Let $ a,b,c,d $ be four real numbers such that $ |ax^3+bx^2+cx+d|\le 1,\forall x\in [0,1] . $ Prove that $ |dx^2+cx^2+bx+a|\le 9/2,\forall x\in [0,1] . $ [i]Lavinia Savu[/i]

Prove that the number permutations $ \alpha$ of $ \{1,2,\dots,n\}$ s.t. there does not exist $ i<j<n$ s.t. $ \alpha(i)<\alpha(j\plus{}1)<\alpha(j)$ is equal to the number of partitions of that set.

Chip and Dale play on a $100 \times 100$ table. In the beginning, a chess king stands in the upper left corner of the table. At each move the king is moved one square right, down or right-down diagonally. A player cannot move in the direction used by his opponent in the previous move. The players move in turn, Chip begins. The player that cannot move loses. Which player has a winning strategy?

$P$ is a monic polynomial of odd degree greater than one such that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] (a) Prove that there are a finite number of natural numbers in range of $f$. (b) Prove that if $f$ is not constant then the equation $P(x)-x=0$ has at least two real solutions. (c) For each natural $n>1$ prove that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ and a monic polynomial of odd degree greater than one $P$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] and range of $f$ contains exactly $n$ different numbers. Time allowed for this problem was 105 minutes.

Let $d > 0$ be an arbitrary real number. Consider the set $S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}$. Prove that $S_{n}(d)$ has a maximum element.

Find all $10$ digit numbers $a_0a_1...a_9$ such that for each $k, a_k$ is the number of times that the digit $k$ appears in the number.

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 $