Let $MATH$ be a square with $MA = 1$. Point $B$ lies on $AT$ such that $\angle MBT = 3.5 \angle BMT$. What is the area of $\vartriangle BMT$?

Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.

Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

A plane is drawn through the height of a regular tetrahedron, which intersects the planes of the lateral faces along $ 3 $ lines that form angles $ \alpha $, $ \beta $, $ \gamma $ with the plane of the tetrahedron's base. Prove that $$ tg^2 \alpha + tg^2 \beta + tg^2 \gamma =12.$$

Find all permutations $(a_1, a_2,...,a_{2021})$ of $(1,2,...,2021)$, such that for every two positive integers $m$ and $n$ with difference bigger than $20^{21}$, the following inequality holds: $GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|$.

Let $ABC$ be an isosceles triangle with $AB = AC$. Let P be a point in the interior of $ABC$ such that $PB > PC$ and $\angle PBA = \angle PCB$. Let $M$ be the midpoint of the side $BC$. Let $O$ be the circumcenter of the triangle $APM$. Prove that $\angle OAC=2 \angle BPM$ .

An integer $a\ge1$ is called [i]Aegean[/i], if none of the numbers $a^{n+2}+3a^n+1$ with $n\ge1$ is prime. Prove that there are at least 500 Aegean integers in the set $\{1,2,\ldots,2018\}$. (Proposed by Gerhard Woeginger, Austria)

Carol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she multiplied the largest with the sum of the other two, but still got the right answer. What is the sum of the three numbers?

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct? [asy]unitsize(3mm); fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; path divider=(-2,2)--(-3,3)--cycle; fill(onewhite,white); fill(rotate(90)*onewhite,white); fill(rotate(180)*onewhite,white); fill(rotate(270)*onewhite,white);[/asy] $ \textbf{(A)}\ B \equal{} W \qquad \textbf{(B)}\ W \equal{} R \qquad \textbf{(C)}\ B \equal{} R \qquad \textbf{(D)}\ 3B \equal{} 2R \qquad \textbf{(E)}\ 2R \equal{} W$

The inner bisections of the $ \angle ABC $ and $ \angle ACB $ angles of the $ ABC $ triangle are at $ I $. The $ BI $ parallel line that runs through the point $ A $ finds the $ CI $ line at the point $ D $. The $ CI $ parallel line for $ A $ finds the $ BI $ line at the point $ E $. The lines $ BD $ and $ CE $ are at the point $ F $. Show that $ F, A $, and $ I $ are collinear if and only if $ AB = AC. $

We have triangle $ABC$ with area $S$. In one step we can move only one vertex at a time so that the area of the triangle during movement remains constant. Prove that we can move this triangle into any other arbitrary triangle $DEF$ with area $S$. [i]Proposed by Marek Maruin, Slovakia[/i]

Source: 2018 Canadian Open Math Challenge Part B Problem 2 ----- Let ABCD be a square with side length 1. Points $X$ and $Y$ are on sides $BC$ and $CD$ respectively such that the areas of triangels $ABX$, $XCY$, and $YDA$ are equal. Find the ratio of the area of $\triangle AXY$ to the area of $\triangle XCY$. [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZi9lLzAzZjhhYzU0N2U0MGY2NGZlODM4MWI4Njg2MmEyMjhlY2M3ZjgzLnBuZw==&rn=YjIuUE5H[/img][/center]

Let $k\geq4$ be an integer. Sunny and Ming play a game with strings. A string is a sequence that every element of it is an integer between $1$ and $k$, inclusive. At first, Sunny chooses two positive integers $N,L\geq2$ and write down $N$ strings, each having length $L$. Then Ming mark at most $\frac{N}{2}$ strings. Then Sunny chooses an unmarked string $s$ and calculate the biggest integer $n$ such that there exists another string satisfying its first $n$ element is the same as the first $n$ element of $s$. Then Sunny burn down all strings which first $n$ element if different from the first $n$ element of $s$, leaving only the ones which have the same first $n$ element of $s$. Finally, Ming chooses an integer $d$ between $1$ and $k$, inclusive, and remove all strings which $(n+1)$th element is $d$. Sunny's score would be the number of strings left. Find the maximum score that Sunny can guarantee to get. [i]Proposed by USJL[/i]

Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Compute \[\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}.\]

If $a,b>0$ then prove: $$\left(\frac{a+b}2-\frac{2ab}{a+b}\right)\operatorname{arctan}\left(\sqrt{ab}\right)+\left(\frac{2ab}{a+b}-\sqrt{ab}\right)\operatorname{arctan}\left(\frac{a+b}2\right)+\left(\sqrt{ab}-\frac{2ab}{a+b}\right)\operatorname{arctan}\left(\sqrt{\frac{a^2+b^2}2}\right)\ge0$$ [i]Proposed by Daniel Sitaru[/i]

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Find the value of $a_2 + a_4 + a_6 + \dots + a_{98}$ if $a_1$, $a_2$, $a_3$, $\dots$ is an arithmetic progression with common difference 1, and $a_1 + a_2 + a_3 + \dots + a_{98} = 137$.

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 $OP = OQ$.

The circles $k_1, k_2, k_3$ with radii ($a>c>b$) $a,b,c$ are tangent to line $d$ at $A,B,C$, respectively. $k_1$ is tangent to $k_2$, and $k_2$ is tangent to $k_3$. The tangent line to $k_3$ at $E$ is parallel to $d$, and it meets $k_1$ at $D$. The line perpendicular to $d$ at $A$ meets line $EB$ at $F$. Prove that $AD=AF$.

Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip? [asy]unitsize(1mm); defaultpen(linewidth(.8pt)); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,2.7)--(0,13.5)); draw((21.6,2.7)--(21.6,13.5));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3$

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

There are given in a table numbers $1,2,...,18$. What is minimal number of numbers we should erase such that the sum of every two remaining numbers is not perfect square of a positive integer.

There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]