Determine all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying $$f(f(x+y)-f(x-y))=y^2f(x)$$ for all $x, y \in \mathbb{R}$.

$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that \[(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3\]

Find all integers $ a$, $ b$, $ n$ greater than $ 1$ which satisfy \[ \left(a^3 \plus{} b^3\right)^n \equal{} 4(ab)^{1995} \]

A geometric sequence consists of $11$ terms. The arithmetic mean of the first $6$ terms is $63$, and the arithmetic mean of the last $6$ terms is $2016$. Find the $7$th term in the sequence. [i]Proposed by Powell Zhang[/i]

Let $\Omega$ be the circumscribed circle of the acute triangle $ABC$ and $ D $ a point the small arc $BC$ of $\Omega$. Points $E$ and $ F $ are on the sides $ AB$ and $AC$, respectively, such that the quadrilateral $CDEF$ is a parallelogram. Point $G$ is on the small arc $AC$ such that lines $DC$ and $BG$ are parallel. Prove that the angles $GFC$ and $BAC$ are equal.

Points $A, B$, and $C$ are on a line in this order. Points $D$ and $E$ lie on the same side of this line, in such a way that triangles $ABD$ and $BCE$ are equilateral. The segments $AE$ and $CD$ intersect in point $S$. Prove that $\angle ASD = 60^o$. [asy] unitsize(1.5 cm); pair A, B, C, D, E, S; A = (0,0); B = (1,0); C = (2.5,0); D = dir(60); E = B + 1.5*dir(60); S = extension(C,D,A,E); fill(A--B--D--cycle, gray(0.8)); fill(B--C--E--cycle, gray(0.8)); draw(interp(A,C,-0.1)--interp(A,C,1.1)); draw(A--D--B--E--C); draw(A--E); draw(C--D); draw(anglemark(D,S,A,5)); dot("$A$", A, dir(270)); dot("$B$", B, dir(270)); dot("$C$", C, dir(270)); dot("$D$", D, N); dot("$E$", E, N); dot("$S$", S, N); [/asy]

There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?

Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and \[ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases} \] Find all values of $a$ such that the sequence is periodical (starting from the beginning).

Let $n>3$ be a natural number . Prove that \[\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3}<\frac{1}{12}.\]

Let $ABC$ be an equilateral triangle. $N$ is a point on the side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side $AB$ such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$ such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$

For each prime $p,$ a polynomial $P(x)$ with rational coefficients is called $p$-[i]good[/i] if and only if there exist three integers $a,b,$ and $c$ such that $0 \le a < b < c < \tfrac{p}{3}$ and $p$ divides all the numerators of $P(a), P(b),$ and $P(c),$ when written in simplest form. Compute the number of ordered pairs $(r,s)$ of rational numbers such that the polynomial $x^3+10x^2+rx+s$ is $p$-good for infinitely many primes $p.$

Prove inequality, with $a$ and $b$ nonnegative real numbers: $\frac{a+b}{1+a+b}\leq \frac{a}{1+a} + \frac{b}{1+b} \leq \frac{2(a+b)}{2+a+b}$

$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.

The internal bisector of angle $A$ in a triangle $ABC$ with $AC > AB$ meets the circumcircle $\Gamma$ of the triangle in $D$. Join$D$ to the center $O$ of the circle $\Gamma$ and suppose that $DO$ meets $AC$ in $E$, possibly when extended. Given that $BE$ is perpendicular to $AD$, show that $AO$ is parallel to $BD$.

Given $x,y,z\in \mathbb{R^+}$. Find all sets of $x,y,z$ that correspond to $$x+y+z=x^2+y^2+z^2+18xyz=1$$ [i](Zhuge Liang)[/i]

A number $ x$ is uniformly chosen on the interval $ [0,1]$, and $ y$ is uniformly randomly chosen on $ [\minus{}1,1]$. Find the probability that $ x>y$.

Let $ G=(V,E)$ be a simple graph. a) Let $ A,B$ be a subsets of $ E$, and spanning subgraphs of $ G$ with edges $ A,B,A\cup B$ and $ A\cap B$ have $ a,b,c$ and $ d$ connected components respectively. Prove that $ a+b\leq c+d$. We say that subsets $ A_1,A_2,\dots,A_m$ of $ E$ have $ (R)$ property if and only if for each $ I\subset\{1,2,\dots,m\}$ the spanning subgraph of $ G$ with edges $ \cup_{i\in I}A_i$ has at most $ n-|I|$ connected components. b) Prove that when $ A_1,\dots,A_m,B$ have $ (R)$ property, and $ |B|\geq2$, there exists an $ x\in B$ such that $ A_1,A_2,\dots,A_m,B\backslash\{x\}$ also have property $ (R)$. Suppose that edges of $ G$ are colored arbitrarily. A spanning subtree in $ G$ is called colorful if and only if it does not have any two edges with the same color. c) Prove that $ G$ has a colorful subtree if and only if for each partition of $ V$ to $ k$ non-empty subsets such as $ V_1,\dots,V_k$, there are at least $ k\minus{}1$ edges with distinct colors that each of these edges has its two ends in two different $ V_i$s. d) Assume that edges of $ K_n$ has been colored such that each color is repeated $ \left[\frac n2\right]$ times. Prove that there exists a colorful subtree. e) Prove that in part d) if $ n\geq5$ there is a colorful subtree that is non-isomorphic to $ K_{1,n-1}$. f) Prove that in part e) there are at least two non-intersecting colorful subtrees.

An observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it takes $5$ seconds for the first car to pass the observer. How long will it take for the $10\text{th}$ car to pass? $\textbf{(A)} \hspace{1mm} 1.07s\\ \textbf{(B)} \hspace{1mm } 0.98s\\ \textbf{(C)}\hspace{1mm} 0.91s\\ \textbf{(D)}\hspace{1mm} 0.86s\\ \textbf{(E)}\hspace{1mm} 0.81s$

Given triangle $ABC$ and the points$D,E\in \left( BC \right)$, $F,G\in \left( CA \right)$, $H,I\in \left( AB \right)$ so that $BD=CE$, $CF=AG$ and $AH=BI$. Note with $M,N,P$ the midpoints of $\left[ GH \right]$, $\left[ DI \right]$ and $\left[ EF \right]$ and with ${M}'$ the intersection of the segments $AM$and $BC$. a) Prove that $\frac{B{M}'}{C{M}'}=\frac{AG}{AH}\cdot \frac{AB}{AC}$. b) Prove that the segments$AM$, $BN$ and $CP$ are concurrent.

Let $a<b<c<d$ be integers. If one of the roots of the equation $(x-a)(x-b)(x-c)(x-d)-9$ is $x=7$, what is $a+b+c+d$? $ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 42 \qquad\textbf{(E)}\ 63 $

Let be four real numbers $ x,y,z,t $ satisfying the following system: $$ \left\{ \begin{matrix} \sin x+\sin y+\sin z +\sin t =0 \\ \cos x+\cos y+\cos z+\cos t=0 \end{matrix} \right. $$ Prove that $$ \sin ((1+2k)x) +\sin ((1+2k)y) +\sin ((1+2k)z) +\sin ((1+2k)t) =0, $$ for any integer $ k. $ [i]Aurel Bârsan[/i]

Call a positive integer $k$ $\textit{pretty}$ if for every positive integer $a$, there exists an integer $n$ such that $n^2+n+k$ is divisible by $2^a$ but not $2^{a+1}$. Find the remainder when the $2021$st pretty number is divided by $1000$. [i]Proposed by i3435[/i]

We say that a positive integer $n$ is $m$[i]-expressible[/i] if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$. Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible? [i]Proposed by Krit Boonsiriseth[/i]

Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$ r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of $r+r_1+ r_2$.

At the $ABC$ triangle the midpoints of $BC, AC, AB$ are respectively $D, E, F$ and the triangle tangent to the incircle at $G$, $H$ and $I$ in the same order.The midpoint of $AD$ is $J$. $BJ$ and $AG$ intersect at point $K$. The $C-$centered circle passing through $A$ cuts the $[CB$ ray at point $X$. The line passing through $K$ and parallel to the $BC$ and $AX$ meet at $U$. $IU$ and $BC$ intersect at the $P$ point. There is $Y$ point chosen at incircle. $PY$ is tangent to incircle at point $Y$. Prove that $D, E, F, Y$ are cyclic.