Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$. Show that there exists a finite sequence of reflections in the sides of $K$, such that $K$ contains the image of $X$ after these reflections.

Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals. [i]Dan Schwarz[/i]

Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the form $\sqrt{m} - n,$ where $m$ and $n$ are positive integers. Find $m+n$.

Two geometric sequences $ a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3\ldots $have the same common ratio, with $a_1=27$,$b_1=99$, and $a_{15}=b_{11}$. Find $a_9.$

A permutation $ (a_1,a_2,a_3,a_4,a_5)$ of $ (1,2,3,4,5)$ is heavy-tailed if $ a_1 \plus{} a_2 < a_4 \plus{} a_5$. What is the number of heavy-tailed permutations? $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 52$

The fraction $ \frac{1}{3}$: $ \textbf{(A)}\ \text{equals 0.33333333} \qquad \textbf{(B)}\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\ \textbf{(C)}\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^9} \\ \textbf{(D)}\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\ \textbf{(E)}\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^9}$

Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1. Prove that there exists a circle with diameter $\sqrt{2}$ such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.

Let $A_n$ denote the answer to the $n$th problem on this contest ($n=1,\dots,30$); in particular, the answer to this problem is $A_1$. Compute $2A_1(A_1+A_2+\dots+A_{30})$. [i]Proposed by Yang Liu[/i]

Let $ABC$ be an acute scalene triangle. Let $D, E$ be points on segments $AB, AC$ respectively, such that $BD=CE$. Prove that the nine-point centers of $ADE$, $ACD$, $ABC$, $AEB$ form a rhombus. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

If $S$ is a sequence of positive integers let $p(S)$ be the product of the members of $S$. Let $m(S)$ be the arithmetic mean of $p(T)$ for all non-empty subsets $T$ of $S$. Suppose that $S'$ is formed from $S$ by appending an additional positive integer. If $m(S) = 13$ and $m(S') = 49$, find $S'$.

In triangle $ABC,$ points $M$ and $N$ are the midpoints of $AB$ and $AC,$ respectively, and points $P$ and $Q$ trisect $BC.$ Given that $A, M, P, N,$ and $Q$ lie on a circle and $BC=1,$ compute the area of triangle $ABC.$

Determine the smallest real constant $c$ such that \[\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2\] for all positive integers $n$ and all positive real numbers $x_1,\cdots ,x_n$.

Prove the following inequality. \[\frac{e-1}{n+1}\leqq\int^e_1(\log x)^n dx\leqq\frac{(n+1)e+1}{(n+1)(n+2)}\ (n=1,2,\cdot\cdot\cdot) \] 1994 Kyoto University entrance exam/Science

A worm is called an [i]adult[/i] if its length is one meter. In one operation, it is possible to cut an adult worm into two (possibly unequal) parts, each of which immediately becomes a worm and begins to grow at a speed of one meter per hour and stops growing once it reaches one meter in length. What is the smallest amount of time in which it is possible to get $n{}$ adult worms starting with one adult worm? Note that it is possible to cut several adult worms at the same time.

Prove that the following polynomial is irreducible in $ \mathbb Z[x,y]$: \[ x^{200}y^5\plus{}x^{51}y^{100}\plus{}x^{106}\minus{}4x^{100}y^5\plus{}x^{100}\minus{}2y^{100}\minus{}2x^6\plus{}4y^5\minus{}2\]

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point \(A\). At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path \(AJABCHCHIJA\), which has \(10\) steps. Let \(n\) be the number of paths with \(15\) steps that begin and end at point \(A\). Find the remainder when \(n\) is divided by \(1000\). [asy] unitsize(32); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } real s = 4; dot(1 * dir( 90), linewidth(s)); dot(1 * dir(162), linewidth(s)); dot(1 * dir(234), linewidth(s)); dot(1 * dir(306), linewidth(s)); dot(1 * dir(378), linewidth(s)); dot(2 * dir(378), linewidth(s)); dot(2 * dir(306), linewidth(s)); dot(2 * dir(234), linewidth(s)); dot(2 * dir(162), linewidth(s)); dot(2 * dir( 90), linewidth(s)); defaultpen(fontsize(10pt)); real r = 0.05; label("$A$", (1-r) * dir( 90), -dir( 90)); label("$B$", (1-r) * dir(162), -dir(162)); label("$C$", (1-r) * dir(234), -dir(234)); label("$D$", (1-r) * dir(306), -dir(306)); label("$E$", (1-r) * dir(378), -dir(378)); label("$F$", (2+r) * dir(378), dir(378)); label("$G$", (2+r) * dir(306), dir(306)); label("$H$", (2+r) * dir(234), dir(234)); label("$I$", (2+r) * dir(162), dir(162)); label("$J$", (2+r) * dir( 90), dir( 90)); [/asy]

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

What is the sum of the roots of $z^{12} = 64$ that have a positive real part? $\textbf{(A) }2 \qquad\textbf{(B) }4 \qquad\textbf{(C) }\sqrt{2} +2\sqrt{3}\qquad\textbf{(D) }2\sqrt{2}+ \sqrt{6} \qquad\textbf{(E) }(1 + \sqrt{3}) + (1+\sqrt{3})i$

A sequence of positive reals $\{ a_n \}$ is defined below. $$a_0 = 1, a_1 = 3, a_{n+2} = \frac{a_{n+1}^2+2}{a_n}$$ Show that for all nonnegative integer $n$, $a_n$ is a positive integer.

Let $A$ be a set of positive integers satisfying the following : $a.)$ If $n \in A$ , then $n \le 2018$. $b.)$ If $S \subset A$ such that $|S|=3$, then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$ What is the maximum cardinality of $A$ ?

[asy] draw((0,1)--(4,1)--(4,2)--(0,2)--cycle); draw((2,0)--(3,0)--(3,3)--(2,3)--cycle); draw((1,1)--(1,2)); label("1",(0.5,1.5)); label("2",(1.5,1.5)); label("32",(2.5,1.5)); label("16",(3.5,1.5)); label("8",(2.5,0.5)); label("6",(2.5,2.5)); [/asy] The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$.

Let $ABCD$ be a cyclic quadrilateral, $E = AC \cap BD$, $F = AD \cap BC$. The bisectors of angles $AFB$ and $AEB$ meet $CD$ at points $X, Y$ . Prove that $A, B, X, Y$ are concyclic.

Given an isosceles $ABC$, which has $2AC = AB + BC$. Denote $I$ the center of the inscribed circle, $K$ the midpoint of the arc $ABC$ of the circumscribed circle. Let $T$ be such a point on the line $AC$ that $\angle TIB = 90 {} ^ \circ$. Prove that the line $TB$ touches the circumscribed circle $\Delta KBI$. (Anton Trygub)