Let $\ell_1, \ell_2$ be two non-parallel lines and $d_1, d_2$ be positive reals. The set of points $X$, such that $dist(X, \ell_i)$ is a multiple of $d_i$ is called a $\textit{grid}$. Let $A$ be finite set of points, not all collinear. A triangle with vertices in $A$ is called $\textit{empty}$ if no points from $A$ lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that $A$ is the intersection of a grid $L$ and a convex polygon $F$.

Let $a,b,c>0.$ Prove that$$\frac{a^3+b^2c+bc^2}{bc}+\frac{b^3+c^2a+ca^2}{ca}+\frac{c^3+a^2b+ab^2}{ab}\geq 3(a+b+c)$$ $$\frac{bc}{a^3+b^2c+bc^2}+\frac{ca}{b^3+c^2a+ca^2}+\frac{ab}{c^3+a^2b+ab^2}\leq \frac{1}{3}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$$

Does there exist an infinite subset $S$ of the natural numbers such that for every $a$, $b \in S$, the number $(ab)^2$ is divisible by $a^2-ab+b^2$?

Let $f$ be a function on $(1,\ldots,n)$ that generates a permutation of $(1,\ldots,n)$. We call a fixed point of $f$ any element in the original permutation such that the element's position is not changed when the permutation is applied. Given that $n$ is a multiple of $4$, $g$ is a permutation whose fixed points are $\left(1,\ldots,\frac n2\right)$, and $h$ is a permutation whose fixed points consist of every element in an even-numbered position. What is the expected number of fixed points in $h(g(1,2,\ldots,104))$?

A $100 \times 100$ table is given. For each $k, 1 \le k \le 100$, the $k$-th row of the table contains the numbers $1,2,\dotsc,k$ in increasing order (from left to right) but not necessarily in consecutive cells; the remaining $100-k$ cells are filled with zeroes. Prove that there exist two columns such that the sum of the numbers in one of the columns is at least $19$ times as large as the sum of the numbers in the other column.

Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

Consider a convex quadrilateral $ABCD$ with $AB=CD$ and $\angle BAC=30^{\circ}$. If $\angle ADC=150^{\circ}$, prove that $\angle BCA= \angle ACD$.

Let $x$ and $y$ be real numbers with $x>y$ such that $x^2y^2+x^2+y^2+2xy=40$ and $xy+x+y=8$. Find the value of $x$.

Find all polynomials $p$ with real coefficients such that for all real $x$ $$(x-8)p(2x)=8(x-1)p(x).$$

Let $ A $ be the set of real solutions of the equation $ 3^x=x+2, $ and let be the set $ B $ of real solutions of the equation $ \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . $ Prove the validity of the following subpoints: [b]a)[/b] $ A\subset B. $ [b]b)[/b] $ B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} . $

Find the minimum value of \[\dfrac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$.

Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and \[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots \] Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.

Fix $n$. Given $n$ points on Cartesian plane such that no pair of points forms a segment that is parallel to either axes, a pair of points is said to be good if their segment gradient is positive. For which $k$ can there exist a set of $n$ points with exactly $k$ good pairs? [i](Proposed by Ivan Chan Kai Chin)[/i]

Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$. A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$. Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$. I. Voronovich

Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satises the following for all $x \in T$: $f(f(x)) = x$ $|f(x) - x| \ge 2$

Find the smallest positive integer that is relatively prime to each of $2, 20, 204,$ and $2048$. [i]Proposed by Yannick Yao[/i]

Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.

Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.

A set of $n$ points in space is given, no three of which are collinear and no four of which are co-planar (on a single plane), and each pair of points is connected by a line segment. Initially, all the line segments are colorless. A positive integer $b$ is given and Alice and Bob play the following game. In each turn Alice colors one segment red and then Bob colors up to $b$ segments blue. This is repeated until there are no more colorless segments left. If Alice colors a red triangle, Alice wins. If there are no more colorless segments and Alice hasn’t succeeded in coloring a red triangle, Bob wins. Neither player is allowed to color over an already colored line segment. 1. Prove that if $b < \sqrt{2n - 2} -\frac32$ , then Alice has a winning strategy. 2. Prove that if $b \ge 2\sqrt{n}$, then Bob has a winning strategy.

Given a concyclic quadrilateral $ABCD$ with circumcenter $O$. Let $E$ be the intersection of $AD$ and $BC$, while $F$ be the intersection of $AC$ and $BD$. A circle $w$ are tangent to $BD$ and $AC$ such that $F$ is the orthocenter of $\triangle QEP$ where $PQ$ is a diameter of $w$. Prove that $EO$ passes through the center of $w$.

Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$. [asy] size(200); defaultpen(fontsize(10)); pair A=origin, B=(14,0), C=(9,12), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), P=centroid(A,B,C); draw(D--A--B--C--A^^B--E^^C--F); dot(A^^B^^C^^P); label("$a$", P--A, dir(-90)*dir(P--A)); label("$b$", P--B, dir(90)*dir(P--B)); label("$c$", P--C, dir(90)*dir(P--C)); label("$d$", P--D, dir(90)*dir(P--D)); label("$d$", P--E, dir(-90)*dir(P--E)); label("$d$", P--F, dir(-90)*dir(P--F)); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$P$", P, 1.8*dir(285));[/asy]

Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: [i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i] Prove that this polynomial is divisible by $x$.

A $ 3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $ 3 \times 3 \times 3$ cube is $ \text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 96$

$512$ persons meet at a meeting[ Under every six of these people there is always at least two who know each other. Prove that there must be six people at this gathering, all mutual know.