Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.

Given a circle $G$ with center $O$ and radius $r$. Let $AB$ be a fixed diameter of $G$. Let $K$ be a fixed point of segment $AO$. Denote by $t$ the line tangent to at $A$. For any chord $CD$ (other than $AB$) passing through $K$. Let $P$ and $Q$ be the points of intersection of lines $BC$ and $BD$ with $t$. Prove that the product $AP\cdot AQ$ remains costant as the chord $CD$ varies.

The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular. (D. Tereshin)

We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers.

A word is a sequence of n letters of the alphabet {a, b, c, d}. A word is said to be complicated if it contains two consecutive groups of identic letters. The words caab, baba and cababdc, for example, are complicated words, while bacba and dcbdc are not. A word that is not complicated is a simple word. Prove that the numbers of simple words with n letters is greater than $2^n$, if n is a positive integer.

Prove that the polynomial $P(X) = (X^2-12X +11)^4+23$ can not be written as the product of three non-constant polynomials with integer coefficients. (Le Anh Vinh)

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Let $a, b \in G.$ Suppose that $ab^{2}= b^{3}a$ and $ba^{2}= a^{3}b.$ Prove that $a = b = 1.$

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$

A line drawn from the vertex $A$ of the equilateral triangle $ABC$ meets the side $BC$ at $D$ and the circumcircle of the triangle at point $Q$. Prove that $\frac{1}{QD} = \frac{1}{QB} + \frac{1}{QC}$.

The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?

Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.

(T.Golenishcheva-Kutuzova, B.Frenkin, 8--11) a) Prove that for $ n > 4$, any convex $ n$-gon can be dissected into $ n$ obtuse triangles.

Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.

There are $N$ red cards and $N$ blue cards. Each card has a positive integer between $1$ and $N$ (inclusive) written on it. Prove that we can choose a (non-empty) subset of the red cards and a (non-empty) subset of the blue cards, so that the sum of the numbers on the chosen red cards equals the sum of the numbers on the chosen blue cards.

Let $a, b, c$ be positive integers. Suppose that $(a + b)(a + c) = 77$ and $(a + b)(b + c) = 56$. Find $(a + c)(b + c)$.

Let $m, n, k$ be positive integers such that $k\le mn$. Let $S$ be the set consisting of the $(m + 1)$-by-$(n + 1)$ rectangular array of points on the Cartesian plane with coordinates $(i, j)$ where $i, j$ are integers satisfying $0\le i\le m$ and $0\le j\le n$. The diagram below shows the example where $m = 3$ and $n = 5$, with the points of $S$ indicated by black dots: [asy] unitsize(1cm); int m=3; int n=5; int xmin=-2; int xmax=7; for (int i=xmin+1; i<=xmax-1; i+=1) { draw((xmin+0.5,i)--(xmax-0.5,i),gray); draw((i,xmin+0.5)--(i,xmax-0.5),gray); } draw((xmin-0.25,0)--(xmax+0.25,0),black,Arrow(2mm)); draw((0,xmin-0.25)--(0,xmax+0.25),black,Arrow(2mm)); for (int i=0; i<=m; ++i) { for (int j=0; j<=n; ++j) { fill(shift(i,j)*scale(.1)*unitcircle); }} label("$x$",(xmax+0.25,0),E); label("$y$",(0,xmax+0.25),N); [/asy]

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$

Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions: [list] [*] $a_1>a_2>\ldots>a_n$; [*] $\gcd (a_1,a_2,\ldots,a_n)=1$; [*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]

Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.

$ABCD$ is a quadrilateral, $E,F,G,H$ are midpoints of $AB,BC,CD,DA$. Find the point P such that $area (PHAE) = area (PEBF) = area (PFCG) = area (PGDH)$.

(D.Shnol, 8--9) The bisectors of two angles in a cyclic quadrilateral are parallel. Prove that the sum of squares of some two sides in the quadrilateral equals the sum of squares of two remaining sides.

In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained? $ \textbf{(A) }2 \qquad \textbf{(B) }3\qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }24 \qquad $

Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.

Let $a$, $b$, $c$ be positive reals for which \begin{align*} (a+b)(a+c) &= bc + 2 \\ (b+c)(b+a) &= ca + 5 \\ (c+a)(c+b) &= ab + 9 \end{align*} If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$? $ \textbf{(A) }105 \qquad \textbf{(B) }112\frac{1}{2} \qquad \textbf{(C) }120 \qquad \textbf{(D) }135 \qquad \textbf{(E) }150 \qquad $