Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$.

A [i]diagonal line[/i] of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon. [i]The Problem Selection Committee[/i]

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.

The convex pentagon $ ABCDE $ has the following properties: $ \text{(i)} AB=BC $ $ \text{(ii)} \angle ABE+\angle CBD =\angle DBE $ $ \text{(iii)} \angle AEB +\angle BDC=180^{\circ} $ Prove that the orthocenter of $ BDE $ lies on $ AC. $

Let $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumference of $C$ such that $PQ \perp OP$, and let $D$ be the circle with diameter $PQ$. Consider a circle tangent to $C$ whose circumference passes through point $P$. Let the curve $\Gamma$ be the locus of the centers of all such circles. If the area enclosed by $\Gamma$ is $1/100$ the area of $C$, then what is the ratio of the area of $C$ to the area of $D$?

Find all natural numbers $n$, such that $3$ divides the number $n\cdot 2^n+1$.

For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$.

A side of an arbitrary triangle has a length greater than $1$. Prove that the given triangle it can be cut into at least $2$ triangles, so that each of them has a side of length equal to $1$.

Prove that for all positive integers $n$, $1^3 + 2^3 + 3^3 +\dots+n^3$ is a perfect square.

The value of the expression $$\sum_{n=2}^{\infty} \frac{\tbinom{n}{2}}{7^{n-2}} = 1+\frac{3}{7}+\frac{6}{49}+\frac{10}{343}+\frac{15}{2401}+\dots+\frac{\binom{n}{2}}{7^{n-2}}+\cdots$$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Individual #11[/i]

Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.

Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that \[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]

A table with $m$ rows and $n$ columns is given. In each cell of the table an integer is written. Heisuke and Oscar play the following game: at the beginning of each turn, Heisuke may choose to swap any two columns. Then he chooses some rows and writes down a new row at the bottom of the table, with each cell consisting the sum of the corresponding cells in the chosen rows. Oscar then deletes one row chosen by Heisuke (so that at the end of each turn there are exactly $m$ rows). Then the next turn begins and so on. Prove that Heisuke can assure that, after some finite amount of turns, no number in the table is smaller than the number to the number on his right. Example: If we begin with $(1,1,1),(6,5,4),(9,8,7)$, Heisuke may choose to swap the first and third column to get $(1,1,1),(4,5,6),(7,8,9)$. Then he chooses the first and second rows to obtain $(1,1,1),(4,5,6),(7,8,9),(5,6,7)$. Then Oscar has to delete either the first or the second row, let's say the second. We get $(1,1,1),(7,8,9),(5,6,7)$ and Heisuke wins.

There are $ 12$ students in a classroom; $6$ of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his/her political alignment to that of the majority at the end of the debate. What is the expected amount of time needed for all $ 12$ students to have the same political alignment, in hours?

Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$. [i]Author: Christopher Bradley, United Kingdom [/i]

Prove that for any integer $n>1$ there are distinct integers $a,b,c$ between $n^2$ and $(n+1)^2$ such that $c$ divides $a^2+b^2$.

Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$. :lol: ([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i])

Let ABC be a triangle whose vertices are inside a circle $\Omega$. Prove that we can choose two of the vertices of ABC such that there are infinitely many circles $\omega$ that satisfy the following properties: 1. $\omega$ is inside of $\Omega$, 2. $\omega$ passes through the two chosen vertices, and 3. the third vertex is in the interior of $\omega$ .

What is the remainder when $2^{1023}$ is divided by $1023$?

The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.

The real numbers $a_1, a_2, a_3$ are greater than $1$ and have the sum equal to $S$. If for any $i = 1, 2, 3$, holds the inequality $\frac{a_i^2}{a_i-1}>S$ , prove the inequality $$\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1$$

$21$ Savage has a $12$ car garage, with a row of spaces numbered $1,2,3,\ldots,12$. How many ways can he choose $6$ of them to park his $6$ identical cars in, if no $3$ spaces with consecutive numbers may be all occupied? [i]2018 CCA Math Bonanza Team Round #9[/i]

Given four points $A,B,C,D$ in space, find a plane, $S$, equidistant from all four points and having $A$ and $C$ on one side, $B$ and $D$ on the other.

Determine all natural numbers $n$ for which it is possible to construct a rectangle of sides $15$ and $n$, with pieces congruent to: [asy] unitsize(0.6 cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,-0.5)--(6,-0.5)); draw((4,0.5)--(7,0.5)); draw((4,1.5)--(7,1.5)); draw((5,2.5)--(6,2.5)); draw((4,0.5)--(4,1.5)); draw((5,-0.5)--(5,2.5)); draw((6,-0.5)--(6,2.5)); draw((7,0.5)--(7,1.5)); [/asy] The squares of the pieces have side $1$ and the pieces cannot overlap or leave free spaces

A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$. .