In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]

The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area. [asy] size(150); defaultpen(linewidth(0.8)); path square=unitsquare; draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square); [/asy]

We have a convex polygon $P$ in the plane and two points $S,T$ in the boundary of $P$, dividing the perimeter in a proportion $1:2$. Three distinct points in the boundary, denoted by $A,B,C$ start to move simultaneously along the boundary, in the same direction and with the same speed. Prove that there will be a moment in which one of the segments $AB, BC, CA$ will have a length smaller or equal than $ST$.

On $5 \times 5$ grid of lattice points, every point is uniformly randomly colored blue, red, or green. Find the expected number of monochromatic triangles T with vertices chosen from the lattice grid, such that some two sides of $T$ are parallel to the axis.

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]

In convex quadrilateral $ABCD$, $\angle BAC=\angle CAD$. $E$ lies on segment $CD$, $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC$.

The sequence of integers $\{ x_{n}\}_{n\ge1}$ is defined as follows: \[x_{1}=1, \;\; x_{n+1}=1+{x_{1}}^{2}+\cdots+{x_{n}}^{2}\;(n=1,2,3 \cdots).\] Prove that there are no squares of natural numbers in this sequence except $x_{1}$.

Jordan is in the center of a circle whose radius is $100$ meters and can move one meter at a time, however, there is a giant who at every step can force you to move in the opposite direction to the one he chose (it does not mean returning to the place of departure, but advance but in the opposite direction to the chosen one). Determine the minimum number of steps that Jordan must give to get out of the circle.

Let $\mathcal{N}_p$ stand for a $p$ dimensional random variable of standard normal distribution. For $a\in\mathbb{R}^p$, let $H_p(a)$ stand for the expectation $E|\mathcal{N}_p+a|$. For $p>1$, prove that $$H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r$$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, such that for arbitrary $x,y \in \mathbb{R}$: \[ (y+1)f(x)+f(xf(y)+f(x+y))=y.\]

Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\] for $t=1,2,\ldots,n-1$ [i]Vasile Brânzănescu[/i]

Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$, and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC}\, .$

Prove that there exists an unique sequence $ \left( c_n \right)_{n\ge 1} $ of real numbers from the interval $ (0,1) $ such that$$ \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } , $$ for all natural numbers $ m, $ and calculate $ \lim_{k\to\infty } kc_k^k. $ [i]Radu Pop[/i]

Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that \[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \] and \[ a_1 + \dots + a_n = b_1 + \dots + b_n. \] Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.

It is known that $a_1, a_2,...a_{108}$ are $108$ different positive integers not exceeding $2015$. Prove that there is a positive integer $k$ such that there are at least four different pairs $(i, j) $satisfying $a_i-a_j =k$.

Let $x_1$ and $x_2$ be the roots of the equation $x^2-px+1=0$ where $p>2$ is a prime number. Prove that $x_1^p+x_2^p$ is an integer divisible by $p^2$. [i]From the folklore[/i]

A positive integer is called [i]decimal palindrome[/i] if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$. Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.

$f(n,k)$ is defined by (1) $f(n,0) = f(n,n) = 1$ and (2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$. How many times do we need to use (2) to find $f(3991,1993)$?

In $3$-dimensional space there is given a point $O$ and a finite set $A$ of segments with the sum of lengths equal to $1988$. Prove that there exists a plane disjoint from $A$ such that the distance from it to $O$ does not exceed $574$.

A $7 \times 7$ square is made up of $16$ $1 \times 3$ tiles and $1$ $1 \times 1$ tile. Prove that the $1 \times 1$ tile lies either at the centre of the square or adjoins one of its boundaries .

Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$. Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$. Must $k$ be a Fibonacci number? [i]Proposed by Fedir Yudin.[/i]

Show that $1994$ divides $10^{900}-2^{1000}$.

Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why? Note: A regular [i]polyhedron [/i] is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with $12$ faces which are regular pentagons and a regular icosahedron is a polyhedron with $20$ faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere. The illustration below shows a dodecahdron and an icosahedron, not necessarily to scale. [img]https://cdn.artofproblemsolving.com/attachments/7/5/9873b42aacf04bb5daa0fe70d4da3bf0b7be38.png[/img]

Andrea and Lauren are $ 20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $ 1$ kilometer per minute. After $ 5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea? $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 80$