ coins. (c) Jar $1$ contains $2015$ coins, jar $2$ contains $2014$ coins, and so on, up to jar $2015$ which contains $1$ coin. Now Max selects in each step a number $n$ from $1$ to $2015$ and adds $n$ to each jar [i]except to the jar $n$[/i]. Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar. (Birgit Vera Schmidt)

Find all pairs of integers $(x,y)$ such that $x \geq 0$ and \[ (6^x-y)^2 = 6^{x+1}-y. \]

Evaluate $$\sum_{k=0}^n (-1)^k \binom{n}{k} (x-k)^n.$$

Let $ ABC$ be a triangle. A circle $ P$ is internally tangent to the circumcircle of triangle $ ABC$ at $ A$ and tangent to $ BC$ at $ D$. Let $ AD$ meets the circumcircle of $ ABC$ agin at $ Q$. Let $ O$ be the circumcenter of triangle $ ABC$. If the line $ AO$ bisects $ \angle DAC$, prove that the circle centered at $ Q$ passing through $ B$, circle $ P$, and the perpendicular line of $ AD$ from $ B$, are all concurrent.

Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations: $$\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}$$

In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.

Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.

On Misha's new phone, a passlock consists of six circles arranged in a $2\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle on the way between two others (e.g. the top left and right circles cannot be adjacent). For example, the left path shown below is allowed but the right path is not. (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$. However, in the diagrams below, the paths are valid/invalid regardless of orientation.) How many passlocks are there consisting of all six circles? [asy] size(270); defaultpen(linewidth(0.8)); real r = 0.3, rad = 0.1, shift = 3.7; pen th = linewidth(5)+gray(0.2); for(int i=0; i<= 2;i=i+1) { for(int j=0; j<= 1;j=j+1) { fill(circle((i,j),r),gray(0.8)); fill(circle((i+shift,j),r),gray(0.8)); } draw((0,1)--(2-rad,1)^^(2,1-rad)--(2,rad)^^(2-rad,0)--(0,0),th); draw(arc((2-rad,1-rad),rad,0,90)^^arc((2-rad,rad),rad,270,360),th); draw((shift+1,0)--(shift+1,1-2*rad)^^(shift+1-rad,1-rad)--(shift+rad,1-rad)^^(shift+rad,1+rad)--(shift+2,1+rad),th); draw(arc((shift+1-rad,1-2*rad),rad,0,90)^^arc((shift+rad,1),rad,90,270),th); } [/asy]

You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is \( S \). Find all possible values of \( S \) that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

Let $(a_n)$ be a sequence of positive real numbers. Show that $$ \limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1$$ and prove that $1$ cannot be replaced by any larger number.

Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose?

Consider the following expression $$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$ Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).

Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.

Let $a,b,c \in \mathbb {R}^{+}$ and $abc=1$ prove that: $\frac {a+b}{(a+b+1)^2}+\frac {b+c}{(b+c+1)^2}+\frac {c+a}{(c+a+1)^2} \geq \frac {2}{a+b+c}$

Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that \begin{align*} x^2-yz &= |y-z|+1, \\ y^2-zx &= |z-x|+1, \\ z^2-xy &= |x-y|+1. \end{align*}

Let $p, u, m, a, c$ be positive real numbers satisfying $5p^5+4u^5+3m^5+2a^5+c^5=91$. What is the maximum possible value of: \[18pumac + 2(2 + p)^2 + 23(1 + ua)^2 + 15(3 + mc)^2?\]

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.