We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.

Let $f(x) = x^4 + 2x^3 - x - 1$. (a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients. (b) Find the exact values of the 4 roots of $f(x)$.

A cubic polynomial $f$ satisfies $f(0)=0, f(1)=1, f(2)=2, f(3)=4$. What is $f(5)$?

Let $ABCD$ be a cyclic quadrilateral such that the ratio of its diagonals is $AC:BD=7:5.$ Let $E$ and $F$ be the intersections of lines $AB$ and $CD$ and lines $BC$ and $AD$, respectively. Let $L$ and $M$ be the midpoints of diagonals $AC$ and $BD$, respectively. Given that $EF=2020,$ the length of $LM$ can be written as $\frac{p}{q}$ where $p,q$ are relatively prime positive integers. Compute $p+q.$

Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.

Prove that the equation $x^x = y^3+z^3$ has infinitely many solutions in positive integers $x,y,z$.

Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.

Let $ABCD$ be a square, $E$ be a point on the side $DC$, $F$ and $G$ be the feet of the altitudes from $B$ to $AE$ and from $A$ to $BE$, respectively. Suppose $DF$ and $CG$ intersect at $H$. Prove that $\angle AHB=90^\circ$.

Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png[/img] Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way. [img]https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png[/img]

Find all a positive integers $a$ and $b$, such that $$\frac{a^b+b^a}{a^a-b^b}$$ is an integer

Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$. [i]J. Szucs[/i]

The graph of the equation $x+y=\lfloor x^2+y^2 \rfloor$ consists of several line segments. Compute the sum of their lengths.

Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size. (AV Savkin)

Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and \[1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 \]

In a triangle $ ABC$ with $ AB\equal{}20, AC\equal{}21$ and $ BC\equal{}29$, points $ D$ and $ E$ are taken on the segment $ BC$ such that $ BD\equal{}8$ and $ EC\equal{}9$. Calculate the angle $ \angle DAE$.

On a planet there are $M$ countries and $N$ cities. There are two-way roads between some of the cities. It is given that: (1) In each county there are at least three cities; (2) For each country and each city in the country is connected by roads with at least half of the other cities in the countries; (3) Each city is connceted with exactly one other city ,that is not in its country; (4) There are at most two roads between cities from cities in two different countries; (5) If two countries contain less than $2M$ cities in total then there is a road between them. Prove that there is cycle of lenght at least $M+\frac{N}{2}$.

Let $A, B$ be two points on circle $\gamma$. At point $A$ and $B$ we construct tangents to $\gamma$, $AC$ and $BD$ respectively such that the tangents are both in the clockwise direction. Let the intersection between $AB$ and $CD$ be $P$ . If $AC = BD$, prove that $P$ bisects the line $CD$.

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $N$ denote the second point of intersection of line $AI$ and $\omega$. The line through $I$ perpendicular to $AI$ intersects line $BC$, segment $[AB]$, and segment $[AC]$ at the points $D$, $E$, and $F$, respectively. The circumcircle of triangle $AEF$ meets $\omega$ again at $P$, and lines $PN$ and $BC$ intersect at $Q$. Prove that lines $IQ$ and $DN$ intersect on $\omega$.

Let \( ABC \) and \( A_1B_1C_1 \) be two triangles such that the segments \( AA_1 \) and \( BC \) intersect, the segments \( BB_1 \) and \( AC \) intersect, and the segments \( CC_1 \) and \( AB \) intersect. If it is known that there exists a point \( X \) inside both triangles such that \[ \begin{aligned} \angle XAB &= \angle XA_1B_1, &\angle XBC &= \angle XC_1A_1, &\angle XCA &= \angle XB_1C_1,\\ \angle XAC &= \angle XB_1A_1, &\angle XBA &= \angle XA_1C_1, &\angle XCB &= \angle XC_1B_1. \end{aligned} \] Prove that the lines \( AC_1 \), \( BB_1 \), and \( CA_1 \) are concurrent or parallel.

Let $n$ and $m$ be natural numbers such that $m+ i=a_ib_i^2$ for $i=1,2, \cdots n$ where $a_i$ and $b_i$ are natural numbers and $a_i$ is not divisible by a square of a prime number. Find all $n$ for which there exists an $m$ such that $\sum_{i=1}^{n}a_i=12$

Suppose $$2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.$$ What is the value of $x?$ $\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}$

Let $a$ , $b$, $c$ and $d$ be positive real numbers such that $a+b+c+d=8$. Prove that $\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\geq8$

Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds. (Poland)

Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.