Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible? [asy] size(100); pair A, B, C, D, E, F; A = (0,0); B = (1,0); C = (2,0); D = rotate(60, A)*B; E = B + D; F = rotate(60, A)*C; draw(Circle(A, 0.5)); draw(Circle(B, 0.5)); draw(Circle(C, 0.5)); draw(Circle(D, 0.5)); draw(Circle(E, 0.5)); draw(Circle(F, 0.5)); [/asy] $\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$

Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

Let $ABCD$ be a square with side 1. On the sides $BC$ and $CD$ are chosen points $P$ and $Q$ where $AP$ and $AQ$ intersect the diagonal $BD$ in points $M$ and $N$ respectively. If $DQ\neq BP$ and the line through $A$ and the intersection point of $MQ$ and $NP$ is perpendicular to $PQ$, prove that $\angle MAN=45^\circ$.

Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] if each $a_i$ equals the following arithmetic mean: \[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\] Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] and $\mathbf b$ is $\mathbf a$-[i]harmonic[/i]. Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.

Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that $ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.

Let $A, E, H, L, T,$ and $V$ be chosen independently and at random from the set $\{0, \tfrac{1}{2}, 1\}.$ Compute the probability that $\lfloor T \cdot H \cdot E \rfloor = L \cdot A \cdot V \cdot A.$

Let $ABC$ be a triangle with $\angle A = 90^o$, $AB = 1$, and $AC = 2$. Let $\ell$ be a line through $A$ perpendicular to $BC$, and let the perpendicular bisectors of $AB$ and $AC$ meet $\ell$ at $E$ and $F$, respectively. Find the length of segment $EF$.

Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?

Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

Let $S$ be the sum of all possible values of $a \cdot c$ such that $$a^3+3ab^2-72ab+432a=4c^3$$ if $a$, $b$, and $c$ are positive integers, $a+b > 11$, $a > b-13$, and $c \le 1000$. Find the sum of all distinct prime factors of $S$. [i]Proposed by Kevin Zhao[/i]

Circle $O$ is the incircle of the square $ABCD$. $O$ is tangent to $AB$ and $AD$ at $E$ and $F$, respectively. Let $K$ be a point on the minor arc $EF$, and let the tangent of $O$ at $K$ intersect $AB$, $AC$, $AD$ at $X$, $Y$, $Z$, respectively. Show that $\displaystyle \frac{AX}{XB} + \frac{AY}{YC} + \frac{AZ}{ZD} =1$.

Consider $n>9$ lines on the plane such that no two lines are parallel. Show that there exist at least $n/9$ lines such that the angle between any two of the lines is at most $20^\circ$.

is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $i) \exists n\in \mathbb{N}:f(n)\neq n$ $ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$

Let $ G$ be a group such that $ G'$ is abelian and each normal and abelian subgroup of $ G$ is finite. Prove that $ G$ is finite.

$ 2\left(1-\frac{1}{2}\right)+3\left(1-\frac{1}{3}\right)+4\left(1-\frac{1}{4}\right)+\cdots+10\left(1-\frac{1}{10}\right)= $ $ \text{(A)}\ 45\qquad\text{(B)}\ 49\qquad\text{(C)}\ 50\qquad\text{(D)}\ 54\qquad\text{(E)}\ 55 $

Find the smallest positive integer $ n$ with the property that the polynomial $ x^4 \minus{} nx \plus{} 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

Let $ ABC$ be a triangle and let $ \mathcal{M}_{a}$, $ \mathcal{M}_{b}$, $ \mathcal{M}_{c}$ be the circles having as diameters the medians $ m_{a}$, $ m_{b}$, $ m_{c}$ of triangle $ ABC$, respectively. If two of these three circles are tangent to the incircle of $ ABC$, prove that the third is tangent as well.

Let $ P$ be a point inside a regular tetrahedron $ T$ of unit volume. The four planes passing through $ P$ and parallel to the faces of $ T$ partition $ T$ into 14 pieces. Let $ f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $ f(P)$ as $ P$ varies over $ T.$

There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in two randomly to each of the $2010$ round table players. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules. If a player holds at least one red card, he passes one red card to the player sitting to his left, otherwise he transfers one white card to the left. The game ends after the round when each player has one red card and one white card. Determine as many rounds as possible.

A regular 2021-gon is divided into 2019 triangles,such that no diagonals intersect. Prove that at least 3 of the 2019 triangles are isoscoles

A convex polygon is given, no two of whose sides are parallel. For each side we consider the angle the side subtends at the vertex farthest from the side. Show that the sum of these angles equals $180^\circ$. [i]M. Smurov[/i]

Each of the $n$ students writes one of the numbers $1,2$ or $3$ on each of the $29$ boards. If any two students wrote different numbers on at least one of the boards and any three students wrote the same number on at least one of the boards, what is the maximum possible value of $n$?

The vertical side of a square is divided onto $n$ segments. The sum of the segments with even numbers lengths equals to the sum of the segments with odd numbers lengths. $n-1$ lines parallel to the horizontal sides are drawn from the segments ends, and, thus, $n$ strips are obtained. The diagonal is drawn from the lower left corner to the upper right one. This diagonal divides every strip onto left and right parts. Prove that the sum of the left parts of odd strips areas equals to the sum of the right parts of even strips areas.