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 $LMT$.

The lengths of the sides of an acute angled triangle are successive integers. Prove that the altitude to the second longest side divides this side into two segments whose difference in length equals $4$.

$a_1, a_2,...,a_{2006}$ is a permutation of $1,2,...,2006$. Prove that $\prod_{i = 1}^{2006} (a_{i}^2-i) $ is a multiple of $3$. ($0$ is counted as a multiple of $3$)

Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments. [i]Vasile Pop[/i]

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$

4. A triangle $A B C$ with angle $A$ equal to $60^{\circ}$ is given. Its incircle is tangent to side $A B$ at point $D$, while its excircle tangent to side $A C$, is tangent to the extension of side $A B$ at point $E$. Prove that the perpendicular to side $A C$, passing through point $D$, meets the incircle again at a point equidistant from points $E$ and $C$. (The excircle is the circle tangent to one side of the triangle and to the extensions of two other sides.) Azamat Mardanov

In five years, Tom will be twice as old as Cindy. Thirteen years ago, Tom was three times as old as Cindy. How many years ago was Tom four times as old as Cindy?

Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.

Show that the polynomial over variables $x,y,z$ \[ x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y) \] can't be written as a finite sum of squares of real polynomials over $x,y,z$.

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? [asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label("$A$",A,SW); label("$B$", B, NW); label("$C$",C,NE); label("$D$",D,SE); label("$P$",P,S); [/asy] $\textbf{(A) }3\sqrt 5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }6\sqrt 5 \qquad \textbf{(D) }20\qquad \textbf{(E) }25$

Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.