$$\int_0^\infty \frac{\sin(20x)\sin(21x)}{x^2}\,dx$$ [i]Proposed by Connor Gordon and Vlad Oleksenko[/i]

4) A $100\times 200$ board has $k$ black cells. An operations consists of choosing a $2\times 3$ or $3\times 2$ sub-board having exactly $5$ black cells and painting of black the remaining cell. Find the least value of $k$ for which exists an initial distribution of the black cells such that after some operations the board is completely black.

Show that for any integer $a\ge 5$ there exist integers $b$ and $c$, $c\ge b\ge a$, such that $a,b,c$ are the lengths of the sides of a right-angled triangle.

A positive integer $x$ satisfies the following: \[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\] Find all possible values of \[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\] where $\{y\}$ denotes the fractional part of $y$.

Let $ABC$ be a triangle with $BC = 5$, $CA = 3$, and $AB = 4$. Variable points $P, Q$ are on segments $AB$, $AC$, respectively such that the area of $APQ$ is half of the area of $ABC$. Let $x$ and $y$ be the lengths of perpendiculars drawn from the midpoint of $PQ$ to sides $AB$ and $AC$, respectively. Find the range of values of $2y + 3x$.

In the figure below, $BDEF$ is a square inscribed in $\triangle ABC$. If $\frac{AB}{BC} = \frac{4}{5}$, what is the area of $BDEF$ divided by the area of $\triangle ABC$? [asy] unitsize(20); pair A = (0, 3); pair B = (0, 0); pair C = (4, 0); draw(A -- B -- C -- cycle); real w = 12.0 / 7; pair D = (w, 0); pair E = (w, w); pair F = (0, w); draw(D -- E -- F); dot(Label("$A$", A, NW), A); dot(Label("$B$", B, SW), B); dot(Label("$C$", C, SE), C); dot(Label("$D$", D, S), D); dot(Label("$E$", E, NE), E); dot(Label("$F$", F, W), F); [/asy]

Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Aidan Duncan[/i]

Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$? $\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20$

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

Ayman wants to color the cells of a $50 \times 50$ chessboard into black and white so that each $2 \times 3$ or $3 \times 2$ rectangle contains an even number of white cells. Determine the number of ways Ayman can color the chessboard.

There are five distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal. (a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers. (b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).

Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$ there exist at least $k$ consecutive composite integers in the set {$n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$}.

An infinite, strictly increasing sequence $\{a_n\}$ of positive integers satisfies the condition $a_{a_n}\le a_n + a_{n + 3}$ for all $n\ge 1$. Prove that there are infinitely many triples $(k, l, m)$ of positive integers such that $k <l <m$ and $a_k + a_m = 2a_l$.

The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.

$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either $$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$ or $$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$ Given $a_{100}\leq a_0$, what is the maximal value of $b_1+b_2+\cdots+b_{100}$?

Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

Find the least positive integer $ N$ such that the sum of its digits is 100 and the sum of the digits of $ 2N$ is 110.

Find the largest integer $N$ satisfying the following two conditions: [b](i)[/b] $\left[ \frac N3 \right]$ consists of three equal digits; [b](ii)[/b] $\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n$ for some positive integer $n.$

The real numbers $x,y,z$ satisfy $x^2+y^2+z^2=3.$ Prove that the inequality $x^3-(y^2+yz+z^2)x+yz(y+z)\le 3\sqrt{3}.$ and find all triples $(x,y,z)$ for which equality holds.

Numbers $a, b, c$ are such that $3a + 4b = 3c$ and $4a - 3b = 4c$. Show that $a^2 + b^2 = c^2$.

$\sqrt{\frac{1}{9} + \frac{1}{16}} =$ $\textbf{(A)}\ \frac15 \qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ \frac27 \qquad \textbf{(D)}\ \frac{5}{12} \qquad \textbf{(E)}\ \frac{7}{12}$

Let $ p$ be the greatest prime factor of 9991. Then, the sum of the digits of $ p$ is $ \text{(A)}\ 4 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 28$

Prove that there exists a positive integer $n$ such that in the decimal representation of each of the numbers $\sqrt{n}$, $\sqrt[3]{n},..., \sqrt[10]{n}$ digits $2015$ stand immediately after the decimal point. [i]A.Golovanov [/i]

Hello all. Post your solutions below. [b]Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.[/b] 4/1/31. A group of $100$ friends stands in a circle. Initially, one person has $2019$ mangos, and no one else has mangos. The friends split the mangos according to the following rules: • sharing: to share, a friend passes two mangos to the left and one mango to the right. • eating: the mangos must also be eaten and enjoyed. However, no friend wants to be selfish and eat too many mangos. Every time a person eats a mango, they must also pass another mango to the right. A person may only share if they have at least three mangos, and they may only eat if they have at least two mangos. The friends continue sharing and eating, until so many mangos have been eaten that no one is able to share or eat anymore. Show that there are exactly eight people stuck with mangos, which can no longer be shared or eaten.