$30$ segments of lengths$$1,\quad \sqrt{3},\quad \sqrt{5},\quad \sqrt{7},\quad \sqrt{9},\quad \ldots ,\quad \sqrt{59} $$ have been drawn on a blackboard. In each step, two of the segments are deleted and a new segment of length equal to the hypotenuse of the right triangle with legs equal to the two deleted segments is drawn. After $29$ steps only one segment remains. Find the possible values of its length.

In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.

Consider a convex quadrilateral \(ABCD\) with \(AC > BD\). In the plane of this quadrilateral, points \(M\) and \(N\) are chosen such that triangles \(ABM\) and \(CDN\) are equilateral, and segments \(MD\) and \(NA\) intersect lines \(AB\) and \(CD\) respectively. Similarly, points \(P\) and \(Q\) are chosen such that triangles \(ADP\) and \(BCQ\) are equilateral, but here segments \(PB\) and \(QA\) do not intersect lines \(AD\) and \(BC\) respectively. Prove that \(MN = AC + BD\) if and only if \(PQ = AC - BD\). [i]Proposed by Zaza Meliqidze, Georgia [/i]

Let $ABC$ be a right triangle with hypotenuse $AB$. Point $E$ is on $AB$ with $AE = 10BE$, and point $D$ is outside triangle $ABC$ such that $DC = DB$ and $\angle CDA = \angle BDE$. Let $[ABC]$ and $[BCD]$ denote the areas of triangles $ABC$ and $BCD$. Determine the value of $\frac{[BCD]}{[ABC]}$.

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ f(xy+f(x)) + f(y) = xf(y) + f(x+y) \] for all real numbers $x$ and $y$.

Suppose $a$ and $b$ are two variables that satisfy $\textstyle\int_0^2(-ax^2+b)dx=0$. What is $\tfrac{a}{b}$?

Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy? $ \textbf {(A) } 33 \qquad \textbf {(B) } 34 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 38 \qquad \textbf {(E) } 39$

Let $ABC$ be an acute triangle and $D$ a point on the side $BC$ such that $\angle BAD = \angle DAC$. The circumcircles of the triangles $ABD$ and $ACD$ intersect the segments $AC$ and $AB$ at $E$ and $F$, respectively. The internal bisectors of $\angle BDF$ and $\angle CDE$ intersect the sides $AB$ and $AC$ at $P$ and $Q$, respectively. Points $X$ and $Y$ are chosen on the side $BC$ such that $PX$ is parallel to $AC$ and $QY$ is parallel to $AB$. Finally, let $Z$ be the point of intersection of $BE$ and $CF$. Prove that $ZX = ZY$.

Let $\alpha, \beta$ be two rational numbers strictly between 0 and 1. Alice and Bob play a game. At the start of the game, Alice chooses a positive integer $n$. Knowing that, Bob then chooses a positive integer $T$. They then do the following for $T$ rounds: at the $i$th round, Bob chooses a set $X_i$ of $n$ positive integers that form a complete residue system modulo $n$. Then Alice chooses a subset $Y_i$ of $X_i$ such that the sum of elements in $Y_i$ is at most $\alpha$ times the sum of elements in $X_i$. After the $T$ rounds, Alice wins if it is possible to pick an integer $s$ between 0 and $n-1$ such that there are at least $\beta T$ positive integers among the elements in $Y_1, Y_2, . . . , Y_T$ (counted with multiplicities) that are equal to $s \pmod n$, and Bob wins otherwise. Find all pairs $(\alpha, \beta)$ of rational numbers strictly between 0 and 1 such that Alice has a winning strategy. [i]Proposed by Hans[/i]

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

For any two convex polygons $P_1$ and $P_2$ with mutually distinct vertices, denote by $f(P_1, P_2)$ the total number of their vertices that lie on a side of the other polygon. For each positive integer $n \ge 4$, determine \[ \max \{ f(P_1, P_2) ~ | ~ P_1 ~ \text{and} ~ P_2 ~ \text{are convex} ~ n \text{-gons} \}. \] (We say that a polygon is convex if all its internal angles are strictly less than $180^\circ$.) [i]Josef Tkadlec (Czech Republic)[/i]

A quadrilateral $ABCD$ without parallel sides is inscribed in a circle $\omega$. We draw a line $\ell_a \parallel BC$ through the point $A$, a line $\ell_b \parallel CD$ through the point $B$, a line $\ell_c \parallel DA$ through the point $C$, and a line $\ell_d \parallel AB$ through the point $D$. Suppose that the quadrilateral whose successive sides lie on these four straight lines is inscribed in a circle $\gamma$ and that $\omega$ and $\gamma$ intersect in points $E$ and $F$. Show that the lines $AC, BD$ and $EF$ intersect in one point. [i]Proposed by A. Kuznetsov[/i]

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.