Let $ABC$ be an acute scalene triangle with incenter $I$ and orthocenter $H$. Let $M$ be the midpoint of $AB$. On the line $AH$ we consider points $D$ and $E$, such that the line $MD$ is parallel to $CI$ and $ME$ is perpendicular to $CI$. Prove that $AE=DH$.

For the real number $a$ it holds that there is exactly one square whose vertices are all on the graph with the equation $y = x^3 + ax$. Find the side length of this square.

Seven classmates are comparing their end-of-year grades in $ 12$ subjects. They observe that for any two of them, there is some subject out of the $ 12$ where the two students got different grades. It is possible to choose n subjects out of the $ 12$ such that if the seven students only compare their grades in these $n$ subjects, it will still be true that for any two, there is some subject out of the n where they got different grades. What is the smallest value of $n$ for which such a selection is surely possible? Note: In Hungarian high schools, students receive an integer grade from $ 1$ to $5$ in each subject at the end of the year.

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

Show that the circumradius $R$ of a triangle $ABC$ equals the arithmetic mean of the oriented distances from its incenter $I$ and three excenters $I_a,I_b, I_c$ to any tangent $\tau$ to its circumcircle. In other words, if $\delta(P)$ denotes the distance from a point $P$ to $\tau$, then with appropriate choices of signs, we have $$\delta(I) \pm \delta_(I_a) \pm \delta_(I_b) \pm \delta_(I_c) = 4R$$ Luis González

The incircle of $\triangle{ABC}$, with incentre $I$, meets $BC, CA$, and $AB$ at $D,E$, and $F$, respectively. The line $EF$ cuts the lines $BI$, $CI, BC$, and $DI$ at $K,L,M$, and $Q$, respectively. The line through the midpoint of $CL$ and $M$ meets $CK$ at $P$. (a) Determine $\angle{BKC}$. (b) Show that the lines $PQ$ and $CL$ are parallel.

For positive integers $a$ and $b$, consider the curve $x^a + y^b = 1$ over real numbers $x$, $y$ and let $S(a, b)$ be the sum $P$ of the number of $x$-intercepts and $y$-intercepts of this curve. Compute $\sum^{10}_{a=1}\sum^5_{b=1} S(a, b).$

In some finite set of positive numbers, each number is expressed as a linear combination of the rest with rational non-negative coefficients. Prove that the ratio of some two numbers in the set is rational.

In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$ [asy] unitsize(40); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0); draw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){ return (0.6*cos(t)+0.64,0.6*sin(t)+2.24); } draw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq); dot((0,0),ds); label("$O$", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label("$A$", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label("$B$", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label("$C$", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label("$E$", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label("$D$", (-0.92,0.7), NE*lsf); label("$\alpha$", (0.48,1.38),NE*lsf); label("$\beta$", (-0.02,0.94),NE*lsf); label("$\gamma$", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle); [/asy]

Spot spotlight located at vertex $B$ of an equilateral triangle $ABC$, illuminates angle $\alpha$. Find all such values of $\alpha$, not exceeding $60^o$, which at any position of the spotlight, when the illuminated corner is entirely located inside the angle $ABC$, from the illuminated and two unlit segments of side $AC$ can be formed into a triangle.

In quadrilateral $ABCD$, let $AB = 7$, $BC = 11$, $CD = 3$, $DA = 9$, $\angle BAD = \angle BCD = 90^o$, and diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. The ratio $\frac{BE}{DE} = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

In the diagram below, $\angle B = 43^\circ$ and $\angle D = 102^\circ$. Find $\angle A + \angle B + \angle C + \angle D + \angle E + \angle F$. [NEEDS DIAGRAM]

Rectangle $ABCD$ with $AB>BC$ has point $P$ inside of it and $Q$ outside of it, such that $PQCD$ is a parallelogram with $PD=AD$. Let $M$ be the midpoint of $CD$. Give that $\angle AMP=\angle BMQ$, prove that $AB=2BC$.

For each subset $S$ of $\mathbb{N}$, with $r_S (n)$ we denote the number of ordered pairs $(a,b)$, $a,b\in S$, $a\neq b$, for which $a+b=n$. Prove that $\mathbb{N}$ can be partitioned into two subsets $A$ and $B$, so that $r_A(n)=r_B(n)$ for $\forall$ $n\in \mathbb{N}$.

Let $P$ be a point on the circumcircle of triangle $ABC$. Let $A_1$ be the reflection of the orthocenter of triangle $PBC$ about the reflection of the perpendicular bisector of $BC$. Points $B_1$ and $C_1$ are defined similarly. Prove that $A_1,B_1,C_1$ are collinear.

There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.) Starting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/i] friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?

Let $ABC$ be an acute-angled triangle with circumscribed circle $k$ and centre of the circumscribed circle $O$. A line through $O$ intersects the sides $AB$ and $AC$ at $D$ and $E$.Denote by $B'$ and $C'$ the reflections of $B$ and $C$ over $O$, respectively. Prove that the circumscribed circles of $ODC'$ and $OEB'$ concur on $k$.

The extensions of the sides $AB$ and $CD$ of the trapezoid $ABCD$ intersect at point $E$. Denote by $H$ and $G$ the midpoints of $BD$ and $AC$. Find the ratio of the area $AEGH$ to the area $ABCD$.

Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.

A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip. Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and a) $n = 2008$, b) $n = 2009$.

A $1995 \times 1995$ square board is given. A coloring of the cells of the board is called [i]good [/i] if the cells can be rearranged so as to produce a colored square board that is symmetric with respect to the main diagonal. Find all values of $k$ for which any $k$-coloring of the given board is [i]good[/i].

A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of groups that can be selected. What is the remainder when $N$ is divided by $100$? $\textbf{(A)}\ 47 \qquad\textbf{(B)}\ 48 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 95 \qquad\textbf{(E)}\ 96$

In an election, there are a total of $12$ candidates. An election committee has $6$ members voting. It is known that at most two candidates voted by any two committee members are the same. Find the maximum number of committee members.

Let be a polynom $ P $ of grade at least $ 2 $ and let be two $ 2\times 2 $ complex matrices such that $$ AB-BA\neq 0=P(AB)-P(BA). $$ Prove that there is a complex number $ \alpha $ having the property that $ P(AB)=\alpha I_2. $ [i]Titu Andreescu[/i] and [i]Dorin Andrica[/i]