Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters. Find the angle between the planes $ABC$' and $ACD'$. (A Zaslavskiy)

A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected? $\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}$

Let $R{}$ and $r{}$ be the radii of the circumscribed and inscribed circles of the acute-angled triangle $ABC{}$ respectively. The point $M{}$ is the midpoint of its largest side $BC.$ The tangents to its circumscribed circle at $B{}$ and $C{}$ intersect at $X{}$. Prove that \[\frac{r}{R}\geqslant\frac{AM}{AX}.\]

Find all functions $ f $ defined on real numbers and taking values in the set of real numbers such that $ f(x+y)+f(y+z)+f(z+x) \geq f(x+2y+3z) $ for all real numbers $ x,y,z $. [hide]There is an infinity of such functions. Every function with the property that $ 3 \inf f \geq \sup f $ is a good one. I wonder if there is a way to find all the solutions. It seems very strange.[/hide]

The number of integral points on the circle with center $(199,0)$, radius of $199$ is________.

Construct a trapezoid given the midpoints of the legs, the point of intersection of the diagonals and the foot of the perpendicular, drawn from this point on the larger base.

Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?

Let $a,b,c$ be the sidelengths of a triangle such that $a^2+b^2 > 5c^2$ holds. Prove that $c$ is the shortest side of the triangle.

A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]

For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$ [list=a] [*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation. [*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation. [/list] [i](Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)[/i]

Let $a, b, c$ be positive reals such that $ab+bc+ca=\frac{3}{4}$. Show that $$(a+b+c)^6 \geq (\frac{9} {8})^3(1+(a+b)^2)(1+(b+c)^2)(1+(c+a)^2).$$ When does equality hold?

In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party? [i]Yudi Satria, Jakarta[/i]

The polynomial $P(x)=x^3+ax+1$ has exactly one solution on the interval $[-2,0)$ and has exactly one solution on the interval $(0,1]$ where $a$ is a real number. Which of the followings cannot be equal to $P(2)$? $ \textbf{(A)}\ \sqrt{17} \qquad\textbf{(B)}\ \sqrt[3]{30} \qquad\textbf{(C)}\ \sqrt{26}-1 \qquad\textbf{(D)}\ \sqrt {30} \qquad\textbf{(E)}\ \sqrt [3]{10} $

Compute $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}.$

In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.

How many subsets containing three different numbers can be selected from the set \[\{ 89,95,99,132, 166,173 \}\] so that the sum of the three numbers is even? $\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24$

Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$. Let $\omega$ be the circumcircle of triangle $ABC$. The tangents of $\omega$ at $B$ and $C$ intersect at $P$. Let $M$ be the midpoint of $PB$. Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$. Point $D$ is on $CM$ such that $ED \parallel BM$. Show that the circumcircle of $CDE$ is tangent to $\omega$.

Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii. (A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.) [color=#45818E]Note.[/color] Weaker results with $2\sqrt{2}$ replaced by some other $c$ may be awarded points depending on the value of $c>2\sqrt{2}$ [i]Proposed by Morteza Saghafian[/i]

Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers. [i]Proposed by Michael Ren[/i]

Let the matrix $S$ be orthogonal and the matrix $I-S$ be invertible, where I is the identity matrix of the same size as $S$. Find $x^T(I-S)^{-1}x$ Where $x$ is a real unit vector.

The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is $ \textbf{(A)}\ \Big\{\frac{5}{2}\Big\}\qquad\textbf{(B)}\ \big\{x\ |\ 2 \le x \le 3\big\}\qquad\textbf{(C)}\ \big\{x\ |\ 2\le x < 3\big\}\qquad \\ \textbf{(D)}\ \Big\{x\ |\ 2 < x \le 3\Big\}\qquad\textbf{(E)}\ \Big\{x\ |\ 2 < x < 3\Big\} $

Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied: $(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity. $(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even . [i]Proposed by Poland.[/i]

Let $k\geqslant 2$ be a natural number. Prove that the natural numbers with an even sum of digits give all the possible residues when divided by $k{}$. [i]Proposed by P. Kozlov and I. Bogdanov[/i]

One hundred million cities lie on Planet MO. Initially, there are no air routes between any two cities. Now an airline company comes. It plans to establish $5050$ two-way routes, each route connects two different cities, and no two routes connect the same two cities. The "degree" of a city is defined to be the number of routes departing from that city. The "benefit" of a route is the product of the "degrees" of the two cities it connects. Find the maximum possible value of the sum of the benefits of these $5050$ routes.