Prove that there are infinitely many integers $m$, $n$, such that $1 < m < n$, and the greatest common divisors $(m, n)$, $(m, n+1)$, $(m+1, n)$ and $(m+1, n+1)$ are all greater than $\sqrt{n}/999$.

Four distinct circles $C,C_1, C_2$, C3 and a line L are given in the plane such that $C$ and $L$ are disjoint and each of the circles $C_1, C_2, C_3$ touches the other two, as well as $C$ and $L$. Assuming the radius of $C$ to be $1$, determine the distance between its center and $L.$

Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$ $$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$

Prove that $$\frac{2}{2!}+\frac{7}{3!}+\frac{14}{4!}+\frac{23}{5!}+...+\frac{k^2-2}{k!}+...+\frac{9998}{100!}<3$$ where $n! = 1 \times 2 \times ... \times n.$ (V Senderov)

A particle is launched from the surface of a uniform, stationary spherical planet at an angle to the vertical. The particle travels in the absence of air resistance and eventually falls back onto the planet. Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion. Spacewoman Kate recalls from Kepler’s laws that every bound orbit around a point mass is an ellipse (or circle), and that the gravitation due to a uniform sphere is identical to that of a point mass. Which of the following best explains the discrepancy? (A) Because the experiment takes place very close to the surface of the sphere, it is no longer valid to replace the sphere with a point mass. (B) Because the particle strikes the ground, it is not in orbit of the planet and therefore can follow a nonelliptical path. (C) Kate disregarded the fact that motions around a point mass may also be parabolas or hyperbolas. (D) Kepler’s laws only hold in the limit of large orbits. (E) The path is an ellipse, but is very close to a parabola due to the short length of the flight relative to the distance from the center of the planet.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

What is the units digit of $2013^{2013}$?

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?

Let $n!_0$ denote the number obtained from $n!$ by deleting all the zeroes in the end of it decimal representation. Find all positive integers $a, b, c$, such that $a!_0+b!_0=c!_0$.

In the triangle $ABC$, the median $BD$ is drawn and through its midpoint and vertex $A$ the line $\ell$. Thus the triangle $ABC$ is divided into three triangles and one quadrilateral. Determine the areas of these figures if the area of ​​triangle $ABC$ is equal to $S$.

Find all positive integers $n$, such that there exist positive integers $a,b$, such that $a+2^b=n^{2022}$ and $a^2+4^b=n^{2023}$.

Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$, where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$.

Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.

Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?

Find the sum of all positive integers $n$ such that $$\frac{n+11}{\sqrt{n-1}}$$ is an integer.

$a,b,c$ are positive real numbers which satisfy $5abc>a^3+b^3+c^3$. Prove that $a,b,c$ can form a triangle.

Let $a,b$ be real numbers. Prove the inequality \[ 2(a^4+a^2b^2+b^4)\ge 3(a^3b+ab^3).\]

Solve the equation $$\log_{10} (x^3+x)=\log_2 x.$$

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ respectively. $A_2$ and $B_2$ are the points symmetric to $A_1$ and $B_1$ respectively, with respect to the midpoints of $CA$ and $CB$. Prove that the points $A,B,A_2,B_2$ are concyclic. [i]I. Bogdanov[/i]

Circles ${w}_{1},{w}_{2}$ intersect at points ${{A}_{1}} $ and ${{A}_{2}} $. Let $B$ be an arbitrary point on the circle ${{w}_{1}}$, and line $B{{A}_{2}}$ intersects circle ${{w}_{2}}$ at point $C$. Let $H$ be the orthocenter of $\Delta B{{A}_{1}}C$. Prove that for arbitrary choice of point $B$, the point $H$ lies on a certain fixed circle.

Assume that $f : [0,1)\to R$ is a function such that $f(x)-x^3$ and $f(x)-3x$ are both increasing functions. Determine if $f(x)-x^2-x$ is also an increasing function.

Find all pairs of positive integers \( a, b \) such that one of the two numbers \( 2(a^2 + b^2) \) and \( (a + b)^2 + 4 \) is divisible by the other. [i]Proposed by Oleksii Masalitin[/i]

Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus. [i]Proposed by Tommy Walker Mackay, United Kingdom[/i]

Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that: \[ AQ+CQ=BP. \]