Let the circles $\omega$ and $\omega^ \prime$ intersect in $A$ and $B$. Tangent to circle$\omega$ at $A$ intersects$\omega^ \prime$ in $C$ and tangent to circle $\omega^ \prime$ at $A$ intersects $\omega$ in $D$. Suppose that $CD$ intersects$\omega$ and $\omega^ \prime$ in $E$ and $F$, respectively (assume that $E$ is between $F$ and $C$). The perpendicular to $AC$ from $E$ intersects $\omega^ \prime$ in point $P$ and perpendicular to $AD$ from $F$ intersects$\omega$ in point $Q$ (The points $A, P$ and $Q$ lie on the same side of the line $CD$). Prove that the points $A, P$ and $Q$ are collinear. Proposed by Mahdi Etesami Fard

Let $p$ be a prime number. We know that each natural number can be written in the form \[\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)\] Uniquely. Now let $T$ be the set of all the sums of the form \[\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).\] (This means to allow numbers with an infinite base $p$ representation). So numbers that for some $N\in \mathbb N$ all the coefficients $a_i, i\ge N$ are zero are natural numbers. (In fact we can consider members of $T$ as sequences $(a_0,a_1,a_2,...)$ for which $\forall_{i\in \mathbb N}: 0\le a_i \le p-1$.) Now we generalize addition and multiplication of natural numbers to this set so that it becomes a ring (it's not necessary to prove this fact). For example: $1+(\sum_{i=0}^{\infty} (p-1)p^i)=1+(p-1)+(p-1)p+(p-1)p^2+...$ $=p+(p-1)p+(p-1)p^2+...=p^2+(p-1)p^2+(p-1)p^3+...$ $=p^3+(p-1)p^3+...=...$ So in this sum, coefficients of all the numbers $p^k, k\in \mathbb N$ are zero, so this sum is zero and thus we can conclude that $\sum_{i=0}^{\infty}(p-1)p^i$ is playing the role of $-1$ (the additive inverse of $1$) in this ring. As an example of multiplication consider \[(1+p)(1+p+p^2+p^3+...)=1+2p+2p^2+\cdots\] Suppose $p$ is $1$ modulo $4$. Prove that there exists $x\in T$ such that $x^2+1=0$. [i]Proposed by Masoud Shafaei[/i]

There exist some number of ordered triples of real numbers $(x,y,z)$ that satisfy the following system of equations: \begin{align*} x+y+2z &= 6\\ x^2+y^2+2z^2 &= 18\\ x^3+y^3+2z^3&=54 \end{align*} Given that the sum of all possible positive values of $x$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$,$b$,$c$, and $d$ are positive integers, $c$ is squarefree, and $\gcd(a,b,d)=1$, find the value of $a+b+c+d$.

Which of the following can be used to distinguish a solid ball from a hollow sphere of the same radius and mass? $\textbf{(A)}$ Measurements of the orbit of a test mass around the object. $\textbf{(B)}$ Measurements of the time it takes the object to roll down an inclined plane. $\textbf{(C)}$ Measurements of the tidal forces applied by the object to a liquid body. $\textbf{(D)}$ Measurements of the behavior of the object as it oats in water. $\textbf{(E)}$ Measurements of the force applied to the object by a uniform gravitational field.

[b]Problem 1[/b] Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable on $\mathbb{R}$. Prove that there exists $x \in [0, 1]$ such that $$\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .$$

Determine the quantity of positive integers $N$ of $10$ digits with the following properties: I- All the digits of $N$ are non-zero. II- $11|N$. III- $N$ and all the permutation(s) of the digits of $N$ are divisible by $12$.

Natural numbers are written in the cells of of a $2014\times2014$ regular square grid such that every number is the average of the numbers in the adjacent cells. Describe and prove how the number distribution in the grid can be.

Suppose that $a$ is an integer and that $n! + a$ divides $(2n)!$ for infinitely many positive integers $n$. Prove that $a = 0$.

Solve the system of equations $$\begin{cases} x^2+arc siny =y^2+arcsin x \\ x^2+y^2-3x=2y\sqrt{x^2-2x-y}+1 \end{cases}$$

Numbers 1 and −1 are written in the cells of a board 2000×2000. It is known that the sum of all the numbers in the board is positive. Show that one can select 1000 rows and 1000 columns such that the sum of numbers written in their intersection cells is at least 1000. [i]D. Karpov[/i]

Let $y^{\alpha}=\sum_{i=1}^n x_i^{\alpha}$ where $\alpha \neq 0, y > 0, x_i > 0$ are real numbers, and let $\lambda \neq \alpha$ be a real number. Prove that $y^{\lambda} > \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) > 0,$ and $y^{\lambda} < \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) < 0.$

$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.

For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that $$ \sum_{k=0}^{3^{1010}-1}(-2)^{f(k)}(z+k)^{2023}=0 $$

In convex quadrilateral ABCD,diagonals AC and BD intersect at S and are perpendicular. a)Prove that midpoints M,N,P,Q of AD,AB,BC,CD form a rectangular b)If diagonals of MNPQ intersect O and AD=5,BC=10,AC=10,BD=11 find value of SO

We call a polynomial $P(x)$ good if the numbers $P(k)$ and $P'(k)$ are integers for all integers $k$. Let $P(x)$ be a good polynomial of degree $d$, and let $N_d$ be the product of all composite numbers not exceeding $d$. Prove that the leading coefficient of the polynomial $N_d \cdot P(x)$ is integer.

$\frac{\sin 10^\circ + \sin 20^\circ}{\cos 10^\circ + \cos 20^\circ}$ equals A. $\tan 10^\circ + \tan 20^\circ$ B. $\tan 30^\circ$ C. $\frac{1}{2} (\tan 10^\circ + \tan 20^\circ)$ D. $\tan 15^\circ$ E. $\frac{1}{4} \tan 60^\circ$

Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids $S_n$ and $T_n$. The pyramid $S_n$ has $n$ layers, where the top layer is a single sphere and the $i^{th}$ layer is an $i\times $i square grid of spheres for each $2 \le i \le n$. Similarly, the pyramid $T_n$ has $n$ layers where the top layer is a single sphere and the $i^{th}$ layer is $\frac{i(i+1)}{2}$ spheres arranged into an equilateral triangle for each $2 \le i \le n$.

The figure on the right shows the map of Squareville, where each city block is of the same length. Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$, respectively. They start walking toward each other's house, leaving at the same time, walking with the same speed, and independently choosing a path to the other's house with uniform distribution out of all possible minimum-distance paths [that is, all minimum-distance paths are equally likely]. What is the probability they will meet? [asy] size(200); defaultpen(linewidth(0.8)); for(int i=0;i<=2;++i) { for(int j=0;j<=4;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } for(int i=3;i<=4;++i) { for(int j=3;j<=6;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } label("$A$",origin,SW); label("$B$",(5,7),SE); [/asy]

Present a family of subsets of the plane such that each one of its members is Lebesgue measurable, each one of its members intersects any circle, and the set of Lebesgue measures of all its members is the set of nonnegative real numbers.

The infinity sequence $r_{1},r_{2},...$ of rational numbers it satisfies that: $\prod_{i=1}^ {k}r_{i}=\sum_{i=1}^{k} r_{i}$. For all natural k. Show that $\frac{1}{r_{n}}-\frac{3}{4}$ is a square of rationale number for all natural $n\geq3$

Let $AC$ be a line segment in the plane and $B$ a points between $A$ and $C$. Construct isosceles triangles $PAB$ and $QAC$ on one side of the segment $AC$ such that $\angle APB = \angle BQC = 120^{\circ}$ and an isosceles triangle $RAC$ on the other side of $AC$ such that $\angle ARC = 120^{\circ}.$ Show that $PQR$ is an equilateral triangle.

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

We are given a triangle $ABC$ such that $\angle BAC < 90^{\circ}$. The point $D$ is on the opposite side of the line $AB$ to $C$ such that $|AD| = |BD|$ and $\angle ADB = 90^{\circ}$. Similarly, the point $E$ is on the opposite side of $AC$ to $B$ such that $|AE| = |CE|$ and $\angle AEC = 90^{\circ}$. The point $X$ is such that $ADXE$ is a parallelogram. Prove that $|BX| = |CX|$.

Suppose $x$ is a non zero real number such that both $x^5$ and $20x+\frac{19}{x}$ are rational numbers. Prove that $x$ is a rational number.

Let $ABC$ be an acute triangle with $E, F$ being the reflections of $B,C$ about the line $AC, AB$ respectively. Point $D$ is the intersection of $BF$ and $CE$. If $K$ is the circumcircle of triangle $DEF$, prove that $AK$ is perpendicular to $BC$. Nguyen Minh Ha, College of Pedagogical University of Hanoi