Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

In a convex hexagon $ABCDEF$ triangles $ABC , CDE , EFA$ are similar. Find conditions on these triangles under which triangle $ACE$ is equilateral if and only if so is $BDF.$

Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.

Find the angles of a triangle $ ABC $ in which $ \frac{\sin A}{\sin B} +\frac{\sin B}{\sin C} +\frac{\sin C}{\sin A} =3. $

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x^2 + y) \ge (\frac{1}{x} + 1)f(y)$$ holds for all $x \in \mathbb{R} \setminus \{0\}$ and all $y \in \mathbb{R}$.

Find all positive integers, such that there exist positive integers $a, b, c$, satisfying $\gcd(a, b, c)=1$ and $n=\gcd(ab+c, ac-b)=a+b+c$.

[b]p1.[/b] An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of $600$ mi/hour, and the second half of the distance is traveled at a a constant speed of $900$ mi/hour. Find the average speed of the plane. [b]p2.[/b] The figure below shows two egg cartons, $A$ and $B$. Carton $A$ has $6$ spaces (cell) and has $3$ eggs. Carton $B$ has $12$ cells and $3$ eggs. Tow cells from the total of $18$ cells are selected at random and the contents of the selected cells are interchanged. (Not that one or both of the selected cells may be empty.) [img]https://cdn.artofproblemsolving.com/attachments/6/7/2f7f9089aed4d636dab31a0885bfd7952f4a06.png[/img] (a) Find the number of selections/interchanges that produce a decrease in the number of eggs in cartoon $A$- leaving carton $A$ with $2$ eggs. (b) Assume that the total number of eggs in cartons $A$ and $B$ is $6$. How many eggs must initially be in carton $A$ and in carton $B$ so that the number of selections/interchanges that lead to an increase in the number of eggs in $A$ equals the number of selections/interchanges that lead to an increase in the number of eggs in $B$. $\bullet$ In other words, find the initial distribution of $6$ eggs between $A$ and $B$ so that the likelihood of an increase in A equals the likelihood of an increase in $B$ as the result of a selection/interchange. Prove your answer. [b]p3.[/b] Divide the following figure into four equal parts (parts should be of the same shape and of the same size, they may be rotated by different angles however they may not be disjoint and reconnected). [img]https://cdn.artofproblemsolving.com/attachments/f/b/faf0adbf6b09b5aaec04c4cfd7ab1d6397ad5d.png[/img] [b]p4.[/b] Find the exact numerical value of $\sqrt[3]{5\sqrt2 + 7}- \sqrt[3]{5\sqrt2 - 7}$ (do not use a calculator and do not use approximations). [b]p5.[/b] The medians of a triangle have length $9$, $12$ and $15$ cm respectively. Find the area of the triangle. [b]p6. [/b]The numbers $1, 2, 3, . . . , 82$ are written in an arbitrary order. Prove that it is possible to cross out $72$ numbers in such a sway the remaining number will be either in increasing order or in decreasing order. PS. You should use hide for answers.

Let $NS$ and $EW$ be two perpendicular diameters of a circle $\mathcal{C}$. A line $\ell$ touches $\mathcal{C}$ at point $S$. Let $A$ and $B$ be two points on $\mathcal{C}$, symmetric with respect to the diameter $EW$. Denote the intersection points of $\ell$ with the lines $NA$ and $NB$ by $A'$ and $B'$, respectively. Show that $|SA'|\cdot |SB'|=|SN|^2$.

Find all pairs $(a,b)$ of real numbers, so that $\sin(2009x)+\sin(ax)+\sin(bx)=0$ holds for any $x\in \mathbf {R}$.

Let $A = \{m : m$ an integer and the roots of $x^2 + mx + 2020 = 0$ are positive integers $\}$ and $B= \{n : n$ an integer and the roots of $x^2 + 2020x + n = 0$ are negative integers $\}$. Suppose $a$ is the largest element of $A$ and $b$ is the smallest element of $B$. Find the sum of digits of $a + b$.

$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.

Draw an equilateral triangle with center $O$. Rotate the equilateral triangle $30^\circ, 60^\circ, 90^\circ$ with respect to $O$ so there would be four congruent equilateral triangles on each other. Look at the diagram. If the smallest triangle has area $1$, the area of the original equilateral triangle could be expressed as $p+q\sqrt r$ where $p,q,r$ are positive integers and $r$ is not divisible by a square greater than $1$. Find $p+q+r$.

Let $ f:[0,1]\longrightarrow\mathbb{R} $ a continuous function. Prove that $$ \int_0^1 f^2\left( x^2 \right) dx\ge \frac{3}{4}\left( \int_0^1 f(x)dx \right)^2 , $$ and find the circumstances under which equality happens.

Austin is at the Lincoln Airport. He wants to take $5$ successive flights whose destinations are randomly chosen among Indianapolis, Jackson, Kansas City, Lincoln, and Milwaukee. The origin and destination of each flight may not be the same city, but Austin must arrive back at Lincoln on the last of his flights. Compute the probability that the cities Austin arrives at are all distinct.

Which of the following are true? $\textbf{(A)}~\exists A\in M_3(\mathbb R)\text{ such that }A^2=-I_3$ $\textbf{(B)}~\exists A,B\in M_3(\mathbb R)\text{ such that }AB-BA=I_3$ $\textbf{(C)}~\forall A\in M_4,\det\left(I_4+A^2\right)\ge0$ $\textbf{(D)}~\text{None of the above}$

Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$ Find all possible values of $a$.

(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$. (b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $5n + 1$ is six times the sum of the digits of $n$.

Let $\alpha > 1$ be an irrational number and $L$ be a integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \cdots$ satisfies that $x_1 > L$ and for all positive integers $n$, \[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \] Prove that (i) $\left\{x_n\right\}$ is eventually periodic. (ii) The eventual fundamental period of $\left\{x_n\right\}$ is an odd integer which doesn't depend on the choice of $x_1$.

Let $A$ and $B$ be points in space such that $AB=1.$ Let $\mathcal{R}$ be the region of points $P$ for which $AP \le 1$ and $BP \le 1.$ Compute the largest possible side length of a cube contained in $\mathcal{R}.$

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

Assume that a real polynomial $P(x)$ has no real roots. Prove that the polynomial $$Q(x)=P(x)+\frac{P''(x)}{2!}+\frac{P^{(4)}(x)}{4!}+\ldots$$ also has no real roots.

Prove that for all positive integers $n$, $n>1$ the number $\sqrt{ \overline{ 11\ldots 44 \ldots 4 }}$, where 1 appears $n$ times, and 4 appears $2n$ times, is irrational.

A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic. [img]http://aycu05.webshots.com/image/5604/2000468517162383885_rs.jpg[/img] For working with ruler, (e.g for calculating $x.y$) we must move the middle arm that the arrow at the beginning of its gradation locate above the $x$ in the lower arm. We find $y$ in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer. 1) Design a ruler for calculating $x^{y}$. Grade first arm ($x$) and ($y$) from 1 to 10. 2) Find all rulers that do the multiplication in the interval $[1,10]$. 3) Prove that there is not a ruler for calculating $x^{2}+xy+y^{2}$, that its first and second arm are grade from 0 to 10.

A divisor $d$ of a positive integer $n$ is said to be a [i]close[/i] divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?

Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd? $\textbf{(A)}\ p<\dfrac{1}{8}\qquad\textbf{(B)}\ p=\dfrac{1}{8}\qquad\textbf{(C)}\ \dfrac{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}$