There is a rectangular field that measures $20\text{m}$ by $15\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?

Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.

A massless string is wrapped around a frictionless pulley of mass $M$. The string is pulled down with a force of 50 N, so that the pulley rotates due to the pull. Consider a point $P$ on the rim of the pulley, which is a solid cylinder. The point has a constant linear (tangential) acceleration component equal to the acceleration of gravity on Earth, which is where this experiment is being held. What is the weight of the cylindrical pulley, in Newtons? [i](Proposed by Ahaan Rungta)[/i] [hide="Note"] This problem was not fully correct. Within friction, the pulley cannot rotate. So we responded: [quote]Excellent observation! This is very true. To submit, I'd say just submit as if it were rotating and ignore friction. In some effects such as these, I'm pretty sure it turns out that friction doesn't change the answer much anyway, but, yes, just submit as if it were rotating and you are just ignoring friction. [/quote]So do this problem imagining that the pulley does rotate somehow. [/hide]

In quadrilateral $ABCD$, angles $A, B, C, D$ form an increasing arithmetic sequence. Also, $\angle ACB = 90^o$ . If $CD = 14$ and the length of the altitude from $C$ to $AB$ is $9$, compute the area of $ABCD$.

Consider a system of inequalities \begin{align*}y-x&\ge|x+1|-|x-1|, \\ |y&-x|-y+x\ge2.\end{align*} Draw solutions of each inequality in the plane separately and highlight solution of the system.

Let $ ABCD$ be a convex quadrilateral. Square $ AB_1A_2B$ is constructed such that the two vertices $ A_2,B_1$ is located outside $ ABCD$. Similarly, we construct squares $ BC_1B_2C$, $ CD_1C_2D$, $ DA_1D_2A$. Let $ K$ be the intersection of $ AA_2$ and $ BB_1$, $ L$ be the intersection of $ BB_2$ and $ CC_1$, $ M$ be the intersection of $ CC_2$ and $ DD_1$, and $ N$ be the intersection of $ DD_2$ and $ AA_1$. Prove that $ KM$ is perpendicular to $ LN$.

Let $ x,y,z\in\mathbb R^{\plus{}}$ and $ x\plus{}y\plus{}z\equal{}3$. Prove that: \[ \frac{x^3}{y^3\plus{}8}\plus{}\frac{y^3}{z^3\plus{}8}\plus{}\frac{z^3}{x^3\plus{}8}\geq\frac19\plus{}\frac2{27}(xy\plus{}xz\plus{}yz)\]

A plant grows in the way we describe below. has a trunk which forks into two branches; each branch of the plant can, in turn, branch off into other two branches, or end in a bud. We will call the [i]load [/i] of a branch the total number of buds it bears, that is, the number of buds fed by the sap that passes by that branch; and we will call the [i]distance [/i] of a bud the number of bifurcations that it sap has to go through to get from the trunk to that bud. If n is the number of bifurcations that a certain plant of that type has, it is asks a) the number of branches of the plant, b) the number of buds, c) show that the sum of the charges of all the branches is equal to the sum of the clearances of all buds. Hint: You can proceed by induction, showing that if some results are correct for a given plant, they remain correct for the plant that is obtained substituting a bud in it for a pair of branches ending in individual buds.

Prove that for every positive integer $n$, the number $A_n = 5^n + 2 \cdot 3^{n-1} + 1$ is a multiple of $8$.

Let $O$ and $I$ be circumcenter and incenter of triangle $ABC$. Let incircle of $ABC$ touches sides $BC$, $CA$ and $AB$ in points $D$, $E$ and $F$, respectively. Lines $FD$ and $CA$ intersect in point $P$, and lines $DE$ and $AB$ intersect in point $Q$. Furthermore, let $M$ and $N$ be midpoints of $PE$ and $QF$. Prove that $OI \perp MN$

Find the number of strictly increasing sequences of nonnegative integers with the first term $0$ and the last term $15$, and among any two consecutive terms, exactly one of them is even. Lê Anh Vinh

If $x , y, z \in \mathbb{R}$ such that $x+y +z =4$ and $x^2 + y^2 +z^2 = 6$, then show that each of $x, y, z$ lies in the closed interval $\left[ \dfrac{2}{3} , 2 \right]$. Can $x$ attain the extreme value $\dfrac{2}{3}$ or $2$?

Is it true that for any permulation $a_1,a_2.....,a_{2002}$ of $1,2....,2002$ there are positive integers $m,n$ of the same parity such that $0<m<n<2003$ and $a_m+a_n=2a_{\frac {m+n}{2}}$

$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

There are $n$ aborigines on an island. Any two of them are either friends or enemies. One day, the chieftain orders that all citizens (including himself) make and wear a necklace with zero or more stones so that: (i) given a pair of friends, there exists a color such that each has a stone of that color; (ii) given a pair of enemies,there does not exist a color such that each a stone of that color. (a) Prove that the aborigines can carry out the chieftain’s order. (b) What is the minimum number of colors of stones required for the aborigines to carry out the chieftain’s order?

Anna and Ben decided to visit Archipelago with $2009$ islands. Some pairs of islands are connected by boats which run both ways. Anna and Ben are playing during the trip: Anna chooses the first island on which they arrive by plane. Then Ben chooses the next island which they could visit. Thereafter, the two take turns choosing an island which they have not yet visited. When they arrive at an island which is connected only to islands they had already visited, whoever's turn to choose next would be the loser. Prove that Anna could always win, regardless of the way Ben played and regardless of the way the islands were connected. [i](12 points for Juniors and 10 points for Seniors)[/i]

Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

Let $m$ be an positive odd integer not divisible by $3$. Prove that $\left[4^m -(2+\sqrt 2)^m\right]$ is divisible by $112.$

Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of ​​this quadrilateral does not depend on the choice of the point $M$.

A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.

In two bowls there are in total ${N}$ balls, numbered from ${1}$ to ${N}$. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, ${x}$. Determine the largest possible value of ${x}$.

An ant is on the bottom edge of a right circular cone with base area $\pi$ and slant length $6$. What is the shortest distance that the ant has to travel to loop around the cone and come back to its starting position?

Prove that if X is a compact $T_2$ space, and X has density d(X), then $X^3$ contains a discrete subspace of cardinality $d(X)$. note: $d(X)$ is the smallest cardinality of a dense subspace of X.

Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]