Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the distance between the circumcenters of triangles $AHB$ and $AHC$.

For which angles $ \theta$, with $ \theta$ a rational number of degrees, is $ {\tan}^{2}\theta\plus{}{\tan}^{2}2\theta$ is irrational?

Nuts are placed in boxes. The mean value of the number of nuts in a box is $10$, and the mean value of the squares of the numbers of nuts in the boxes is less than $1000$. Prove that at least $10\%$ of the boxes are not empty. (AY Belov)

Given $a_n = (n^2 + 1) 3^n,$ find a recurrence relation $a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0.$ Hence evaluate $\sum_{n\geq0} a_n x^n.$

In triangle $ABC$ with $\overline{AB}=\overline{AC}=3.6$, a point $D$ is taken on $AB$ at a distance $1.2$ from $A$. Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$. Then $\overline{AE}$ is: $ \textbf{(A)}\ 4.8 \qquad\textbf{(B)}\ 5.4\qquad\textbf{(C)}\ 7.2\qquad\textbf{(D)}\ 10.8\qquad\textbf{(E)}\ 12.6 $

Given a triangle $ABC$, the line passing through the vertex $A$ symmetric to the median $AM$ wrt the line containing the bisector of the angle $\angle BAC$ intersects the circle circumscribed around the triangle $ABC$ at points $A$ and $K$. Let $L$ be the midpoint of the segment $AK$. Prove that $\angle BLC=2\angle BAC$.

David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform? [i]Ray Li[/i]

Show that $n^n + (n + 1)^{n + 1}$ is composite for infinitely many positive integers $n$.

If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$ ${{ \textbf{(A)}\ \frac{a+b}{a+1} \qquad\textbf{(B)}\ \frac{2a+b}{a+1} \qquad\textbf{(C)}\ \frac{a+2b}{1+a} \qquad\textbf{(D)}\ \frac{2a+b}{1-a} }\qquad\textbf{(E)}\ \frac{a+2b}{1-a}} $

In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$. [asy]unitsize(2.5mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); //draw(O2--O1); //dot(O1); //dot(O2); draw(Q--R); label("$Q$",Q,N); label("$P$",P,dir(80)); label("$R$",R,E); //label("12",waypoint(O1--O2,0.4),S);[/asy]

Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,$ and $61$.

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f+6f'+12f''+8f'''$ has at least two distinct real zeros.

Solve the equation $\sqrt {a-\sqrt{a+x}}=x$ in real numbers in terms of the real number $a>1$.

Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$

Show that for every integer $n \geq 1$, it is possible to express $5^n$ as the sum of two nonzero squares.

There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?

Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad

Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.

Consider a circle with diameter $AB$ and let $C,D$ be points on its circumcircle such that $C,D$ are not in the same side of $AB$.Consider the parallel line to $AC$ passing from $D$ and let it intersect $AB$ at $E$.Similarly consider the paralell line to $AD$ passing from $C$ and let it intersect $AB$ at $F$.The perpendicular line to $AB$ at $E$ intersects $BC$ at $X$ and the perpendicular line to $AB$ at $F$ intersects $DB$ at $Y$.Prove that the permiter of triangle $AXY$ is twice $CD$. [b]Remark:[/b]This problem is proved to be wrong due to a typo in the exam papers you can find the correct version [url=https://artofproblemsolving.com/community/c6h1832731_geometry__iran_mo_2019]here[/url].

On the plane there are two cylindrical towers with radii of bases $r$ and $R$. Find the set of all those points of the plane from which these towers are visible at the same angle. Consider the case of more towers.

Consider the sequence $(a_n)_{n\geq 1}$ given by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+\sqrt{1+a_n}}$, for all $n\geq 1$. Show that $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = \lim_{n\rightarrow\infty}\sum_{k=1}^n \log_2(1+a_k)=2.$$ [i]Mathematical Gazette[/i]

NASA’s Perseverance Rover was launched on July $30,$ $2020.$ After traveling $292{,}526{,}838$ miles, it landed on Mars in Jezero Crater about $6.5$ months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour? $\textbf{(A) } 6{,}000 \qquad \textbf{(B) } 12{,}000 \qquad \textbf{(C) } 60{,}000 \qquad \textbf{(D) } 120{,}000 \qquad \textbf{(E) } 600{,}000$

Six assassins, numbered 1-6, stand in a circle. Each assassin is randomly assigned a target such that each assassin has a different target and no assassin is their own target. In increasing numerical order, each assassin, if they are still alive, kills their target. Find the expected number of assassins still alive at the end of this process. [i]Proposed by Allen Yang[/i]

Let $\mathcal{G}$ be a simple graph with $n$ vertices and $m$ edges such that no two cycles share an edge. Prove that $2m < 3n$. [i]Note[/i]: A [i]simple graph[/i] is a graph with at most one edge between any two vertices and no edges from any vertex to itself. A [i]cycle[/i] is a sequence of distinct vertices $v_1, \dots, v_n$ such that there is an edge between any two consecutive vertices, and between $v_n$ and $v_1$. [i]Proposed by Ethan Tan[/i]

Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.) [i]Proposed by Steven Liu[/i]