Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.

For nonnegative integer $n$, the following are true: [list] [*]$f(0)=0$ [*]$f(1)=1$ [*]$f(n)=f(n-\tfrac{m(m-1)}2)-f(\tfrac{m(m+1)}2-n)$ for integer $m$ satisfying $m\geq 2$ and $\tfrac{m(m-1)}2<n\leq\tfrac{m(m+1)}2$.[/list] Find the smallest integer $n$ such that $f(n)=4$.

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

Given is a pyramid $SABCD$ whose base is a convex quadrilateral $ ABCD $ with perpendicular diagonals $ AC $ and $ BD $, and the orthogonal projection of vertex $S$ onto the base is the point $0$ of the intersection of the diagonals of the base. Prove that the orthogonal projections of point $O$ onto the lateral faces of the pyramid lie on the circle.

Let $ a,b,c$ be positive real numbers. Prove that \[ \frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} . \]

Let $ABCD$ be a trapezium such that $AB\parallel CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB = BE$ and $AC = DE$. Prove that the internal angle bisector of $\angle BAC$ is perpendicular to $AD$.

A list of integers has mode $ 32$ and mean $ 22$. The smallest number in the list is $ 10$. The median $ m$ of the list is a member of the list. If the list member $ m$ were replaced by $ m \plus{} 10$, the mean and median of the new list would be $ 24$ and $ m \plus{} 10$, respectively. If $ m$ were instead replaced by $ m \minus{} 8$, the median of the new list would be $ m \minus{} 4$. What is $ m$? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 17\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 20$

Let $a,b,c$ be the sidelengths of a triangle. Prove that $$2<\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}<\sqrt{6}.$$

Inside a circular column with a bottom surface with radius of $6$, there are two balls with radius of $6$ as well. The distance betwen their centers is $13$. Draw a plane that is tangent to both spherical surface, intersect the circular column at a curve $C$. $C$ is a ellipse, then the sum of its short axis and long axis is________.

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?

A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

A sequence of natural numbers $a_1,a_2,...$ satisfies $a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2$ for $n \in N$. Prove that for every natural $n$ there exists a natural $m$ such that $a_na_{n+1} = a_m$.

10.2 Prove for all $x>0$ and $n\in\mathbb{N}$ the following inequality \[1+x^{n+1}\geq \frac{(2x)^n}{(1+x)^{n-1}}.\] ([i]A. Khrabrov[/i])

Real numbers $a,b,c$ with $0\le a,b,c\le 1$ satisfy the condition $$a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}.$$ Prove that $$\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.$$ [i] (Nora Gavrea)[/i]

Let $p(k)$ be the probability that if we choose a uniformly random subset $S$ of $\{1, 2, \ldots, 18\},$ then $|S| \equiv k \pmod{5}.$ Evaluate $$\sum_{k=0}^4 \left|p(k)-\frac{1}{5}\right|.$$

[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.

How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$?

A [i]permutation[/i] of the integers $1901, 1902, \cdots, 2000$ is a sequence $a_1, a_2, \cdots, a_{100}$ in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums \[s_1 = a_1,\;\;s_2 = a_1 + a_2,\;\;s_3 = a_1 + a_2 + a_3, \; \ldots\;, \; s_{100} = a_1 + a_2 + \cdots + a_{100}.\] How many of these permutations will have no terms of the sequence $s_1, \ldots, s_{100}$ divisible by three?

Prove that for every $ \vartheta$, $ 0<\vartheta<1$, there exist a sequence $ \lambda_n$ of positive integers and a series $ \sum_{n=1}^{\infty} a_n$ such that (i)$ \lambda_{n+1}-\lambda_n > (\lambda_n)^{\vartheta}$, (ii) $ \lim_{r\rightarrow 1^-} \sum_{n=1}^{\infty} a_nr^{\lambda_n}$ exists, (iii) $ \sum _{n=1}^{\infty} a_n$ is divergent. [i]P. Turan[/i]

a) In a $ XYZ$ triangle, the incircle tangents the $ XY $ and $ XZ $ sides at the $ T $ and $ W $ points, respectively. Prove that: $$ XT = XW = \frac {XY + XZ-YZ} {2} $$ Let $ ABC $ be a triangle and $ D $ is the foot of the relative height next to $ A. $ Are $ I $ and $ J $ the incentives from triangle $ ABD $ and $ ACD $, respectively. The circles of $ ABD $ and $ ACD $ tangency $ AD $ at points $ M $ and $ N $, respectively. Let $ P $ be the tangency point of the $ BC $ circle with the $ AB$ side. The center circle $ A $ and radius $ AP $ intersect the height $ D $ at $ K. $ b) Show that the triangles $ IMK $ and $ KNJ $ are congruent c) Show that the $ IDJK $ quad is inscritibed

Let $a, b, c>0$ such that $a^{2}+b^{2}+c^{2}=3abc.$ Prove the following inequality: \[ \frac{a}{b^{2}c^{2}}+\frac{b}{c^{2}a^{2}}+\frac{c}{a^{2}b^{2}}\geq\frac{9}{a+b+c} \]

Find the prime number $p$ for which $p + 2500$ is a perfect square.

Antonio and Bernardo play the following game. They are given two piles of chips, one with $m$ and the other with $n$ chips. Antonio starts, and thereafter they make in turn one of the following moves: (i) take a chip from one pile; (ii) take a chip from each of the piles; (ii) remove a chip from one of the piles and put it onto the other. Who cannot make any more moves, loses. Decide, as a function of $m$ and $n$ if one of the players has a winning strategy, and in the case of the affirmative answer describe that strategy.

Let $k>1$. Fix a circle $\omega$ with center $O$ and radius $r$, and fix a point $A$ with $OA=kr$. Let $AB$, $AC$ be tangents to $\omega$. Choose a variable point $P$ on the minor arc $BC$ in $\omega$. Lines $AB$ and $CP$ intersect at $X$ and lines $AC$ and $BP$ intersect at $Y$. The circles $(BPX)$ and $(CPY)$ meet at another point $Z$. Prove that the line $PZ$ always passes through a fixed point except for one value of $k>1$, and determine this value. [i]Proposed by Ivan Chan Kai Chin[/i]