Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that: \[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}. \] Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.

Find all triples $(x,y,z)$ of real numbers that satisfy \[ \begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned} \]

When a bucket is two-thirds full of water, the bucket and water weigh $ a$ kilograms. When the bucket is one-half full of water the total weight is $ b$ kilograms. In terms of $ a$ and $ b$, what is the total weight in kilograms when the bucket is full of water? $ \textbf{(A)}\ \frac23a\plus{}\frac13b\qquad \textbf{(B)}\ \frac32a\minus{}\frac12b\qquad \textbf{(C)}\ \frac32a\plus{}b$ $ \textbf{(D)}\ \frac32a\plus{}2b\qquad \textbf{(E)}\ 3a\minus{}2b$

In each $ 2007^{2}$ unit squares on chess board whose size is $ 2007\times 2007$, there lies one coin each square such that their "heads" face upward. Consider the process that flips four consecutive coins on the same row, or flips four consecutive coins on the same column. Doing this process finite times, we want to make the "tails" of all of coins face upward, except one that lies in the $ i$th row and $ j$th column. Show that this is possible if and only if both of $ i$ and $ j$ are divisible by $ 4$.

A box with weight $W$ will slide down a $30^\circ$ incline at constant speed under the influence of gravity and friction alone. If instead a horizontal force $P$ is applied to the box, the box can be made to move up the ramp at constant speed. What is the magnitude of $P$? $\textbf{(A) } P = W/2 \\ \textbf{(B) } P = 2W/\sqrt{3}\\ \textbf{(C) } P = W\\ \textbf{(D) } P = \sqrt{3}W \\ \textbf{(E) } P = 2W$

Given a triangle \(ABC\), an arbitrary point \(D\) is chosen on the side \(AC\). In triangles \(ABD\) and \(CBD\), the angle bisectors \(BK\) and \(BL\) are drawn, respectively. The point \(O\) is the circumcenter of \(\triangle KBL\). Prove that the second intersection point of the circumcircles of triangles \(ABL\) and \(CBK\) lies on the line \(OD\). [i]Proposed by Anton Trygub[/i]

Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$. When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that?

Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n}\right)^b$ , where $m, n$ and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.

Let $S$ be a subset of $\{1,2,3,4,5,6,7,8,9,10\}$. If $S$ has the property that the sums of three elements of $S$ are all different, find the maximum number of elements of $S$.

We call a bar of width ${w}$ on the surface of the unit sphere ${\Bbb{S}^2}$, a spherical segment, centered at the origin, which has width ${w}$ and is symmetric with respect to the origin. Prove that there exists a constant ${c>0}$, such that for any positive integer ${n}$ the surface ${\Bbb{S}^2}$ can be covered with ${n}$ bars of the same width so that any point is contained in no more than ${c\sqrt{n}}$ bars.

In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city $A$ can be[i] k -directionally[/i] connected to the city $B$, if : we can orient at most $k$ roads such that after[i] arbitrary[/i] orientation of remaining roads for any fixed road $l$ (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at $A$, passing through $l$ and ending at $B$ and visiting each city at most once. Suppose that in a country with $n$ cities, any two cities can be[i] k - directionally[/i] connected. What is the minimal value of $k$?

A circle is drawn through vertices $A$ and $B$ of triangle $ABC$, intersecting sides $AC$ and $BC$ at points $M$ and $P$. It is known that the segment $MP$ contains the center of the circle inscribed in $ABC$. Find $MP$ if $AB = c$, $BC = a$, $CA=b$.

a) In how many ways can we read the word SARAJEVO from the table below, if it is allowed to jump from cell to an adjacent cell (by vertex or a side) cell? b) After the letter in one cell was deleted, only $525$ ways to read the word SARAJEVO remained. Find all possible positions of that cell.

In $\triangle ABC$, $\angle ACB=3\angle BAC$, $BC=5$, $AB=11$. Find $AC$

For how many different primes $p$, there exists an integer $n$ such that $ p\mid n^3+3$ and $p\mid n^5+5$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{Infinitely many} $

The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

There are $n$ non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$. Show that $A,X,Y$ are collinear.

The circle inscribed in triangle $ABC$ touches $AC$ at point $F$. The perpendicular from point $F$ on $BC$ intersects the bisector of angle $C$ at point $N$. Prove that segment $FN$ is equal to the radius of the circle inscribed in triangle $ABC$. (Oleksii Karliuchenko)

Two players, the builder and the destroyer, plays the following game. Builder starts and players chooses alternatively different elements from the set $\{0,1,\ldots,10\}.$ Builder wins if some four integer of those six integer he chose forms an arithmetic sequence. Destroyer wins if he can prevent to form such an arithmetic four-tuple. Which one has a winning strategy?

Find all positive integers $n$ that have precisely $\sqrt{n+1}$ natural divisors.

For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold: Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as \[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\] where $\#S$ is the number of elements in set $S$, and $\pi_j$ is the projection $\mathbb{R}^k\to \mathbb{R}^{k-1}$ omitting the $j$-th coordinate. Then the $t$-th power mean of the scores of all $a^{(i)}$'s is at most $n$. Note: The $t$-th power mean of positive real numbers $x_1,\ldots,x_n$ is defined as \[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\] when $t\neq 0$, and it is $\sqrt[n]{x_1\cdots x_n}$ when $t=0$. [i]Proposed by Cheng-Ying Chang and usjl[/i]

Which of the followings is false for the sequence $9,99,999,\dots$? $\textbf{(A)}$ The primes which do not divide any term of the sequence are finite. $\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence. $\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers. $\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence. $\textbf{(E)}$ None of above

Let $ABC$ be a triangle with circumcircle $\Gamma$. If the internal angle bisector of $\angle A$ meets $BC$ and $\Gamma$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $A$ and $D$ tangent to $BC$, let the external angle bisector of $\angle A$ meet $\Gamma$ at $F$, and let $FO_1$ meet $\Gamma$ at some point $P \neq F$. Show that the circumcircle of $DEP$ is tangent to $BC$.