find all function $f:\mathbb{R} \to \mathbb{R}$ such that \[f(x^3-xf(y)^2)=xf(x+y)f(x-y)\] holds for all real number $x$, $y$. [i]Proposed by chengbilly[/i]

Circle with center $I$, inscribed in a triangle $ABC$ , touches the sides $BC$ and $AC$ at points $A_1$ and $B_1$ respectively. On rays $A_1I$ and $B_1I$, respectively, let be the points $A_2$ and $B_2$ such that $IA_2=IB_2=R$, where $R$is the radius of the circumscribed circle of the triangle $ABC$. Prove that: a) $AA_2 = BB_2 = OI$ where $O$ is the center of the circumscribed circle of the triangle $ABC$, b) lines $AA_2$ and $BB_2$ intersect on the circumcircle of the triangle $ABC$.

For all $n \in \mathbb{N}$, show that the number of integral solutions $(x, y)$ of \[x^{2}+xy+y^{2}=n\] is finite and a multiple of $6$.

There is an integer in each cell of a $2m\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessary and sufficient condition, in terms of $m$, $n$ and the initial numbers on the table, for which there exists a sequence of operations that makes all the numbers on the table equal.

On the plane a square is given, and $1993$ equilateral triangles are inscribed in this square. All vertices of any of these triangles lie on the border of the square. Prove that one can find a point on the plane belonging to the borders of no less than $499$ of these triangles. (N Sendrakyan)

Prove for positive reals $a,b,c$ that $(ab+bc+ca+1)(a+b)(b+c)(c+a) \ge 2abc(a+b+c+1)^2$

Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence HH) or flips tails followed by heads (the sequence TH). What is the probability that she will stop after flipping HH?

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD =DP = 3$, and $GC = 4$, find $AB^2$. [i]Proposed by Muztaba Syed[/i]

In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

Determine the smallest positive integer $n \ge 3$ for which \[ A \equiv 2^{10n} \pmod{2^{170}} \] where $A$ denotes the result when the numbers $2^{10}$, $2^{20}$, $\dots$, $2^{10n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A = 10241048576$).

Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.

Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .

For each positive integer $n$, let $s_n$ be the number of permutations $(a_1, a_2, \cdots, a_n)$ of $(1, 2, \cdots, n)$ such that $\dfrac{a_1}{1} + \dfrac{a_2}{2} + \cdots + \dfrac{a_n}{n}$ is a positive integer. Prove that $s_{2n} \ge n$ for all positive integer $n$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

In fencing, you win a round if you are the first to reach $15$ points. Suppose that when $A$ plays against $B$, at any point during the round, $A$ scores the next point with probability $p$ and $B$ scores the next point with probability $q=1-p$. (However, they never can both score a point at the same time.) Suppose that in this round, $A$ already has $14-k$ points, and $B$ has $14-\ell$ (where $0\le k,\ell\le 14$). By how much will the probability that $A$ wins the round increase if $A$ scores the next point?

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.

Determine what day of the week day was: June $6$, $1944$. Note: Leap years are those that are multiples of $4$ and do not end in $00$ or that are multiples of $400$, for example $1812$, $1816$, $1820$, $1600$, $2000$, but $1800$, $1810$, $2100$ are not leaps. Giving the answer without any mathematical justification will not award points.

Prove that for all $x,y,z>0$, the inequality \[\frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}} \ge 5 \sqrt[3]{\frac{xyz}{16}}\] holds. Determine if equality can hold and if so, in which cases it occurs.

Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$. [i]Proposed by Oleksiy Masalitin[/i]

Factor the polynomial $x^3+x^2z+xyz+y^2z-y^3$.

Nigli looks at the $6$ pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting $6$ numbers? [asy] import graph; size(8cm); // Draw the outer circle draw(circle((0,0), 1)); // Add the hour notches for (int i = 1; i <= 12; ++i) { real angle = (90 - i * 30) * pi / 180; pair outer = (cos(angle), sin(angle)); // Outer point of the notch pair inner = 0.9 * outer; // Inner point of the notch draw(inner -- outer); // Draw the notch // Add the hour numbers pair textPos = 1.15 * outer; // Position slightly outside the circle label(format("%d", i), textPos, align=(0,0)); } // Calculate the positions for 2 and 8 real angle2 = (90 - 2 * 30) * pi / 180; // 2 o'clock position real angle8 = (90 - 8 * 30) * pi / 180; // 8 o'clock position pair pos2 = (cos(angle2), sin(angle2)); // Position for 2 o'clock pair pos8 = (cos(angle8), sin(angle8)); // Position for 8 o'clock // Draw a dashed line from 2 to 8 draw(pos2 -- pos8, dashed); [/asy] $\textbf{(A) }5 \qquad\textbf{(B) } 6.5\qquad\textbf{(C) }8\qquad\textbf{(D) }9.5 \qquad\textbf{(E) }12$\\

Let $0 < a_1 < a_2 <... < a_{16} < 122$ be $16$ integers. Prove that there exist integers $(p, q, r, s)$, with $1 \le p < r \le s < q \le 16$, such that $a_p + a_q = a_r + a_s$. An additional $2$ points will be awarded for this problem, if you can find a larger bound than $122$ (with proof).

In an acute-angled triangle $ABC$, $M$ is a point on the side $BC$, the line $AM$ meets the circumcircle $\omega$ of $ABC$ at the point $Q$ distinct from $A$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to the diameter $AK$ of $\omega$ at the point $P$. Let $L$ be the point on $\omega$ distinct from $Q$ such that $PL$ is tangent to $\omega$ at $L$. Prove that $L,M$ and $K$ are collinear.

On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?