Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$. [i]Proposed by [b]AOPS12142015[/b][/i]

Of $4n$ points in a row, $2n$ are colored white and $2n$ are colored black. Swot tha tthere are $2n$ consecutive points of which exactly $n$ are white and $n$ are black.

Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).

Let $O$ be the circumcenter and $G$ be the centroid of a triangle $ABC$. If $R$ and $r$ are the circumcenter and incenter of the triangle, respectively, prove that \[ OG \leq \sqrt{ R ( R - 2r ) } . \] [i]Greece[/i]

Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$. Let $A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}$. Prove that $|A| \leqslant 5050$ and determine when the equality holds. [i]Proposed by Yunhao Fu[/i]

$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible value for the sum of all the written numbers.

Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?

Let $n\geq 3$ be a positive integer. Alice and Bob are playing a game in which they take turns colouring the vertices of a regular $n$-gon. Alice plays the first move. Initially, no vertex is coloured. Both players start the game with $0$ points. In their turn, a player colours a vertex $V$ which has not been coloured and gains $k$ points where $k$ is the number of already coloured neighbouring vertices of $V$. (Thus, $k$ is either $0$, $1$ or $2$.) The game ends when all vertices have been coloured and the player with more points wins; if they have the same number of points, no one wins. Determine all $n\geq 3$ for which Alice has a winning strategy and all $n\geq 3$ for which Bob has a winning strategy.

A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: $\textbf{(A) }\dfrac2{n-1}\qquad \textbf{(B) }\dfrac{n-1}2\qquad \textbf{(C) }\dfrac2n\qquad \textbf{(D) }2n\qquad \textbf{(E) }2(n-1)$

Aaron has a coin that is slightly unbalanced. The odds of getting heads are $60\%$. What are the odds that if he flips it endlessly, at some point during his flipping he has a total of three more tails than heads?

Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds: $$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

If $ r$ and $ s$ are the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$, the value of $ \frac {1}{r^2} \plus{} \frac {1}{s^2}$ is: $ \textbf{(A)}\ b^2 \minus{} 4ac \qquad\textbf{(B)}\ \frac {b^2 \minus{} 4ac}{2a} \qquad\textbf{(C)}\ \frac {b^2 \minus{} 4ac}{c^2} \qquad\textbf{(D)}\ \frac {b^2 \minus{} 2ac}{c^2}$ $ \textbf{(E)}\ \text{none of these}$

The symbol $ 25_b$ represents a two-digit number in the base $ b$. If the number $ 52_b$ is double the number $ 25_b$, then $ b$ is: $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

Of the vertices of a cube, $7$ of them have assigned the value $0$, and the eighth the value $1$. A [i]move[/i] is selecting an edge and increasing the numbers at its ends by an integer value $k > 0$. Prove that after any finite number of [i]moves[/i], the g.c.d. of the $8$ numbers at vertices is equal to $1$. Russian M.O.