Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

In the figure, the chord $[CD]$ is perpendicular to the diameter $[AB]$ and intersects it at $H$. Length of $AB$ is a two-digit natural number. Changing the order of these two digits gives length of $CD$. Knowing that distance from $H$ to the center $O$ is a positive rational number, calculate $AB$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/eb9c61579a38118b4f753bbc19a9a50e0732dc.png[/img]

A sector with acute central angle $ \theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is $ \textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \ne A$. If $AB = 10, AD = 24, XA = 17$, and $XB = 21$, then the area of $\vartriangle DEY$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.

One of Michael's responsibilities in organizing the family vacation is to call around and find room rates for hotels along the root the Kubik family plans to drive. While calling hotels near the Grand Canyon, a phone number catches Michael's eye. Michael notices that the first four digits of $987-1234$ descend $(9-8-7-1)$ and that the last four ascend in order $(1-2-3-4)$. This fact along with the fact that the digits are split into consecutive groups makes that number easier to remember. Looking back at the list of numbers that Michael called already, he notices that several of the phone numbers have the same property: their first four digits are in descending order while the last four are in ascending order. Suddenly, Michael realizes that he can remember all those numbers without looking back at his list of hotel phone numbers. "Wow," he thinks, "that's good marketing strategy." Michael then wonders to himself how many businesses in a single area code could have such phone numbers. How many $7$-digit telephone numbers are there such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order?

Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters $M, A, T$, and $H$ in that order, not necessarily consecutively (for example, one valid string might be $MAMMSHTH$.) What is the expected length of the string Clayton wrote? [i]Proposed by Andrew Milas and Andrew Wu[/i]

Is it true that when all the faces of a tetrahedron have the same area, they are congruent triangles?

An equilateral triangle $ABC$ is in between two parallel lines $x, y$ that pass through points $A$ and $B$ respectively. Given that $C$ is twice as far from $y$ as $x$, the acute angle that $CA$ makes with $x$ is $\theta$. Then $(\tan \theta)^2$ is of the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

Given a real number $a$, we define a sequence by $x_0 = 1$, $x_1 = x_2 = a$, and $x_{n+1} = 2x_nx_{n-1} - x_{n-2}$ for $n \ge 2$. Prove that if $x_n = 0$ for some $n$, then the sequence is periodic.

The incircle $I$ of $\triangle ABC$ is tangent to sides $AB,BC,CA$ at $D,E,F$ respectively. Line $EF$ intersects lines $AI,BI,DI$ at $M,N,K$ respectively. Prove that $DM\cdot KE=DN\cdot KF$.

The problem committee of a mathematical olympiad prepares some variants of the contest. Each variant contains $4$ problems, chosen from a shortlist of $n$ problems, and any two variants have at most one problem in common. (a) If $n = 14$, determine the largest possible number of variants the problem committee can prepare. (b) Find the smallest value of n such that it is possible to prepare ten variants of the contest.

All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.

Let $a,b,c $ be real positive numbers such that $abc(a+b+c)=3$ Prove that $(a+b)(b+c)(c+a) \geq 8$

Determine whether it's possible to cover a $K_{2012}$ with a) 1000 $K_{1006}$'s; b) 1000 $K_{1006,1006}$'s. [i]David Yang.[/i]

Solve the system of equations $$\begin{cases} \dfrac{1}{x+y}+ x = 3 \\ \\ \dfrac{x}{x+y}=2 \end{cases}$$

Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is: $\textbf{(A)}\ 159\qquad \textbf{(B)}\ 131\qquad \textbf{(C)}\ 95\qquad \textbf{(D)}\ 79\qquad \textbf{(E)}\ 50$

A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance $d$ from it. By distance is meant the shortest distance measured over the curved survace of the lampshade. Prove that the area of the lampshade is $d^2(2\theta + \sqrt 3)$ where $\cot \frac {\theta}{2} = \frac{3}{\theta}.$

Prove that in acute-angled traingle ABC if $r$ is inradius and $R$ is radius of circumcircle then: \[a^2+b^2+c^2\geq 4(R+r)^2\]

Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.

Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?

Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le3$, let $D_i $, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$. Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.

Let be three positive real numbers $ a,b,c $ whose sum is $ 1. $ Prove: $$ 0\le\sum_{\text{cyc}} \log_a\frac{(abc)^a}{a^2+b^2+c^2} $$

Let $ABC$ be a triangle and $\Gamma$ be the circle with diameter $[AB]$. The bisectors of $\angle BAC$ and $\angle ABC$ cut the circle $\Gamma$ again at $D$ and $E$, respectively. The incicrcle of the triangle $ABC$ cuts the lines $BC$ and $AC$ in $F$ and $G$, respectively. Show that the points $D, E, F$ and $G$ lie on the same line.

Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$

Find the last two digits of $2023+202^3+20^{23}$. [i]Proposed by Anthony Yang[/i]