Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

If $ n$ is a real number, then the simultaneous system $ nx \plus{} y \equal{} 1$ $ ny \plus{} z \equal{} 1$ $ x \plus{} nz \equal{} 1$ has no solution if and only if $ n$ is equal to $ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \text{ or } 1 \qquad \textbf{(E)}\ \frac12$

In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws). Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ($i = 1,2,...,n$), respectively. Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$

The total number of languages used in KAUST is $n$. For each positive integer $k \le n$, let $A_k$ be the set of all those people in KAUST who can speak at least $k$ languages; and let $B_k$ be the set of all people $P$ in KAUST with the property that, for any $k$ pairwise different languages (used in KAUST), $P$ can speak at least one of these $k$ languages. Prove that (a) If $2k \ge n + 1$ then $A_k \subseteq B_k$ (b) If $2k \le n + 1$ then $A_k \supseteq B_k.$ Nguyễn Duy Thái Sơn

Let $S$ be the set of all lattice points $(x, y)$ in the plane satisfying $|x|+|y|\le 10$. Let $P_1,P_2,\ldots,P_{2013}$ be a sequence of 2013 (not necessarily distinct) points such that for every point $Q$ in $S$, there exists at least one index $i$ such that $1\le i\le 2013$ and $P_i = Q$. Suppose that the minimum possible value of $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. (A [i]lattice point[/i] is a point with all integer coordinates.) [hide="Clarifications"] [list] [*] $k = 2013$, i.e. the problem should read, ``... there exists at least one index $i$ such that $1\le i\le 2013$ ...''. An earlier version of the test read $1 \le i \le k$.[/list][/hide] [i]Anderson Wang[/i]

João calculated the product of the non zero digits of each integer from $1$ to $10^{2009}$ and then he summed these $10^{2009}$ products. Which number did he obtain?

For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that: $\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.

For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

Let $ S \equal{}\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^\plus{}$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a\minus{}b|\equal{}\{1,n\}$

Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.

2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

Let $d$ be a natural number. Given two natural numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if the $d$ numbers obtained substituting each one of the digits of $M$ by the digit of $N$ which is on the same position are all multiples of $7$. Find all the values of $d$ for which the following condition is valid: For any two numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if $N$ is a friend of $M$.

Let $a$, $b$, $c$ be real numbers, such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Determine the value of $ab+bc+ca$.

Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, respectively. Let $O$ be the circumcenter of $ABC$. If $BI_B$ is perpendicular to $AO$, $AI_C = 3$ and $AC = 4\sqrt2$, then $AB^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Note: In triangle $\vartriangle ABC$, the $A$-excenter is the intersection of the exterior angle bisectors of $\angle ABC$ and $\angle ACB$. The $B$-excenter and $C$-excenter are defined similarly.

Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define \[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\] Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum \[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\] $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

Let $n,m$ be positive integers. Let $A_1,A_2,A_3, ... ,A_m$ be sets such that $A_i \subseteq \{1, 2, 3, . . . , n\}$ and $|A_i| = 3$ for all $i$ (i.e. $A_i$ consists of three different positive integers each at most $n$). Suppose for all $i < j$ we have $|A_i \cap A_j | \le 1$ (i.e. $A_i$ and $A_j$ have at most one element in common). (a) Prove that $m \le \frac{n(n-1)}{ 6}$ . (b) Show that for all $n \ge3$ it is possible to have $m \ge \frac{(n-1)(n-2)}{ 6}$ .

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

Comparing the numbers $ 10^{\minus{}49}$ and $ 2\cdot 10^{\minus{}50}$ we may say: $ \textbf{(A)}\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}1}}\qquad\\ \textbf{(B)}\ \text{the first exceeds the second by }{2\cdot 10^{\minus{}1}}\qquad \\ \textbf{(C)}\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}50}}\qquad \\ \textbf{(D)}\ \text{the second is five times the first}\qquad \\ \textbf{(E)}\ \text{the first exceeds the second by }{5}$

Suppose $A>B>0$ and A is $x\%$ greater than $B$. What is $x$? $ \textbf {(A) } 100\left(\frac{A-B}{B}\right) \qquad \textbf {(B) } 100\left(\frac{A+B}{B}\right) \qquad \textbf {(C) } 100\left(\frac{A+B}{A}\right)\qquad \textbf {(D) } 100\left(\frac{A-B}{A}\right) \qquad \textbf {(E) } 100\left(\frac{A}{B}\right)$

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

The parallelogram $ABCD$ isn't a diamond. The ratio of the diagonal lengths $|AC|/|BD|$ equals to $k$. The $[AM)$ ray is symmetric to the $[AD)$ ray with respect to the $(AC)$ line. The $[BM)$ ray is symmetric to the $[BC)$ ray with respect to the $(BD)$ line. ($M$ point is those rays intersection.) Find the ratio $|AM|/|BM|$ .

Find all couples $(m,n)$ of positive integers such that satisfied $m^2+1=n^2+2016$ .

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

$2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k}$ is equal to $\textbf{(A) }2^{-2k}\qquad\textbf{(B) }2^{-(2k-1)}\qquad\textbf{(C) }-2^{-(2k+1)}\qquad\textbf{(D) }0\qquad \textbf{(E) }2$