An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]

Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$ [i]Proposed by Navid Safaei[/i]

Let $\tau(n)$ be the number of distinct positive divisors of $n$ (including $1$ and itself). Find the sum of all positive integers $n$ satisfying $n=\tau(n)^3.$

The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$ \[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\] Show that $2003$ is not divisor of any of the terms in these two sequences.

Consider $\vartriangle ABC$ such that $CA + AB = 3BC$. Let the incircle $\omega$ touch segments $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$, respectively, and define $P$ and $Q$ such that segments $\overline{P E}$ and $\overline{QF}$ are diameters of $\omega$. Define the function $D$ of a point $K$ to be the sum of the distances from $K$ to $P$ and $Q$ (i.e. $D(K) = KP + KQ$). Let $W, X, Y$ , and $Z$ be points chosen on lines $\overleftrightarrow {BC}$, $\overleftrightarrow {CE}$, $\overleftrightarrow {EF}$, and $\overleftrightarrow {F B}$, respectively. Given that $BC =\sqrt{133}$ and the inradius of $\vartriangle ABC$ is $\sqrt{14}$, compute the minimum value of $D(W) + D(X) + D(Y ) + D(Z)$.

Integers are placed on every cell of an infinite checkerboard. For each cell if it contains integer $a$ then the sum of the numbers in the cell under it and the cell right to it is $2a+1$. Prove that in every infinite diagonal row of direction [i] top-right down-left [/i] all numbers are different.

Let $\mathbb{Z}/n\mathbb{Z}$ denote the set of integers considered modulo $n$ (hence $\mathbb{Z}/n\mathbb{Z}$ has $n$ elements). Find all positive integers $n$ for which there exists a bijective function $g: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, such that the 101 functions \[g(x), \quad g(x) + x, \quad g(x) + 2x, \quad \dots, \quad g(x) + 100x\] are all bijections on $\mathbb{Z}/n\mathbb{Z}$. [i]Ashwin Sah and Yang Liu[/i]

Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

As shown in figure , $\odot O$ is the inscribed circle of trapezoid $ABCD$, and the tangent points are $E, F, G, H$, $AB \parallel CD$. The line passing through$ B$, parallel to $AD$ intersects extension of $DC$ at point $P$. The extension of $AO$ intersects $CP$ at point $Q$. If $AE=BE$ , prove that $\angle CBQ = \angle PBQ$. [img]https://cdn.artofproblemsolving.com/attachments/d/2/7c3a04bb1c59bc6d448204fd78f553ea53cb9e.png[/img]

Let $c$ be a constant. The simultaneous equations \begin{align*} x-y = &\ 2 \\ cx+y = &\ 3 \\ \end{align*} have a solution $(x, y)$ inside Quadrant I if and only if $ \textbf{(A)}\ c=-1 \qquad\textbf{(B)}\ c>-1 \qquad\textbf{(C)}\ c<\frac{3}{2} \qquad\textbf{(D)}\ 0<c<\frac{3}{2}\\ \qquad\textbf{(E)}\ -1<c<\frac{3}{2} $

If one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$, then $x$ equals $\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }1/2\qquad\textbf{(D) }2\qquad \textbf{(E) }3$

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

In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry$?$ [asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy] $\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

Find all triples $(x, y,z)$ such that $x, y, z \in (0,1)$ and $$\left(x+\frac{1}{2x}-1\right) \left(y+\frac{1}{2y}-1\right) \left(z+\frac{1}{2z}-1\right) = \left(1-\frac{xy}{z}\right)\left(1-\frac{yz}{x}\right)\left(1-\frac{zx}{y}\right)$$ (D. Bazylev)