In triangle $\triangle ABC$ with median from $B$ not perpendicular to $AB$ nor $BC$, we call $X$ and $Y$ points on $AB$ and $BC$, which lie on the axis of the median from $B$. Find all such triangles, for which $A,C,X,Y$ lie on one circumrefference.

A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter as shown. It is then tipped over by some angle $\gamma$ with respect to the table. At what minimum angle $\gamma$ will the disk lose balance and tumble over? Express your answer in degrees, rounded to the nearest integer. [asy] draw(arc((2,0), 1, 0,180)); draw((0,0)--(4,0)); draw((0,-2.5)--(4,-2.5)); draw(arc((3-sqrt(2)/2, -4+sqrt(2)/2+1.5), 1, -45, 135)); draw((3-sqrt(2), -4+sqrt(2)+1.5)--(3, -4+1.5)); draw(anglemark((3-sqrt(2), -4+sqrt(2)+1.5), (3, -4+1.5), (0, -4+1.5))); label("$\gamma$", (2.8, -3.9+1.5), WNW, fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

Prove that $n!\cdot(n+1)!\cdot(n+2)!$ divides $(3n)!$ for every integer $n \geq 3$.

Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$ [i]Proposed by Krit Boonsiriseth[/i]

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

The square was cut into $n^2$ rectangles with sides $a_i \times b_j, i , j= 1,..., n$. For what is the smallest $n$ in the set $\{a_1, b_1, ..., a_n, b_n\}$ all the numbers can be different?

The parabola $P$ has focus a distance $m$ from the directrix. The chord $AB$ is normal to $P$ at $A.$ What is the minimum length for $AB?$

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Semicircle $O$ has diameter $AB = 12$. Arc $AC = 135^o$ . Let $D$ be the midpoint of arc $AC$. Compute the region bounded by the lines $CD$ and $DB$ and the arc $CB$.

Pythagoras drew some points in the plane and and connected some of these with segments. Now Tortillagoras wants to write a positive integer next to every point, such that one number divides another number if and only if these numbers are written next to points that Pythagoras has connected.Can Tortillagoras do this for the following drawings? [i]In part b), vertices in the same row or column but not adjacent are not connected.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/e/7356e39e44e45e3263275292af3719595e2dd2.png[/img]

(a) Let $x$ and $y$ be integers. Prove that $x = y$ if $x^n \equiv y^n$ mod $n$ for all positive integers $n$. (b) For which pairs of integers $(x, y)$ are there infinitely many positive integers $n$ such that $x^n \equiv y^n$ mod $n$?

Let $ p, r $ be prime numbers, and $ q $ natural. Solve the equation $ (p+q+r)^2=2p^2+2q^2+r^2 $.

In the given figure, $ABCD$ is a square sheet of paper. It is folded along $EF$ such that $A$ goes to a point $A'$ different from $B$ and $C$, on the side $BC$ and $D$ goes to $D'$. The line $A'D'$ cuts $CD$ in $G$. Show that the inradius of the triangle $GCA'$ is the sum of the inradii of the triangles $GD'F$ and $A'BE$. [asy] size(5cm); pair A=(0,0),B=(1,0),C=(1,1),D=(0,1),Ap=(1,0.333),Dp,Ee,F,G; Ee=extension(A,B,(A+Ap)/2,bisectorpoint(A,Ap)); F=extension(C,D,(A+Ap)/2,bisectorpoint(A,Ap)); Dp=reflect(Ee,F)*D; G=extension(C,D,Ap,Dp); D(MP("A",A,W)--MP("E",Ee,S)--MP("B",B,E)--MP("A^{\prime}",Ap,E)--MP("C",C,E)--MP("G",G,NE)--MP("D^{\prime}",Dp,N)--MP("F",F,NNW)--MP("D",D,W)--cycle,black); draw(Ee--Ap--G--F); dot(A);dot(B);dot(C);dot(D);dot(Ap);dot(Dp);dot(Ee);dot(F);dot(G); draw(Ee--F,dashed); [/asy]

A cylinder with length $\ell$ has a radius of $6$ meters, and three spheres with radii $3, 4$, and $5$ meters are placed inside the cylinder. If the spheres are packed into the cylinder such that $\ell$ is minimized, determine the length $\ell$.

Let $f:[1,\infty)\to(0,\infty)$ be a non-increasing function such that $$\limsup_{n\to\infty}\frac{f(2^{n+1})}{f(2^n)}<\frac12.$$Prove that $$\int^\infty_1f(x)\text dx<\infty.$$

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x,y>0$ we have $$f(xy+f(x))=yf(x)+x.$$ [i]Proposed by Nikola Velov[/i]

Teresa the bunny has a fair $8$-sided die. Seven of its sides have fixed labels $1, 2, \cdots , 7,$ and the label on the eighth side can be changed and begins as $1$. She rolls it several times, until each of $1, 2, \dots, 7$ appears at least once. After each roll, if $k$ is the smallest positive integer that she has not rolled so far, she relabels the eighth side with $k$. The probability that $7$ is the last number she rolls is $\tfrac ab,$ where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

The rectangle is cut into 6 squares, as shown on the figure below. The gray square in the middle has a side equal to 1. What is the area of the rectangle? [img]https://i.ibb.co/gg1tBTN/Kyiv-MO-2023-7-1.png[/img]

Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?

Joe, who is already feared by all bandits in the Wild West, would like to officially become a sheriff. To do that, he has to take a special exam where he has to demonstrate his talent in three different areas: tracking, shooting and lasso throwing. He successfully completes each task with a given probability, independently of each other. He passes the exam if he can complete at least two of the tasks successfully. Joe calculated that in case he starts with tracking and completes it successfully, his chance of passing the exam is $32\%$. If he starts with successful shooting, the chance of passing is $49\%$, whereas if he starts with successful lasso throwing, he passes with probability $52\%$. The overall probability of passing (calculated before the start of the exam) is $X/1000$ . What is the value of $X$?

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $ 4$, and the second number is divided by $ 5$. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square? [asy]unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(15pt)); draw(Circle(origin,1)); for(int i = 0;i < 6; ++i) { draw(origin--dir(60i+30)); } label("$7$",midpoint(origin--(dir(0))),E); label("$1$",midpoint(origin--(dir(60))),NE); label("$6$",midpoint(origin--(dir(120))),NW); label("$3$",midpoint(origin--(dir(180))),W); label("$9$",midpoint(origin--(dir(240))),SW); label("$2$",midpoint(origin--(dir(300))),SE); draw((2,0)--(3.5,0)--(3.5,1)--(2,1)--cycle); draw((2,0)--(3.5,0)--(3.5,-1)--(2,-1)--cycle); pair[] V = {(2.5,0.5),(2,0),(3,0),(2.5,-0.5),(2,-1),(3,-1)}; for(int i = 0; i <= 5; ++i) { pair A = V[i]; path p = A--(A.x,A.y + 0.5)--(A.x + 0.5,A.y + 0.5)--(A.x + 0.5, A.y)--cycle; fill(p,mediumgray); draw(p); } path pointer = (-2.5,-0.125)--(-2.5,0.125)--(-1.2,0.125)--(-1.05,0)--(-1.2,-0.125)--cycle; fill(pointer,mediumgray); draw(pointer); label("Pointer",(-1.85,0),fontsize(10pt)); label("$4$",(2,0.5),2N + 2W); label("$3$",(2,0),2N + 2W); label("$2$",(2,-0.5),2N + 2W); label("$1$",(2,-1),2N + 2W); label("$1$",(2,-1),2S + 2E); label("$2$",(2.5,-1),2S + 2E); label("$3$",(3,-1),2S + 2E);[/asy]$ \textbf{(A)}\ \frac {1}{3}\qquad \textbf{(B)}\ \frac {4}{9}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {5}{9}\qquad \textbf{(E)}\ \frac {2}{3}$

If $ x^{2} \plus{}y^{2} \plus{}z\equal{}15$, $ x\plus{}y\plus{}z^{2} \equal{}27$ and $ xy\plus{}yz\plus{}zx\equal{}7$, then $\textbf{(A)}\ 3\le \left|x\plus{}y\plus{}z\right|\le 4 \\ \textbf{(B)}\ 5\le \left|x\plus{}y\plus{}z\right|\le 6 \\ \textbf{(C)}\ 7\le \left|x\plus{}y\plus{}z\right|\le 8 \\ \textbf{(D)}\ 9\le \left|x\plus{}y\plus{}z\right|\le 10 \\ \textbf{(E)}\ \text{None}$

Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$

An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.