Show that for any rational number $a$ the equation $y =\sqrt{x^2 +a}$ has infinitely many solutions in rational numbers $x$ and $y$.

The positive differences $a_i-a_j$ of five different positive integers $a_1, a_2, a_3, a_4, a_5$ are all different (there are altogether $10$ such differences). Find the least possible value of the largest number among the $a_i$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.

$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$.

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

Segment $\overline{PQ}$ is drawn and squares $ABPQ$ and $CDQP$ are constructed in the plane such that they lie on opposite sides of segment $\overline{PQ}.$ If $PQ=1,$ find $BD.$

If $a,b,c$ are triangle sides then prove that $(\sum_{cyc}\sqrt{\frac{a}{-a+b+c}} \geq 3$

The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively. Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$

[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?

$ABC$ is a triangle. A circle through $A$ touches the side $BC$ at $D$ and intersects the sides $AB$ and $AC$ again at $E, F$ respectively. $EF$ bisects $\angle AFD$ and $\angle ADC = 80^o$. Find $\angle ABC$.

For real numbers $x,y$ satisfying $x^2+y^2-4x-2y+4=0$, what is the greatest value of $$16\cos^2\sqrt{x^2+y^2}+24\sin\sqrt{x^2+y^2}?$$

Let $\triangle ABC$ be a triangle with $\angle A \neq 60^\circ$. Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, let $B^\prime$ be the reflection of $B$ with respect to $AC$, and let $C^\prime$ be the reflection of $C$ with respect to $AB$. Let $P$ be the intersection of $I_C B^\prime$ and $I_B C^\prime$. Denote by $P_A, P_B, P_C$ the reflections of the point $P$ with respect to $BC, CA, AB$. Show that the three lines $A P_A, B P_B, C P_C$ meet at a single point.

The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is [asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S); label("$10$",(9,8),N); label("$6$",(7,0),S); label("$8$",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy] $\text{(A)}\ 24 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80$

For every $n \in \mathbb{N}$, there exist integers $k$ such that $n | k$ and $k$ contains only zeroes and ones in its decimal representation. Let $f(n)$ denote the least possible number of ones in any such $k$. Determine whether there exists a constant $C$ such that $f(n) < C$ for all $n \in \mathbb{N}$.

How many positive five-digit integers are there that have the product of their five digits equal to $900$? (Karl Czakler)

Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $|a + b\omega + c\omega^2|$.  

Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$. Line $EF$ meets $BC$ at $P$, and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$, respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$. Prove that $BN>CN$.

Peter buys a lottery ticket on which he enters an $n$-digit number, none of the digits being $0$. On the draw date, the lottery administrators will reveal an $n\times n$ table, each cell containing one of the digits from $1$ to $9$. A ticket wins a prize if it does not match any row or column of this table, read in either direction. Peter wants to bribe the administrators to reveal the digits on some cells chosen by Peter, so that Peter can guarantee to have a winning ticket. What is the minimum number of digits Peter has to know?

An infinite sequence of real numbers $a_1,a_2,a_3,\dots$ is called $\emph{spooky}$ if $a_1=1$ and for all integers $n>1$, \[\begin{array}{c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c} na_1&+&(n-1)a_2&+&(n-2)a_3&+&\dots&+&2a_{n-1}&+&a_n&<&0,\\ n^2a_1&+&(n-1)^2a_2&+&(n-2)^2a_3&+&\dots&+&2^2a_{n-1}&+&a_n&>&0. \end{array}\]Given any spooky sequence $a_1,a_2,a_3,\dots$, prove that \[2013^3a_1+2012^3a_2+2011^3a_3+\cdots+2^3a_{2012}+a_{2013}<12345.\]

Suppose each element $i\in S=\{1,2,\ldots,n\}$ is assigned a nonempty set $S_i\subseteq S$ so that the following conditions are fulfilled: (i) for any $i,j\in S$, if $j\in S_i$ then $i\in S_j$; (ii) for any $i,j\in S$, if $|S_i|=|S_j|$ then $S_i\cap S_j=\emptyset$. Prove that there exists $k\in S$ for which $|S_k|=1$.

Find the formula for the general term of the sequence an, for which $a_1 = 1$, $a_2 = 3$, $a_{n+1} = 3a_n-2a_{n-1}$ (you need to express an in terms of $n$).

Assume that for pairwise distinct real numbers $a,b,c,d$ holds: $$ (a^2+b^2-1)(a+b)=(b^2+c^2-1)(b+c)=(c^2+d^2-1)(c+d).$$ Prove that $ a+b+c+d=0.$

Let $\Omega=\{z=x+iy~\in\mathbb{C}~:~|y|\leqslant 1\}$. If $f(z)=z^2+2$, then draw a sketch of $$f\Big(\Omega\Big)=\{f(z):z\in\Omega\}$$ Justify your answer.

Does there exist on the Cartesian plane a convex $2023$-gon with vertices at integer points, such that the lengths of all its sides are equal? [i]Proposed by Anton Trygub[/i]