[b]Q.[/b]Let $ABC$ be a triangle with incenter $I$ and orthocenter $H$. Let $A_1,A_2$ lie on $\overline{BC}$ such that $\overline{IA_1}\perp \overline{IB},\overline{IA_2}\perp\overline{IC}$. $\overline{AA_1},\overline{AA_2}$ cut $\odot(ABC)$ again at $A_3,A_4$. $\overline{A_3A_4}$ cuts $\overline{BC}$ at $A_0$. Similarly, we have $B_0,C_0$. Prove that $A_0,B_0,C_0$ are collinear on a line which is perpendicular to line $\overline{IH}$.

Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? [i]A. Golovanov[/i]

Let $n = 2^k$. Prove that we can select $n$ integers from any $2n-1$ integers such that their sum is divisible by $n$.

The price of an article was increased $ p\%$. Later the new price was decreased $ p\%$. If the last price was one dollar, the original price was: $ \textbf{(A)}\ \frac {1 \minus{} p^2}{200} \qquad\textbf{(B)}\ \frac {\sqrt {1 \minus{} p^2}}{100} \qquad\textbf{(C)}\ \text{one dollar} \qquad\textbf{(D)}\ 1 \minus{} \frac {p^2}{10000 \minus{} p^2}$ $ \textbf{(E)}\ \frac {10000}{10000 \minus{} p^2}$

Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$, and $AB/CD=k>1$. The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$. The lines $KL$ and $MN$ meet at $P$, and the lines $KN$ and $LM$ meet at $Q$. If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$.

Inside an equilateral triangle there is a point whose distances from the sides of the triangle are $3, 4$ and $5$. Find the area of the triangle.

Given a triangle $ABC$, points $X,Y,$ and $Z$ are the midpoints of $BC,CA,$ and $AB$ respectively. The perpendicular bisector of $AB$ intersects line $XY$ and line $AC$ at $Z_1$ and $Z_2$ respectively. The perpendicular bisector of $AC$ intersects line $XZ$ and line $AB$ at $Y_1$ and $Y_2$ respectively. Let $K$ be a point such that $KZ_1 = KZ_2$ and $KY_1 = KY_2$. Prove that $KB=KC$.

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(x)f(y)=f(x-y)$. Find all possible values of $f(2017)$.

There are $n$ rectangles drawn on the rectangular sheet of paper with the sides of the rectangles parallel to the sheet sides. The rectangles do not have pairwise common interior points. Prove that after cutting out the rectangles the sheet will split into not more than $n+1$ part.

Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Prove that if a polynomial with integer coefficients takes a value equal to $1$ in absolute value at three different integer points, then it has no integer zeros.

Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.

[asy] real t=pi/8;real u=7*pi/12;real v=13*pi/12; real ct=cos(t);real st=sin(t);real cu=cos(u);real su=sin(u); draw(unitcircle); draw((ct,st)--(-ct,st)--(cos(v),sin(v))); draw((cu,su)--(cu,st)); label("A",(-ct,st),W);label("B",(ct,st),E); label("M",(cu,su),N);label("P",(cu,st),S); label("C",(cos(v),sin(v)),W); //Credit to Zimbalono for the diagram[/asy] In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to $\textbf{(A) }3x+2\qquad\textbf{(B) }3x+1\qquad\textbf{(C) }2x+3\qquad\textbf{(D) }2x+2\qquad \textbf{(E) }2x+1$

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$. Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $ .there are at least $2n-1$ pairs with the property that $a_i+a_j\ge 0$.

Let $A$ be the set $\{1,2,\ldots,n\}$, $n\geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ with the property: \[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \] [i]Marcel Chirita[/i]

A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that $i$ of each $a$ consecutive integers at least one is coloured red; $ii$ of each $b$ consecutive integers at least one is coloured blue; $iii$ of each $c$ consecutive integers at least one is coloured green; $iiii$ of each $d$ consecutive integers at least one is coloured purple. Determine all good quadruples with $a = 2.$

In an isosceles trapezoid $ABCD$ with bases $AD$ and $BC$, diagonals intersect at point $P$, and lines $AB$ and $CD$ intersect at point $Q$. $O_1$ and $O_2$ are the centers of the circles circumscribed around the triangles $ABP$ and $CDP$, $r$ is the radius of these circles. Construct the trapezoid ABCD given the segments $O_1O_2$, $PQ$ and radius $r$.

A semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as shown below. How many non-degenerate triangles can be drawn from the given points? [asy] draw((0,-2)--arc((0,0),1,0,180)--cycle); dot((-0.8775,-0.245)); dot((-0.735,-0.53)); dot((-0.5305,-0.939)); dot((-0.3875,-1.225)); dot((-0.2365,-1.527)); dot((0.155,-1.69)); dot((0.306,-1.388)); dot((0.4,-1.2)); dot((0.551,-0.898)); dot((0.837,-0.326)); dot(dir(25)); dot(dir(50)); dot(dir(65)); dot(dir(100)); dot(dir(115)); dot(dir(140)); [/asy]

The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.

Let $r$ be the remainder when $2017^{2025!}-1$ is divided by $2025!.$ Compute $\tfrac{r}{2025!}.$ (Note that $2017$ is prime.)

Let $p$ be a monic cubic polynomial such that $p(0)=1$ and such that all the zeroes of $p^\prime (x)$ are also zeroes of $p(x)$. Find $p$. Note: monic means that the leading coefficient is $1$.

Two players play the following game, using a round table $4$ feet in diameter, and a large pile of quarters. Each player can put in his turn one quarter on the table, but the one who cannot put a quarter (because there is no free space on the table) loses the game. Is there a winning strategy for the first or for the second player?