Solve in integers the system of equations: $$x^2-y^2=z$$ $$3xy+(x-y)z=z^2$$

Let $ABC$ be an acute triangle, and let $H$ be its orthocentre. Denote by $H_A$, $H_B$, and $H_C$ the second intersection of the circumcircle with the altitudes from $A$, $B$, and $C$ respectively. Prove that the area of triangle $H_A H_B H_C$ does not exceed the area of triangle $ABC$.

Let $ \overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB\equal{}\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segment whose endpoints are the intersections of diameter $ \overline{MN}$ with the chords $ \overline{AC}$ and $ \overline{BC}$. The largest possible value of $ d$ can be written in the form $ r\minus{}s\sqrt{t}$, where $ r$, $ s$, and $ t$ are positive integers and $ t$ is not divisible by the square of any prime. Find $ r\plus{}s\plus{}t$.

The lateral sidelines $AB$ and $CD$ of trapezoid $ABCD$ meet at point $S$. The bisector of angle $ASC$ meets the bases of the trapezoid at points $K$ and $L$ ($K$ lies inside segment $SL$). Point $X$ is chosen on segment $SK$, and point $Y$ is selected on the extension of $SL$ beyond $L$ such a way that $\angle AXC - \angle AYC = \angle ASC$. Prove that $\angle BXD - \angle BYD = \angle BSD$.

Albert and Kevin are playing a game. Kevin has a $10\%$ chance of winning any given round in the match. If Kevin wins the first game, he wins the match. If not, he requests that the match be extended to a best of $3$. If he wins the best of $3$, he wins the match. If not, then he requests the match be extended to a best of $5$, and so forth. What is the probability that Kevin eventually wins the match? (A best of $2n+ 1$ match consists of a series of rounds. The first person to reach $n + 1$ winning games wins the match)

The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers. Find $p+q.$ [asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); }[/asy]

You can tile a $2 \times5$ grid of squares using any combination of three types of tiles: single unit squares, two side by side unit squares, and three unit squares in the shape of an L. The diagram below shows the grid, the available tile shapes, and one way to tile the grid. In how many ways can the grid be tiled? [asy] import graph; size(15cm); pen dps = linewidth(1) + fontsize(10); defaultpen(dps); draw((-3,3)--(-3,1)); draw((-3,3)--(2,3)); draw((2,3)--(2,1)); draw((-3,1)--(2,1)); draw((-3,2)--(2,2)); draw((-2,3)--(-2,1)); draw((-1,3)--(-1,1)); draw((0,3)--(0,1)); draw((1,3)--(1,1)); draw((4,3)--(4,2)); draw((4,3)--(5,3)); draw((5,3)--(5,2)); draw((4,2)--(5,2)); draw((5.5,3)--(5.5,1)); draw((5.5,3)--(6.5,3)); draw((6.5,3)--(6.5,1)); draw((5.5,1)--(6.5,1)); draw((7,3)--(7,1)); draw((7,1)--(9,1)); draw((7,3)--(8,3)); draw((8,3)--(8,2)); draw((8,2)--(9,2)); draw((9,2)--(9,1)); draw((11,3)--(11,1)); draw((11,3)--(16,3)); draw((16,3)--(16,1)); draw((11,1)--(16,1)); draw((12,3)--(12,2)); draw((11,2)--(12,2)); draw((12,2)--(13,2)); draw((13,2)--(13,1)); draw((14,3)--(14,1)); draw((14,2)--(15,2)); draw((15,3)--(15,1));[/asy]

Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$, $B$. $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$. $AC$ tangents $C(O_1)$ at $A$. Draw $AE \bot CD$, and $AE$ intersects $C(O_1)$ at $E$. Draw $AF \bot DE$, and $AF$ intersects $DE$ at $F$. Prove that $BD$ bisects $AF$.

For each positive integer $k$, let $S_k$ be the set of real numbers that can be expressed in the form \[\frac{1}{n_1}+\frac{1}{n_2}+\dots+\frac{1}{n_k},\] where $n_1,n_2\dots,n_k$ are positive integers. Prove that $S_k$ does not contain an infinite strictly increasing sequence.

For each positive integer $n$ write the sum $\sum_{i=}^{n}\frac{1}{i}=\frac{p_n}{q_n}$ with $\text{gcd}(p_n,q_n)=1$. Find all such $n$ such that $5\nmid q_n$.

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.

Let $ABC$ be a triangle with $AB = AC$ with centroid $G$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$ respectively and $O$ be the circumcenter of triangle $BCN$ . Prove that $MBOG$ is a cyclic quadrilateral .

A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.

Let $O$ be the incenter of a tangential quadrilateral $ABCD$. Prove that the orthocenters of $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$, $\vartriangle DOA$ lie on a line.

Find the sum of all positive integers $n$ such that $1+2+\cdots+n$ divides \[15\left[(n+1)^2+(n+2)^2+\cdots+(2n)^2\right].\]

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}$$

Given $m\in\mathbb{N}$ and a prime number $p$, $p>m$, let \[M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\,\, \text{is a perfect square} \} \] Prove that $|M|$ does not depend on $p$. [i]Proposed by Aleksandar Ivanov[/i]

Two distinct points $A$ and $B$ are marked on the left half of the parabola $y = x^2$. Consider any pair of parallel lines which pass through $A$ and $B$ and intersect the right half of the parabola at points $C$ and $D$. Let $K$ be the intersection point of the diagonals $AC$ and $BD$ of the obtained trapezoid $ABCD$. Let $M, N$ be the midpoints of the bases of $ABCD$. Prove that the difference $KM - KN$ depends only on the choice of points $A$ and $B$ but does not depend on the pair of parallel lines described above. (I. Voronovich)

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

Prove that the polynomial $x^{1999}+x^{1998}+...+x^3+x^2+ax+b$ for any real values of the coefficients $a>b>0$ does not have an integer root.

Paul wrote the list of all four-digit numbers such that the hundreds digit is $5$ and the tens digit is $7$. For example, $1573$ and $7570$ are on Paul's list, but $2754$ and $571$ are not. Find the sum of all the numbers on Pablo's list. $Note$. The numbers on Pablo's list cannot start with zero.

Find all pairs of real numbers $(x, y)$ that satisfy the equation $\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}$

Find all sequences of positive integers $a_1,a_2,\dots$ such that $$(n^2+1)a_n = n(a_{n^2}+1)$$ for all positive integers $n$.

There are $2500$ people in Lexington High School, who all start out healthy. After $1$ day, $1$ person becomes infected with coronavirus. Each subsequent day, there are twice as many newly infected people as on the previous day. How many days will it be until over half the school is infected?

Let $ S $ be the set of positive integers which are not divisible by perfect squares greater than $ 1.$ Prove that for every $n\in\mathbb{N}$ the following equality is true $$\sum_{k\in S}\left[\sqrt{\frac{n}{k}}\right]=n,$$ where $[x]$ is the integer part of $x\in\mathbb{R}.$