Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product $$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$ has at most $n$ distinct prime divisors. [i]Proposed by Géza Kós[/i]

A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.

In every $1\times1$ square of an $m\times n$ table we have drawn one of two diagonals. Prove that there is a path including these diagonals either from left side to the right side, or from the upper side to the lower side.

Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$?

Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula \[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\] Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.

Let $S = \{-17, -16, ..., 16, 17\}$. We call a subset $T$ of $S$ a good set if $-x \in T$ for all $x \in T$ and if $x, y, z \in T (x, y, z$ may be equal) then $x + y + z \ne 0$. Find the largest number of elements in a good set.

Michael wants to arrange a doubles tennis tournament among his friends. However, he has some peculiar conditions: the total number of matches should equal the total number of players, and every pair of friends should play as either teammates or opponents in at least one match. The number of players in a single match is four. What is the largest number of people who can take part in such a tournament?

If $a_1, a_2, \cdots, a_7$ are not necessarily distinct real numbers such that $1 < a_i < 13$ for all $i$, then show that we can choose three of them such that they are the lengths of the sides of a triangle.

Let $ABC$ be a triangle and let the tangent at $B{}$ to its circumcircle meet the internal bisector of the angle $A{}$ at $P{}$. The line through $P{}$ parallel to $AC$ meets $AB$ at $Q{}$. Assume that $Q{}$ lies in the interior of segment $AB$ and let the line through $Q{}$ parallel to $BC$ meet $AC$ at $X{}$ and $PC$ at $Y{}$. Prove that $PX$ is tangent to the circumcircle of the triangle $XYC$.

Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$ Is $N$ even or odd? Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.

If $b>1$, $x>0$ and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is $\text{(A)}\ \frac{1}{216} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ \text{not uniquely determined}$

Let $A$ be a set of functions $f : R\to R$. For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$. Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $ABCDEF$ be an equiangular hexagon. Let $P$ be the point that is a distance of 6 from $BC$, $DE$, and $FA$. If the distances from $P$ to $AB$, $CD$, and $EF$ are $8$, $11$, and $5$ respectively, find $(DE-AB)^2$. [i]Proposed by Andy Xu[/i]

Find the total length of the set of real numbers satisfying \[\frac{x^2 - 80x + 1500}{x^2 - 55x + 700} < 0.\]

Determine the largest possible number of groups one can compose from the integers $1,2,3,..., 19,20$, so that the product of the numbers in each group is a perfect square. (The group may contain exactly one number, in that case the product equals this number, each number must be in exactly one group.) (E. Barabanov, I. Voronovich)

In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? [asy] unitsize(150); pair A,B,C,D,E,F,G,H,J,K; A=(1,0); B=(0,0); C=(0,1); D=(1,1); draw(A--B--C--D--A); E=(2-sqrt(3),0); F=(2-sqrt(3),1); draw(E--F); G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1); K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0); draw(G--H--K--J--G); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N); label("$G$",G,E); label("$H$",H,N); label("$K$",K,W); label("$J$",J,S); [/asy] $ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $

A $9 \times 9$ chessboard with the standard checkered pattern has white squares at its four corners. What is the least number of rooks that can be placed on this board so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)

Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$. [i]Proposed by Nathan Xiong[/i]

The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?

Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$.

Find all positive integers $d$ for which there exists a degree $d$ polynomial $P$ with real coefficients such that there are at most $d$ different values among $P(0),P(1),P(2),\cdots,P(d^2-d)$ .

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f \colon \mathbb{N} \to \mathbb{Z}$ such that \[\left\lfloor \frac{f(mn)}{n} \right\rfloor=f(m)\] for all positive integers $m,n$. [i]Merlijn Staps[/i]

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Problem 4. Let$\left( A,+,\cdot \right)$ be a ring with the property that $x=0$ is the only solution of the ${{x}^{2}}=0,x\in A$ecuation. Let $B=\left\{ a\in A|{{a}^{2}}=1 \right\}$. Prove that: (a) $ab-ba=bab-a$, whatever would be $a\in A$ and $b\in B$. (b) $\left( B,\cdot \right)$ is a group