Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

Find all possible integer values of the sum: $$\frac{a}{b}+ \frac{2023 \times b}{4 \times a},$$ where $a$ and $b$ are positive integers with no prime factors in common.

Let $ABC$ be a triangle with $AB = AC$ and $\angle BAC = 30^{\circ}$, Let $A'$ be the reflection of $A$ in the line $BC$; $B'$ be the reflection of $B$ in the line $CA$; $C'$ be the reflection of $C$ in line $AB$, Show that $A'B'C'$ is an equilateral triangle.

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Determine if there is a non-natural natural number $n$ with the property that $\sqrt{n + 1} + \sqrt{n - 1}$ is rational.

(a) Let $f: \mathbb{R}^{n\times n}\rightarrow\mathbb{R}$ be a linear mapping. Prove that $\exists ! C\in\mathbb{R}^{n\times n}$ such that $f(A)=Tr(AC), \forall A \in \mathbb{R}^{n\times n}$. (b) Suppose in addtion that $\forall A,B \in \mathbb{R}^{n\times n}: f(AB)=f(BA)$. Prove that $\exists \lambda \in \mathbb{R}: f(A)=\lambda Tr(A)$

In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? $ \textbf{(A)}\ \dfrac{2}{5} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{4}{7} \qquad\textbf{(D)}\ \dfrac{3}{5} \qquad\textbf{(E)}\ \dfrac{4}{5} $

A football league has $n$ teams. Each team plays one game with every other team. Each win is awarded $2$ points, each tie $1$ point, and each loss $0$ points. After the league is over, the following statement is true: for every subset $S$ of teams in the league, there is a team (which may or may not be in $S$) such that the total points the team obtained by playing all the teams in $S$ is odd. Prove that $n$ is even.

Let $ABCD$ be a square pyramid of height $\frac{1}{2}$ with square base $ABCD$ of side length $AB=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $ABCD$ is the center of square $ABCD$). The faces $ADE$ and $CDE$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$). Find $\tan \alpha$.

Let $ABC$ be an acute triangle with $AB<AC < BC$, inscirbed in circle $\Gamma_1$, with center $O$. Circle $\Gamma_2$, with center point $A$ and radius $AC$ intersects $BC$ at point $D$ and the circle $\Gamma_1$ at point $E$. Line $AD$ intersects circle $\Gamma_1$ at point $F$. The circumscribed circle $\Gamma_3$ of triangle $DEF$, intersects $BC$ at point $G$. Prove that: a) Point $B$ is the center of circle $\Gamma_3$ b) Circumscribed circle of triangle $CEG$ is tangent to $AC$.

A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find the sum $a + b + c$.

The vertices of a $50\text{-gon}$ divide a circumference into $50$ arcs, whose lengths are $1,2,\ldots 50$ in some order. It is known that any two opposite arcs (corresponding to opposite sides) differ by $25$. Prove that the polygon has two parallel sides.

A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$.

A combination lock has a $3$ number combination, with each number an integer between $0$ and $39$ inclusive. Call the numbers $n_1$, $n_2$, and $n_3$. If you know that $n_1$ and $n_3$ leave the same remainder when divided by $4$, and $n_2$ and $n_1 + 2$ leave the same remainder when divided by $4$, how many possible combinations are there?

Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland

How many triangles are in this figure? (Some triangles may overlap other triangles.) [asy] draw((0,0)--(42,0)--(14,21)--cycle); draw((14,21)--(18,0)--(30,9));[/asy] $ \text{(A)}\ 9\qquad\text{(B)}\ 8\qquad\text{(C)}\ 7\qquad\text{(D)}\ 6\qquad\text{(E)}\ 5 $

p1. A set of cards contains $100$ cards, each of which is written with a number from $1$ up to $100$. On each of the two sides of the card the same number is written, side one is red and the other is green. First of all Leny arranges all the cards with red writing face up. Then Leny did the following three steps: I. Turn over all cards whose numbers are divisible by $2$ II. Turn over all the cards whose numbers are divisible by $3$ III. Turning over all the cards whose numbers are divisible by $5$, but didn't turn over all cards whose numbers are divisible by $5$ and $2$. Find the number of Leny cards now numbered in red and face up, p2. Find the area of ​​three intersecting semicircles as shown in the following image. [img]https://cdn.artofproblemsolving.com/attachments/f/b/470c4d2b84435843975a0664fad5fee4a088d5.png[/img] p3. It is known that $x+\frac{1}{x}=7$ . Determine the value of $A$ so that $\frac{Ax}{x^4+x^2+1}=\frac56$. p4. There are $13$ different gifts that will all be distributed to Ami, Ima, Mai,and Mia. If Ami gets at least $4$ gifts, Ima and Mai respectively got at least $3$ gifts, and Mia got at least $2$ gifts, how many possible gift arrangements are there? p5. A natural number is called a [i]quaprimal [/i] number if it satisfies all four following conditions: i. Does not contain zeros. ii. The digits compiling the number are different. iii. The first number and the last number are prime numbers or squares of an integer. iv. Each pair of consecutive numbers forms a prime number or square of an integer. For example, we check the number $971643$. (i) $971643$ does not contain zeros. (ii) The digits who compile $971643$ are different. (iii) One first number and one last number of $971643$, namely $9$ and $3$ is a prime number or a square of an integer. (iv) Each pair of consecutive numbers, namely $97, 71, 16, 64$, and $43$ form prime number or square of an integer. So $971643$ is a quadratic number. Find the largest $6$-digit quaprimal number. Find the smallest $6$-digit quaprimal number. Which digit is never contained in any arbitrary quaprimal number? Explain.

Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ such that \[f(xy + x + y) + f \left( \frac1x \right) f\left( \frac1y \right) = 1\] for every $x$, $y\in \mathbb{R}_+$. [i]Proposed by Li4 and Untro368.[/i]

Prove that for every natural $n$ $$\frac{1}{3} + \frac{2}{3\cdot 5} + \frac{3}{3 \cdot 5 \cdot 7} + ...+ \frac{n}{3 \cdot 5 \cdot 7 \cdot ...\cdot (2n+1)} < \frac{1}{2}.$$

A circle is drawn on the plane. How to use only a ruler to draw a perpendicular from a given point outside the circle to a given line passing through the center of this circle?

Find all positive integers $n$ such that \[2n = \varphi(n)^{\frac{2}{3}}(\varphi(n)^{\frac{2}{3}}+1)\]

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$. $(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$. $(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.

Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)