Let a convex quadrilateral $ ABCD$ fixed such that $ AB \equal{} BC$, $ \angle ABC \equal{} 80, \angle CDA \equal{} 50$. Define $ E$ the midpoint of $ AC$; show that $ \angle CDE \equal{} \angle BDA$ [i](Paolo Leonetti)[/i]

Find all pariwise distinct real numbers $x,y,z$ such that $\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}$. (It means, those three fractions make a permutation of $x, y$, and $z$.)

In a convex $12$-gon, all angles are equal. It is known that the lengths of some $10$ of its sides are equal to $1$, and the length of one more equals $2$. What can be the area of ​​this $12$-gon?

Suppose that $a^{\log_{b}c}+b^{\log_{c}a}=m$. Find the value of $c^{\log_{b}a}+a^{\log_{c}b}$ .

Find the sum of all positive integers $x$ such that $$|x^2-x-6|$$ has exactly $4$ positive integer divisors. [i]Proposed by Evan Chang (squareman), USA[/i]

Several pupils with different heights are standing in a row. If they were arranged according to their heights, such that the highest would stand on the right, then each pupil would move for at most 8 positions. Prove that every pupil has no more than 8 pupils lower then him on his right.

A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.

Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$

Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that $$(a - b)(b - c)(a- c) \le 2.$$ When does equality hold? [i](Karl Czakler)[/i]

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

Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number.

Mongolia TST 2011 Test 1 #2 Let $p$ be a prime number. Prove that: $\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$ (proposed by B. Batbayasgalan, inspired by Putnam olympiad problem) Note: I believe they meant to say $p>2$ as well.

Consider a regular triangular array of $n(n+1)/2$ points.Let $f(n)$ denote the number of equilateral triangles formed by taking some $3$ points in the array as vertices.Prove that $f(n)=\frac{(n-1)n(n+1)(n+2)}{24}$.

Prove that the following inequality holds with the exception of finitely many positive integers $n$: \[\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.\]

A square in the [i]xy[/i]-plane has area [i]A[/i], and three of its vertices have [i]x[/i]-coordinates $2,0,$ and $18$ in some order. Find the sum of all possible values of [i]A[/i].

Given a real number $q$, $1 < q < 2$ define a sequence $ \{x_n\}$ as follows: for any positive integer $n$, let \[x_n=a_0+a_1 \cdot 2+ a_2 \cdot 2^2 + \cdots + a_k \cdot 2^k \qquad (a_i \in \{0,1\}, i = 0,1, \cdots m k)\] be its binary representation, define \[x_k= a_0 +a_1 \cdot q + a_2 \cdot q^2 + \cdots +a_k \cdot q^k.\] Prove that for any positive integer $n$, there exists a positive integer $m$ such that $x_n < x_m \leq x_n+1$.

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$ Circle $\Gamma_{1}$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\Gamma_{2}$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\Gamma_{1}$ and $\Gamma_{2}$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that [b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $

Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. Then it's again Betül's turn. And the game continues until two stones remain. When two stones remain, Ali confesses the marked stone. At least in how many moves can Betül guarantee to find out the marked stone? $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 19 $

For how many integer values of $ k$ do the graphs of $ x^2 \plus{} y^2 \equal{} k^2$ and $ xy \equal{} k$ [u]not[/u] intersect? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

What is the period of the function $f(x)=\cos(\cos(x))$?

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Let $A_i$ and $A_i^{'}$ $(i=1,2,3,4)$ be diametrically opposite vertexes of a rectangular cuboid and $M{}$ a point inside it. Prove that $S\leq\sum_{i=1}^{4}MA_i\cdot MA_i^{'}$, where $S{}$ is the total surface area of the rectangular cuboid.

a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$ Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary. b) $ABC$ is a triangle where $\angle A$ is [b]obtuse[/b]. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$ 1) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ . 2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.

Draw a circle $k$ with diameter $\overline{EF}$, and let its tangent in $E$ be $e$. Consider all possible pairs $A,B\in e$ for which $E\in \overline{AB}$ and $AE\cdot EB$ is a fixed constant. Define $(A_1,B_1)=(AF\cap k,BF\cap k)$. Prove that the segments $\overline{A_1B_1}$ all concur in one point.