Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.

Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates $(2,2)$. What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$? [asy] pair O,P,Q,R,T; O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0); draw((-5,0)--(3,0)); draw((0,-1)--(0,3)); draw(P--Q--R); draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8)); draw((-0.2,2.8)--(0,3)--(0.2,2.8)); draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2)); draw((2.8,-0.2)--(3,0)--(2.8,0.2)); draw(Q--T); label("$O$",O,SW); label("$P$",P,S); label("$Q$",Q,NE); label("$R$",R,W); label("$T$",T,S); [/asy] NOT TO SCALE $\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)$

We have four white equilateral triangles of $3$ cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?

Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-[i]inoceronte[/i] sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

Two spheres in contact have a common tangent cone. These three surfaces divide the space into various parts, only one of which is bounded by all three surfaces, it is "ring-shaped." Being given the radii of the spheres, $r$ and $R$, find the volume of the "ring-shaped" part. (The desired expression is a rational function of $r$ and $R.$)

$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$. Prove that some two of the marked points are the ends of one diameter.

Triangle $ABC$ with it's circumcircle $\Gamma$ is given. Points $D$ and $E$ are chosen on segment $BC$ such that $\angle BAD=\angle CAE$. The circle $\omega$ is tangent to $AD$ at $A$ with it's circumcenter lies on $\Gamma$. Reflection of $A$ through $BC$ is $A'$. If the line $A'E$ meet $\omega$ at $L$ and $K$. Then prove either $BL$ and $CK$ or $BK$ and $CL$ meet on $\Gamma$.

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$ (2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$ Find $a_{n},\ b_{n}.$

Let $ABC$ be a triangle with circumcenter $O$. Points $P$ and $Q$ are interior to sides $CA$ and $AB$, respectively. Circle $\omega$ passes through the midpoints of segments $BP$, $CQ$, $PQ$. Prove that if line $PQ$ is tangent to circle $\omega$, then $OP = OQ$.

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.) [i]Proposed by Sergey Berlov, Russia[/i]

Let $S_1, S_2, \ldots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \subseteq \{S_1, S_2, \ldots, S_{100}\},$ the size of the intersection of the sets in $T$ is a multiple of the number of sets in $T$. What is the least possible number of elements that are in at least $50$ sets? [i]Proposed by Rishabh Das[/i]

Three runners, $X, Y$ and $Z$, participated in a race. $Z$ got held up at the start and began running last, while $Y$ was second from the start. During the race $Z$ exchanged positions with other contestants $6$ times, while $X$ did that $5$ times. It is known that $Y$ finished ahead of $X$. In what order did they finish?

Let $I$ be the centre of the inscribed circle of the non-isosceles triangle $ABC$, and let the circle touch the sides $BC, CA, AB$ at the points $A_1, B_1, C_1$ respectively. Prove that the centres of the circumcircles of $\vartriangle AIA_1,\vartriangle BIB_1$ and $\vartriangle CIC_1$ are collinear.

The evil sorceress Morgana lives in a square-shaped palace divided into a \(101 \times 101\) grid of rooms, each initially at a temperature of \(20^\circ\)C. Merlin attempts to freeze Morgana by casting a spell that permanently sets the central cell's temperature to \(0^\circ\)C. At each subsequent nanosecond, the following steps occur in order: 1. For every cell except the central one, the new temperature is computed as the arithmetic mean of the temperatures of its adjacent cells (those sharing a side). 2. All new temperatures (except the central cell) are updated simultaneously to the calculated values. Morgana can freely move between rooms but will freeze if all room temperatures drop below \(10^{-2025}\) degrees. The ice spell lasts for \(10^{75}\) nanoseconds, after which temperatures revert to their initial values. Will Merlin succeed in freezing Morgana?

What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?

Find all prime numbers whose base $b$ representations (for some $b$) contain each of the digits $0,1,\ldots,b-1$ exactly once. (Digit $0$ may appear as the first digit.)

There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^2$ small triangles. Once a second each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. For which natural numbers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?

Triangle $\vartriangle ABC$ has side lengths $AB = 8$, $BC = 15$, and $CA = 17$. Circles $\omega_1$ and $\omega_2$ are externally tangent to each other and within $\vartriangle ABC$. The radius of circle $\omega_2$ is four times the radius of circle $\omega_1$. Circle $\omega_1$ is tangent to $\overline{AB}$ and $\overline{BC}$, and circle $\omega_2$ is tangent to $\overline{BC}$ and $\overline{CA}$. Compute the radius of circle $\omega_1$.

Andrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was $47$. If Andrea is $29$ years older than Jim, what is the sum of their ages now?

What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\] $\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$

Xonathan Jue goes to the casino with exactly \$1000. Each week, he has a $1/3$ chance of breaking even and $2/3$ chance of losing \$500. Evaluate the expected amount of weeks before he loses everything. [i]2022 CCA Math Bonanza Lightning Round 1.2[/i]

Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.