Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.

Given a convex quadrilateral $ABCD$, such that $OA = \frac{OB \cdot OD}{OC + CD}$ where $O$ is the intersection of the two diagonals. The circumcircle of triangle $ABC$ intersects $BD$ at point $Q$. Prove that $CQ$ bisects $\angle ACD$

Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$ [i]Proposed by Henry Jiang[/i]

Find $0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6$

Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying the equality $P(Q(x))=P(x)Q(x)-P(x)$. [i](I. Voronovich)[/i]

A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $x=f(t)$ (where $f(t)$ is not constant) gives the horizontal distance in terms of the time $t$. Show that the vertical distance $y$ is given by $$y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt +Af(t)+B$$ where $A$ and $B$ are constants and $g$ is the acceleration due to gravity.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Let $K(O,R)$ be a circle with center $O$ and radious $R$ and $(e)$ to be a line thst tangent to $K$ at $A$. A line parallel to $OA$ cuts $K$ at $B, C$, and $(e)$ at $D$, ($C$ is between $B$ and $D$). Let $E$ to be the antidiameric of $C$ with respect to $K$. $EA$ cuts $BD$ at $F$. i)Examine if $CEF$ is isosceles. ii)Prove that $2AD=EB$. iii)If $K$ si the midlpoint of $CF$, prove that $AB=KO$. iv)If $R=\frac{5}{2}, AD=\frac{3}{2}$, calculate the area of $EBF$

Let $n$ be a positive integer. Let $A,B,$ and $C$ be three $n$-dimensional vector subspaces of $\mathbb{R}^{2n}$ with the property that $A \cap B = B \cap C = C \cap A = \{0\}$. Prove that there exist bases $\{a_1,a_2, \dots, a_n\}$ of $A$, $\{b_1,b_2, \dots, b_n\}$ of $B$, and $\{c_1,c_2, \dots, c_n\}$ of $C$ with the property that for each $i \in \{1,2, \dots, n\}$, the vectors $a_i,b_i,$ and $c_i$ are linearly dependent.

Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$. [i](nooonuii)[/i]

Prove that for every positive integer $m$ there exists a positive integer N such that $S(2^n) > m$ for every positive integer $n > N$, where by $S(x)$ we denote the sum of digits of a positive integer $x$.

Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.

Triangle $ABC$ is such that $\angle BAC>\angle ABC>60^o$. The perpendicular bisector of $\overline{AB}$ intersects the segment $\overline {BC}$ at $O$. Suppose there exists a point $D$ on the segment $\overline{AC}$ such that $OD=AB$ and $\angle ODA=30^o$. Find $\angle BAC$. [i](Vlad Robu)[/i]

Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.

Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB})=12^\circ$, and $m(\widehat{DAE})=72^\circ$. What is $m(\widehat{CDE})$ in degrees? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 66 \qquad\textbf{(C)}\ 68 \qquad\textbf{(D)}\ 70 \qquad\textbf{(E)}\ 72 $

A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds: For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials. $a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$. $b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials. [i]Proposed by Alireza Shavali[/i]

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

Which family of elements has solid, liquid, and gaseous members at $25^{\circ} \text{C}$ and $1 \text{atm}$ pressure? $ \textbf{(A)}\hspace{.05in}\text{alkali metals (Li-Cs)} \qquad\textbf{(B)}\hspace{.05in}\text{pnictogens (N-Bi)} \qquad$ $\textbf{(C)}\hspace{.05in}\text{chalcogens (O-Te)} \qquad\textbf{(D)}\hspace{.05in}\text{halogens (F-I)}\qquad$

Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.

In a triangle $ABC$ we have $\angle B = 2 \cdot \angle C$ and the angle bisector drawn from $A$ intersects $BC$ in a point $D$ such that $|AB| = |CD|$. Find $\angle A$.

Prove that $2024!$ is divisible by a) $2024^2$; b) $2024^8$. ($n!=1\cdot 2 \cdot 3 \cdot ... \cdot n$) [i]Z.Smysl[/i]

Let $ABC$ be a triangle which it not right-angled. Define a sequence of triangles $A_iB_iC_i$, with $i \ge 0$, as follows: $A_0B_0C_0$ is the triangle $ABC$ and, for $i \ge 0$, $A_{i+1},B_{i+1},C_{i+1}$ are the reflections of the orthocentre of triangle $A_iB_iC_i$ in the sides $B_iC_i$,$C_iA_i$,$A_iB_i$, respectively. Assume that $\angle A_m = \angle A_n$ for some distinct natural numbers $m,n$. Prove that $\angle A = 60^{\circ}$.

The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.