Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of \[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\] Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$

Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then $P(a^2+a) \geq a.P(a+1)$

Let $ABCD$ be a convex quadrilateral with $AB=AD, m\angle A = 40^{\circ}, m\angle C = 130^{\circ},$ and $m\angle ADC - m\angle ABC = 20^{\circ}.$ Find the measure of the non-reflex angle $\angle CDB$ in degrees.

Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are (a) congruent polygons? (b) regular polygons?

Consider the permutation of $1,2,...,n$, which we denote as $\{a_1,a_2,...,a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions: (1)$a_1=1$ (2)$|a_i-a_{i-1}|\le2, i=1,2,...,n-1$ what is the residue when we divide $f(2015)$ by $4$ ?

Acute triangle $\triangle ABC$ with $AB<AC$, circumcircle $\Gamma$ and circumcenter $O$ is given. Midpoint of side $AB$ is $D$. Point $E$ is chosen on side $AC$ so that $BE=CE$. Circumcircle of triangle $BDE$ intersects $\Gamma$ at point $F$ (different from point $B$). Point $K$ is chosen on line $AO$ satisfying $BK \perp AO$ (points $A$ and $K$ lie in different half-planes with respect to line $BE$). Prove that the intersection of lines $DF$ and $CK$ lies on $\Gamma$.

Except for the first two terms, each term of the sequence $1000, x, 1000-x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length?

Let $n$ and $k$ be positive integers. A simple graph $G$ does not contain any cycle whose length be an odd number greater than $1$ and less than $ 2k + 1$. If $G$ has at most $n + \frac{(k-1) (n-1) (n+2)}{2}$ vertices, prove that the vertices of $G$ can be painted with $n$ colors in such a way that any edge of $G$ has its ends of different colors.

$x, y, z$ are positive reals which their product is not smaller then their sum. Prove the inequality: $$\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9$$

Find the sum of all positive integers $a$ such that there exists an integer $n$ that satisfies the equation: \[a! \cdot 2^{\lfloor \sqrt{a} \rfloor}=n!.\] [i]Proposed by Ivy Zheng[/i]

A classroom has $30$ students and $30$ desks arranged in $5$ rows of $6$. If the class has $15$ boys and $15$ girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl?

Let $ U\subset\mathbb R^n$ be an open set, and let $ L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $ U\times (\mathbb R^n\setminus\{0\})$ and of $ C^2$-class Langrange function, such that for all $ p\in U$ the Gauss-curvature of the hyper surface \[ \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}\] is nowhere zero. Determine the extremals of $ L$ if it satisfies the following system \[ \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})\] of partial differetial equations, where $ y^k(u,v) : \equal{} v^k$ for $ (u,v)\in U\times\mathbb R^k$, $ v \equal{} (v^1,\dots,v^k)$.

Find the value of $645$. [i]Proposed by George Xing, et al.[/i]

Let \( I \) and \( O \) be the incenter and circumcenter of the right triangle \( ABC \) (\( \angle C = 90^\circ \)), and let \( K \) be the tangency point of the incircle with \( AC \). Let \( P \) and \( Q \) be the points where the circumcircle of triangle \( AOK \) intersects \( OC \) and the circumcircle of triangle \( ABC \), respectively. Prove that points \( C, I, P, \) and \( Q \) are concyclic. [i]Proposed by Mykhailo Sydorenko[/i]

Let $X=(x_1,x_2,......,x_9)$ be a permutation of the set $\{1,2,\ldots,9\}$ and let $A$ be the set of all such $X$ . For any $X \in A$, denote $f(X)=x_1+2x_2+\cdots+9x_9$ and $ M=\{f(X)|X \in A \}$. Find $|M|$. ($|S|$ denotes number of members of the set $S$.)