Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$. [i]Proposed by pog[/i]

Yor and Fiona are playing a match of tennis against each other. The first player to win $6$ games wins the match (while the other player loses the match). Yor has currently won $2$ games, while Fiona has currently won $0$ games. Each game is won by one of the two players: Yor has a probability of $\frac23$ to win each game, while Fiona has a probability of $\frac13$ to win each game. Then, $\frac{m}{n}$ is the probability Fiona wins the tennis match, for relatively prime integers $m,n$. Compute $m$.

Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius $1$ is regular (i.e., has equal sides).

At the wedding of two Bulgarian nationals in mathematics, every guest who gave a positive integer \(n\), not yet given by another guest, which divides \(3^n-3\) but does not divide \(2^n-2\), received a prize. If there were an infinite number of guests, would the newlyweds theoretically need an infinite number of gifts?

There are $2023$ distinct points on a plane, which are coloured in white or red, such that for each white point, there are exactly two red points whose distance is $2023$ to that white point. Find the minimum number of red points.

Let $X$ be the sum of all divisors of the number $(3\cdot 2009)^{((2\cdot 2009)^{2009}-1)}$ . Find the last digit of $X$.

Triangle \(ABC\) has \(AB > AC\) and \(\angle A = 60^\circ \). Let \(M\) be the midpoint of \(BC\), \(N\) be the point on segment \(AB\) such that \(\angle BNM = 30^\circ\). Let \(D,E\) be points on \(AB, AC\) respectively. Let \(F, G, H\) be the midpoints of \(BE, CD, DE\) respectively. Let \(O\) be the circumcenter of triangle \(FGH\). Prove that \(O\) lies on line \(MN\).

Given the isosceles triangle $ABC$ with $| AB | = | AC | = 10$ and $| BC | = 15$. Let points $P$ in $BC$ and $Q$ in $AC$ chosen such that $| AQ | = | QP | = | P C |$. Calculate the ratio of areas of the triangles $(PQA): (ABC)$.

For real numbers $x$, let \[P(x)=1+\cos (x)+i \sin (x)-\cos (2 x)-i \sin (2 x)+\cos (3 x)+i \sin (3 x)\] where $i=\sqrt{-1}$. For how many values of $x$ with $0 \leq x<2 \pi$ does $P(x)=0 ?$ $\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 4$

Let $ABCD$ be a unit square, and let $E$ be a point on segment $AC$ such that $AE = 1$. Let $DE$ meet $AB$ at $F$ and $BE$ meet $AD$ at $G$. Find the area of $CFG$.

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.

In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Let $u(t)$ be a continuous function in the system of differential equations $$\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).$$ Show that, regardless of the choice of $u(t)$, the solution of the system which satisfies $x=x_0 , y=y_0$ at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0 $, show that, for any positive value $t_0$ of $t$, it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$

Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.

Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}$.

(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property: \[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\] [i]Proposed by Nikolay Nikolov[/i]

There are $22$ chairs in a round table. Find the minimum n such that for any group of $n$ people sitting in the table, we always can find two people with exactly $2$ or $8$ chairs between them. (Le Anh Vinh)

Consider the paths of length $16$ that go from the lower left corner to the upper right corner of an $8\times 8$ grid. Find the number of such paths that change direction exactly $4$ times.

In a group of $n > 20$ people, there are some (at least one, and possibly all) pairs of people that know each other. Knowing is symmetric; if Alice knows Blaine, then Blaine also knows Alice. For some values of $n$ and $k,$ this group has a peculiar property: If any $20$ people are removed from the group, the number of pairs of people that know each other is at most $\frac{n-k}{n}$ times that of the original group of people. (a) If $k = 41,$ for what positive integers $n$ could such a group exist? (b) If $k = 39,$ for what positive integers $n$ could such a group exist?

At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by $4$. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She think she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5? $\textbf{(A) }\frac{11}{72}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{3}{16}\qquad\textbf{(D) }\frac{11}{24}\qquad\textbf{(E) }\frac{1}{2}$

In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.

If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)? (A) From A to B (B) From B to C only (C) From B to D (D) From C to D only (E) From D to E