Positive integers $a,b,c$ satisfy the equations $a^2=b^3+ab$ and $c^3=a+b+c$. Prove that $a=bc$.

Let $AH_A$, $BH_B$, $CH_C$ be the altitudes of the acute-angled $\Delta ABC$. Let $X$ be an arbitrary point of segment $CH_C$, and $P$ be the common point of circles with diameters $H_CX$ and BC, distinct from $H_C$. The lines $CP$ and $AH_A$ meet at point $Q$, and the lines $XP$ and $AB$ meet at point $R$. Prove that $A, P, Q, R, H_B$ are concyclic.

Let $A$ be an infinite set of points in the plane such that inside each circle there are only a finite number of points of $A$, with the following properties: $\bullet$ $(0, 0)$ belongs to $A$. $\bullet$ If $(a, b)$ and $(c, d)$ belong to $A$, then $(a-c, b-d)$ belongs to $A$. $\bullet$ There is a value of $\alpha$ such that by rotating the set $A$ with center at $(0, 0)$ and angle $\alpha$, the set $A$ is obtained again. Prove that $\alpha$ must be equal to $\pm 60^{\circ}$ or $\pm 90^{\circ}$ or $\pm 120^{\circ}$ or $\pm 180^{\circ}$.

Given a straight line with points $A, B, C$ and $D$. Construct using $AB$ and $CD$ regular triangles (in the same half-plane). Let $E,F$ be the third vertex of the two triangles (as in the figure) . The circumscribed circles of triangles $AEC$ and $BFD$ intersect in $G$ ($G$ is is in the half plane of triangles). Prove that the angle $AGD$ is $120^o$ [img]https://1.bp.blogspot.com/-66akc83KSs0/X9j2BBOwacI/AAAAAAAAM0M/4Op-hrlZ-VQRCrU8Z3Kc3UCO7iTjv5ZQACLcBGAsYHQ/s0/2011%2BDurer%2BC5.png[/img]

In how many ways can Eve fill each of the six squares of a $2 \times 3$ grid with either a $0$ or a $1$, such that Anne can then divide the grid into three congruent rectangles: one containing two $0$s, one containing two $1$s, and one containing a $0$ and a $1$? [i]Proposed by Michael Tang[/i]

For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

Sofiya and Marquis are playing a game. Sofiya announces to Marquis that she's thinking of a polynomial of the form $f(x)=x^3+px+q$ with three integer roots that are not necessarily distinct. She also explains that all of the integer roots have absolute value less than (and not equal to) $N$, where $N$ is some fixed number which she tells Marquis. As a "move" in this game, Marquis can ask Sofiya about any number $x$ and Sofiya will tell him whether $f(x)$ is positive negative, or zero. Marquis's goal is to figure out Sofiya's polynomial. If $N=3\cdot 2^k$ for some positive integer $k$, prove that there is a strategy which allows Marquis to identify the polynomial after making at most $2k+1$ "moves".

Diagonals of an isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ are perpendicular. Let $DE$ be the perpendicular from $D$ to $AB$, and let $CF$ be the perpendicular from $C$ to $DE$. Prove that angle $DBF$ is equal to half of angle $FCD$.

Let $\ast$ be the symbol denoting the binary operation on the set $S$ of all non-zero real numbers as follows: For any two numbers $a$ and $b$ of $S$, $a\ast b=2ab$. Then the one of the following statements which is not true, is $\textbf{(A) }\ast\text{ is commutative over }S \qquad\textbf{(B) }\ast\text{ is associative over }S\qquad$ $\textbf{(C) }\frac{1}{2}\text{ is an identity element for }\ast\text{ in }S\qquad\textbf{(D) }\text{Every element of }S\text{ has an inverse for }\ast\qquad$ $\textbf{(E) }\dfrac{1}{2a}\text{ is an inverse for }\ast\text{ of the element }a\text{ of }S$

Let $f(x):\mathbb {Q} \rightarrow \mathbb {Q}$ be a function satisfying $f(x+2y)+f(2x-y)=5f(x)+5f(y)$ Find all such functions.

A square is drawn in the Cartesian coordinate plane with vertices at $(2,2)$, $(-2,2)$, $(-2,-2)$, and $(2,-2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x,y)$ to each of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, $(x-1,y+1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 39$

Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$

Denote by $\lceil x \rceil$ the greatest integer less than or equal to $x$ and by $S(x)$ the sequence $\lceil x \rceil, \lceil 2x \rceil, \lceil 3x \rceil, \dots.$ Prove that there are distinct real solutions $\alpha$ and $\beta$ of the equation $$x^3-10x^2+29x-25=0$$ such that infinitely many positive integers appear both in $S(\alpha)$ and in $S(\beta).$

Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation \[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\] \[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\] \[+(x - b)(x - c)(x - d)(x - e) = 0\] has four distinct real solutions.

Let $ABCD$ be a quadrilateral inscribed in a circle such that $\angle ABC=60^{\circ}.$ a) Prove that if $BC=CD$ then $AB= CD+DA.$ b) Is it true that if $AB= CD+DA$ then $BC=CD$?

In scalene $\triangle ABC$, the tangent from the foot of the bisector of $\angle A$ to the incircle of $\triangle ABC$, other than the line $BC$, meets the incircle at point $K_a$. Points $K_b$ and $K_c$ are analogously defined. Prove that the lines connecting $K_a$, $K_b$, $K_c$ with the midpoints of $BC$, $CA$, $AB$, respectively, have a common point on the incircle.

Two ants sit on the surface of a tetrahedron. Prove that they can meet by breaking the sum of a distance not exceeding the diameter of a circle is circumscribed around the edge of a tetrahedron.

Suppose that $x,y,z$ are positive real numbers that satisfy \begin{align*} x+y+z&=xyz\\ \frac{x^2}{16(1+x^2)}=\frac{y^2}{25(1+y^2)}&=\frac{z^2}{36(1+z^2)}. \end{align*} Compute $$\frac{x^2(1+x^2)^2}{x^2(1+z^2)^2}.$$

Given a triangle $ABC$, let its incircle touch the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $G$ be the midpoint of the segment $DE$. Prove that $\angle EFC = \angle GFD$.

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

Ana & Bruno decide to play a game with the following rules.: a) Ana has cards $1, 3, 5,7,..., 2n-1$ b) Bruno has cards $2, 4,6, 8,...,2n$ During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after $n$ turns. Determine the highest number of points Ana can earn, and how she manages to do this.

Let $ p \equiv 3 \pmod{4}$ be a prime. Define $T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \} $. For arbitrary subset $ S ( \ne \emptyset ) \subset T $, prove that there exist subset $ A \subset S $ satisfying following conditions: (a) $ (x_i , y_i ) \in A ( 1 \le i \le 3) $ then $ p \not | x_1 + x_2 - y_3 $ or $ p \not | y_1 + y_2 + x_3 $. (b) $ 8 n(A) > n(S) $

As shown in figure , $\odot O$ is the inscribed circle of trapezoid $ABCD$, and the tangent points are $E, F, G, H$, $AB \parallel CD$. The line passing through$ B$, parallel to $AD$ intersects extension of $DC$ at point $P$. The extension of $AO$ intersects $CP$ at point $Q$. If $AE=BE$ , prove that $\angle CBQ = \angle PBQ$. [img]https://cdn.artofproblemsolving.com/attachments/d/2/7c3a04bb1c59bc6d448204fd78f553ea53cb9e.png[/img]

Let $a, b, c, d, e$ $(a < b < c < d < e)$be positive integers. FInd the greatest possible value of the expression $\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]}$, where $[x,y]$ denotes the least common multiple of $x{}$ and $y{}$.

Compute the remainder when the product of all positive integers less than and relatively prime to $2019$ is divided by $2019$.