On a circle $\Gamma$, points $A, B, N, C, D, M$ are chosen in a clockwise order in such a way that $N$ and $M$ are the midpoints of clockwise arcs $BC$ and $AD$ respectively. Let $P$ be the intersection of $AC$ and $BD$, and let $Q$ be a point on line $MB$ such that $PQ$ is perpendicular to $MN$. Point $R$ is chosen on segment $MC$ such that $QB = RC$, prove that the midpoint of $QR$ lies on $AC$.

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

Let $a,b$ and $c$ be the sides of a triangle, prove that $\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}<2$.

Let $a_1, a_2, ..., a_{2004}$ be integer numbers such that for all positive integers $n$ the number $A_n = a^n_1 + a^n_2 + ...+ a^n_{2004}$ is a perfect square. What is the minimal number of zeros within the $2004$ numbers?

Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?

Kimia has a weird clock; the clock's second hand moves 34 or 47 seconds forward instead of each regular second, at random. As an example, if the clock displays the time as $\text{12:23:05}$, the following times could be displayed in this order: $$\text{12:23:39, 12:24:13, 12:25:00, 12:25:34, 12:26:21,\dots}$$ Prove that the clock's second hand would eventually land on a perfect square.

How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$) of length $2013$ have letters in alphabetic order?

Find all real $x, y$, satisfying $$(x+1)^2(y+1)^2=27xy$$ and $$(x^2+1)(y^2+1)=10xy.$$

Prove that the product of five consecutive positive integers is never a perfect square.

* On a plane, given points $A, B, C$ and angles $\angle D, \angle E, \angle F$ each less than $180^o$ and the sum equal to $360^o$, construct with the help of ruler and protractor a point $O$ such that $\angle AOB = \angle D, \angle BOC = \angle E$ and $\angle COA = \angle F.$

Let $ ABC$ be an acute, scalene triangle, and let $ M$, $ N$, and $ P$ be the midpoints of $ \overline{BC}$, $ \overline{CA}$, and $ \overline{AB}$, respectively. Let the perpendicular bisectors of $ \overline{AB}$ and $ \overline{AC}$ intersect ray $ AM$ in points $ D$ and $ E$ respectively, and let lines $ BD$ and $ CE$ intersect in point $ F$, inside of triangle $ ABC$. Prove that points $ A$, $ N$, $ F$, and $ P$ all lie on one circle.

In each cell of a $4\times 4$ table, one of the numbers $1$ or $2$ is written. For each row, calculate the sum of the numbers written in it, and for each column, calculate the product of the numbers written in it. Show that some two of the eight results obtained are equal.

Given a line $\ell$ and a ray $p$ on a plane with its origin on this line. Two fixed circles (not necessarily equal) are constructed, inscribed in the two formed angles. On ray $p$, point $A$ is taken so that the tangents from $A$ to the given circles, different from $p$, intersect line $\ell$ at points $B$ and $C$, and at the same time triangle $ABC$ contains the given circles. Find the locus of the centers of the circles inscribed in triangle $ABC$ (as $A$ moves).

Show that \[\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)\] for all $a,b,c$ greater than 1.

$ (6?3)+4-(2-1) = 5. $ To make this statement true, the question mark between the 6 and the 3 should be replaced by $ \text{(A)}\div\qquad\text{(B)}\ \times\qquad\text{(C)}+\qquad\text{(D)}\ -\qquad\text{(E)}\ \text{None of these} $

Three tokens $A$, $B$, $C$ are, each one in a vertex of an equilateral triangle of side $n$. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case $n=3$ Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules: [b](1) [/b]First $A$ moves, after that $B$ moves, and then $C$, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other. [b](2)[/b] Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn. Show that for every positive integer $n$ it is possible to paint red all the sides of the little triangles.

Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ is a perfect square for $ 2p \minus{} 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 \minus{} 4ac.$

An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.

The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ , $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that $3a=2b+c$.

Let $ABCD$ be an isosceles trapezoid with bases $AB = 92$ and $CD = 19$. Suppose $AD = BC = x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2 = $ $ \textbf{(A)}\ 1369\qquad\textbf{(B)}\ 1679\qquad\textbf{(C)}\ 1748\qquad\textbf{(D)}\ 2109\qquad\textbf{(E)}\ 8825 $

Determine all pairs $ (x,y)$ of positive integers satisfying the equation \[ x!\plus{}y!\equal{}x^{y}.\]

Compute the smallest positive integer that gives a remainder of $1$ when divided by $11$, a remainder of $2$ when divided by $21$, and a remainder of $5$ when divided by $51$. [i]2021 CCA Math Bonanza Lightning Round #3.3[/i]

Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by William Yue[/i]

Find the sum of the squares of the values $x$ that satisfy $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$.

For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying $$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$