Let $I$ and $J$ be intervals. Let $\varphi,\psi:I\to\mathbb{R}$ be strictly increasing continuous functions and let $\Phi,\Psi:J\to\mathbb{R}$ be continuous functions. Suppose that $\varphi(x)+\psi(x)=x$ and $\Phi(u)+\Psi(u)=u$ holds for all $x\in I$ and $u\in J$. Show that if $f:I\to J$ is a continuous solution of the functional inequality $$f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),$$then $\Phi\circ f\circ \varphi^{-1}$ and $\Psi\circ f\circ \psi^{-1}$ are convex functions.

On the line which contains diameter $PQ$ of circle $k(S,r)$, point $A$ is chosen outside the circle such that tangent $t$ from point $A$ touches the circle in point $T$. Tangents on circle $k$ in points $P$ and $Q$ are $p$ and $q$, respectively. If $PT \cap q={N}$ and $QT \cap p={M}$, prove that points $A$, $M$ and $N$ are collinear.

Mary typed a six-digit number, but the two $1$s she typed didn't show. What appeared was $2002$. How many different six-digit numbers could she have typed? $\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }15\qquad\textbf{(E) }20$

Consider a triangle $ABC$ with $AB=4$, $BC=3$, and $AC=2$. Let $D$ be the midpoint of line $BC$. Find the length of $AD$.

If a convex set of points in the line has at least two diameters, say $AB$ and $CD$, prove that $AB$ and $CD$ have a common point.

Which of the five "T-like shapes" would be symmetric to the one shown with respect to the dashed line? [asy] unitsize(48); for (int a=0; a<3; ++a) { fill((2a+1,1)--(2a+.8,1)--(2a+.8,.8)--(2a+1,.8)--cycle,black); } draw((.8,1)--(0,1)--(0,0)--(1,0)--(1,.8)); draw((2.8,1)--(2,1)--(2,0)--(3,0)--(3,.8)); draw((4.8,1)--(4,1)--(4,0)--(5,0)--(5,.8)); draw((.2,.4)--(.6,.8),linewidth(1)); draw((.4,.6)--(.8,.2),linewidth(1)); draw((2.4,.8)--(2.8,.4),linewidth(1)); draw((2.6,.6)--(2.2,.2),linewidth(1)); draw((4.4,.2)--(4.8,.6),linewidth(1)); draw((4.6,.4)--(4.2,.8),linewidth(1)); draw((7,.2)--(7,1)--(6,1)--(6,0)--(6.8,0)); fill((6.8,0)--(7,0)--(7,.2)--(6.8,.2)--cycle,black); draw((6.2,.6)--(6.6,.2),linewidth(1)); draw((6.4,.4)--(6.8,.8),linewidth(1)); draw((8,.8)--(8,0)--(9,0)--(9,1)--(8.2,1)); fill((8,1)--(8,.8)--(8.2,.8)--(8.2,1)--cycle,black); draw((8.4,.8)--(8.8,.8),linewidth(1)); draw((8.6,.8)--(8.6,.2),linewidth(1)); draw((6,1.2)--(6,1.4)); draw((6,1.6)--(6,1.8)); draw((6,2)--(6,2.2)); draw((6,2.4)--(6,2.6)); draw((6.4,2.2)--(6.4,1.4)--(7.4,1.4)--(7.4,2.4)--(6.6,2.4)); fill((6.4,2.4)--(6.4,2.2)--(6.6,2.2)--(6.6,2.4)--cycle,black); draw((6.6,1.8)--(7,2.2),linewidth(1)); draw((6.8,2)--(7.2,1.6),linewidth(1)); label("(A)",(0,1),W); label("(B)",(2,1),W); label("(C)",(4,1),W); label("(D)",(6,1),W); label("(E)",(8,1),W); [/asy]

Let be two real numbers $ a<b $ and a function $ f:[a,b]\longrightarrow\mathbb{R} $ having the property that if the sequence $ \left(f\left( x_n \right)\right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]a)[/b] Prove that if $ f $ admits antiderivatives, then $ f $ is integrable. [b]b)[/b] Is the converse of [b]a)[/b] true? [i]Marcelina Popa[/i]

Determine all polynomials $P(x,y)$ with real coefficients such that \[P(x+y,x-y)=2P(x,y) \qquad \forall x,y\in\mathbb{R}.\]

Let $N$ be the number of ordered triples of 3 positive integers $(a,b,c)$ such that $6a$, $10b$, and $15c$ are all perfect squares and $abc = 210^{210}$. Find the number of divisors of $N$. [i]Proposed by Andy Xu[/i]

Let $D$ be a unique point on segment $BC$, in $ABC$. If $AD^2 = BD \cdot CD$, show that $AB + AC = \sqrt{2}BC$.

Let \(D\) be the midpoint of side \(BC\) of triangle \(ABC\). Let \(E, F\) be points on sides \(AB, AC\) respectively such that \(DE = DF\). Prove that \(AE + AF = BE + CF \iff \angle EDF = \angle BAC\).

Find the sum of all possible values of $ab$, given that $(a,b)$ is a pair of real numbers satisfying \[a + \dfrac2b = 9~\text{ and }~b + \dfrac2a = 1.\] $\textbf{(A) }\dfrac{10}9\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad\textbf{(E) }9$

Suppose that positive numbers $x$ and $y$ satisfy $x^{3}+y^{4}\leq x^{2}+y^{3}$. Prove that $x^{3}+y^{3}\leq 2.$

Two boxes contain some number of red, yellow, and blue balls. The first box has $3$ red, $4$ yellow, and $5$ blue balls, and the second box has $6$ red, $2$ yellow, and $7$ blue balls. There are two ways to select a ball from these boxes; one could first randomly choose a box and then randomly select a ball or one could put all the balls in the same box and simply randomly select a ball from there. How much greater is the probability of drawing a red ball using the second method than the first?

Prove that the number $99999+111111\sqrt{3}$ cannot be written in the form $(A+B\sqrt{3})^2$, where $A$ and $B$ are integers.

In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.

Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x - 1008)^2 \cdot (x- 1009)^2 = c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot ... \cdot 3\cdot 1)^4}{2^{2020}}$ .

What is the least number of rooks that can be placed on a standard $8 \times 8$ chessboard so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)

A frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only allowed to hop to point $(x,y)$ if $|y-x|\leq1$. Compute the number of distinct valid paths the frog can take.

Let $n$ be the smallest integer such that the sum of digits of $n$ is divisible by $5$ as well as the sum of digits of $(n+1)$ is divisible by $5$. What are the first two digits of $n$ in the same order.

Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $1,2,\ldots,n^2$ as digits. That is, the equation\[ r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0 \]is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m\ne 0$. We write \[ r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i} \]to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions \[ k=(a_3a_2a_1a_0)_{-3+i}~~~~a_3\ne 0. \]Find the sum of all such $k$.

In triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$, respectively and $D$ is the projection of $A$ into $BC$. Point $O$ is the circumcenter of $ABC$ and circumcircles of $BOC$, $DMN$ intersect at points $R, T$. Lines $DT$, $DR$ intersect line $MN$ at $E$ and $F$, respectively. Lines $CT$, $BR$ intersect at $K$. A point $P$ lies on $KD$ such that $PK$ is the angle bisector of $\angle BPC$. Prove that the circumcircles of $ART$ and $PEF$ are tangent. [i]Proposed by Mehran Talaei - Iran[/i]

Find the minimal positive integer $n$ such that no matter what $n$ distinct numbers from $1$ to $1000$ you choose, such that no two are divisible by a square of the same prime, one of the chosen numbers is a square of prime. [i]D. Zmiaikou[/i]

There are $n$ assassins numbered from $1$ to $n,$ and all assasins are initially alive. The assassins play a game in which they take turns in increasing order of number, with assassin $1$ getting the first turn, then assassin $2$, etc., with the order repeating after assassin $n$ has gone; if an assassin is dead when their turn comes up, then their turn is skipped and it goes to the next assassin in line. On each assassin’s turn, they can choose to either kill the assassin who would otherwise move next or to do nothing. Each assassin will kill on their turn unless the only option for guaranteeing their own survival is to do nothing. If there are $2023$ assassins at the start of the game, after an entire round of turns in which no one kills, how many assassins must remain?

The number $2019$ is written on a blackboard. Every minute, if the number $a$ is written on the board, Evan erases it and replaces it with a number chosen from the set $$ \left\{ 0, 1, 2, \ldots, \left\lceil 2.01 a \right\rceil \right\} $$ uniformly at random. Is there an integer $N$ such that the board reads $0$ after $N$ steps with at least $99\%$ probability?