When two stars are very far apart their gravitational potential energy is zero; when they are separated by a distance $d$ the gravitational potential energy of the system is $U$. If the stars are separated by a distance $2d$ the gravitational potential energy of the system is $ \textbf{(A)}\ U/4\qquad\textbf{(B)}\ U/2 \qquad\textbf{(C)}\ U \qquad\textbf{(D)}\ 2U\qquad\textbf{(E)}\ 4U $

In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.

Team Stanford has a $ \frac{1}{3}$ chance of winning any given math contest. If Stanford competes in 4 contests this quarter, what is the probability that the team will win at least once?

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

The circle inscribed in the triangle $A_1A_2A_3$ is tangent to its sides $A_1A_2, A_2A_3, A_3A_1$ at points $T_1, T_2, T_3$, respectively. Denote by $M_1, M_2, M_3$ the midpoints of the segments $A_2A_3, A_3A_1, A_1A_2$, respectively. Prove that the perpendiculars through the points $M_1, M_2, M_3$ to the lines $T_2T_3, T_3T_1, T_1T_2$ meet at one point.

Let $S_1$ and $S_2$ be two circumferences, with centers $O_1$ and $O_2$ respectively, and secants on $M$ and $N$. The line $t$ is the common tangent to $S_1$ and $S_2$ closer to $M$. The points $A$ and $B$ are the intersection points of $t$ with $S_1$ and $S_2$, $C$ is the point such that $BC$ is a diameter of $S_2$, and $D$ the intersection point of the line $O_1O_2$ with the perpendicular line to $AM$ through $B$. Show that $M$, $D$ and $C$ are collinear.

On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$. Babis

Let \(\mathbb{Z}^2\) denote the set of points in the Euclidean plane with integer coordinates. Find all functions \(f : \mathbb{Z}^2 \to [0,1]\) such that for any point \(P\), the value assigned to \(P\) is the average of all the values assigned to points in \(\mathbb{Z}^2\) whose Euclidean distance from \(P\) is exactly 2020.

Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.

An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

Let $n\geq 4$ be an integer, and let $a_1,a_2,\ldots,a_n$ be positive real numbers such that \[ a_1^2+a_2^2+\cdots +a_n^2=1 . \] Prove that the following inequality takes place \[ \frac{a_1}{a_2^2+1}+\cdots +\frac{a_n}{a_1^2+1} \geq \frac{4}{5}\left( a_1 \sqrt{a_1}+\cdots +a_n \sqrt{a_n} \right)^2 . \] [i]Bogdan Enescu, Mircea Becheanu[/i]

Find all isosceles trapezoids that are divided into $2$ isosceles triangles by a diagonal.

Find all rational solutions to: \begin{eqnarray*} t^2 - w^2 + z^2 &=& 2xy \\ t^2 - y^2 + w^2 &=& 2xz \\ t^2 - w^2 + x^2 &=& 2yz . \end{eqnarray*}

Alfonso teaches Francis how to draw a spiral in the plane: First draw half of a unit circle. Starting at one of the ends, draw half a circle with radius $1/2$. Repeat this process at the endpoint of each half circle, where each time the radius is half of the previous half-circle. Assuming you can’t stop Francis from drawing the entire spiral, compute the total length of the spiral.

Let $ABC$ be an acute triangle, which is not equilateral. Denote by $O$ and $H$ its circumcenter and orthocenter, respectively. The circle $k$ passes through $B$ and touches the line $AC$ at $A$. The circle $l$ with center on the ray $BH$ touhes the line $AB$ at $A$. The circles $k$ and $l$ meet in $X$ ($X\ne A$). Show that $\angle HXO=180^\circ-\angle BAC$. [i]Proposed by Josef Tkadlec[/i]

Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$, and let $P$ be the midpoint of the minor arc $BC$ of $\Gamma$. Let $AP$ and $BC$ meet at $D$, and let $M$ be the midpoint of $AB$. Also, let $E$ be the point such that $AE\perp AB$ and $BE\perp MP$. Prove that $AE=DE$.

Given the pyramid $ABCD$. Let $O$ be the midpoint of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.

In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.

How many different solutions on the interval $[0, \pi]$ does the equation $$6\sqrt2 \sin x \cdot tgx - 2\sqrt2 tgx +3\sin x -1=0$$ have?

Triangle $ABC$ has $AB = 8$, $BC = 12$, and $AC = 16$. Point $M$ is on $\overline{AC}$ so that $AM = MC$. Then, $\overline{BM}$ has length $x$. Find $x^2$

Let $AB$ be a diameter of a semicircle $\Gamma$. Two circles, $\omega_1$ and $\omega_2$, externally tangent to each other and internally tangent to $\Gamma$, are tangent to the line $AB$ at $P$ and $Q$, respectively, and to semicircular arc $AB$ at $C$ and $D$, respectively, with $AP<AQ$. Suppose $F$ lies on $\Gamma$ such that $ \angle FQB = \angle CQA $ and that $ \angle ABF = 80^\circ $. Find $ \angle PDQ $ in degrees.

In decimal representation $$\text {34!=295232799039a041408476186096435b0000000}.$$ Find the numbers $a$ and $b$.

Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.

Points $C,D$ are on side $BE$ of triangle $ABE$, such that $BC=CD=DE$. Points $X,Y,Z,T$ are circumcenters of $ABE,ABC,ADE,ACD$. Prove, that $T$ - centroid of $XYZ$

Let $S_n$ denote the set of permutations of the sequence $(1,2,\dots, n)$. For every permutation $\pi=(\pi_1, \dots, \pi_n)\in S_n$, let $\mathrm{inv}(\pi)$ be the number of pairs $1\le i < j \le n$ with $\pi_i>\pi_j$; i. e. the number of inversions in $\pi$. Denote by $f(n)$ the number of permutations $\pi\in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n+1$. Prove that there exist infinitely many primes $p$ such that $f(p-1)>\frac{(p-1)!}{p}$, and infinitely many primes $p$ such that $f(p-1)<\frac{(p-1)!}{p}$. (Proposed by Fedor Petrov, St. Petersburg State University)