Find all pairs $(a, b)$ of positive rational numbers such that $$\sqrt{a}+\sqrt{b} = \sqrt{2+\sqrt{3}}.$$

Let $AH_1$, $BH_2$ be two altitudes of an acute-angled triangle $ABC$ , $D$ be the projection of $H_1$ to $AC$, $E$ be the projection of $D$ to $AB$, $F$ be the common point of $ED$ and $AH_1$. Prove that $H_2F \parallel BC$. [i](Proposed by E.Diomidov)[/i]

Let $ABC$ be a given triangle. Let $A_1$ and $A_2$ be points on the side $BC$. Let $B_1$ and $B_2$ be points on the side $CA$. Let $C_1$ and $C_2$ be points on the side $AB$. Suppose that the points $A_1,A_2,B_1,B_2,C_1$ and $C_2$ lie on a circle. Prove that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $AA_2, BB_2$ and $CC_2$ are concurrent.

Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$. Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.

Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which \[ (f(n))^p \equiv n\quad {\rm mod}\; f(p) \] for all $ n \in {\bf N}$ and all prime numbers $ p$.

Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$. A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$, respectively. Let $X$ be the intersection of the line $BE$ and $DS$, and $Y$ be the intersection of the line $CE$ and $DT$. Prove that, if the line $AD$ is tangent to the circle $\odot(DXY)$, then the line $AE$ is tangent to the circle $\odot(EXY)$. [i]Proposed by ltf0501.[/i]

There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$. There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$. Show that the sphere touches the circumsphere of $ABCP$.

Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?

[list=1] [*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$ [*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$ [/list]

What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 36 $

Let $X=\{1,2,\ldots,2001\}$. Find the least positive integer $m$ such that for each subset $W\subset X$ with $m$ elements, there exist $u,v\in W$ (not necessarily distinct) such that $u+v$ is of the form $2^{k}$, where $k$ is a positive integer.

Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$ $$f(x)f(f(x)+y) = f(x^2) + f(xy).$$ [i]Proposed by Culver Kwan[/i]

Without using tables, find the exact value of the product: \[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]

$ABCD$ is a unit square. If $\angle PAC =\angle PCD$, find the length $BP$.

Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.

Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board? Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$. Since Lisa gets $21$ for the second time, the turn order ends. [i](Stephan Pfannerer)[/i]

Given two non-negative integers $m>n$, let's say that $m$ [i]ends in[/i] $n$ if we can get $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9. Determine how many three-digit numbers end in the product of their digits.

A sequence of positive real numbers $a_1, a_2, \dots$ is called $\textit{phine}$ if it satisfies $$a_{n+2}=\frac{a_{n+1}+a_{n-1}}{a_n},$$ for all $n\geq2$. Is there a $\textit{phine}$ sequence such that, for every real number $r$, there is some $n$ for which $a_n>r$?

[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

There are an even number of points in a plane. No three of them lie on one straight line. Half of the points are red, the other half are blue. Prove that there exists a connecting line of a red and a blue point such that in each of the half-planes bounded by that line the number of red points is equal to the number of blue points.

Find for every value of $n$ a set of numbers $p$ for which the following statement is true: Any convex $n$-gon can be divided into $p$ isosceles triangles.

Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Determine the number of positive divisors of $2008^8$ that are less than $2008^4$.

Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.