Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1}$, $\overline{PA_2}$, and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac17$, while the region bounded by $\overline{PA_3}$, $\overline{PA_4}$, and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac 19$. There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6}$, $\overline{PA_7}$, and the minor arc $\widehat{A_6A_7}$ is equal to $\tfrac18 - \tfrac{\sqrt 2}n$. Find $n$.

Let $q$ be a fixed odd prime. A prime $p$ is said to be [i]orange [/i] if for every integer $a$ there exists an integer $r$ such that $r^q \equiv a$ (mod $p$). Prove that there are infinitely many [i]orange [/i] primes.

Six gamers play a round-robin tournament where each gamer plays one game against each of the other five gamers. In each game there is one winner and one loser where each player is equally likely to win that game, and the result of each game is independent of the results of the other games. The probability that the tournament will end with exactly one gamer scoring more wins than any other player is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Prove that for any positive integer $n$ there exist infinitely many triples $(a,b,c)$ of pairwise distinct positive integers such that $ab+n,bc+n,ac+n$ are all perfect squares

If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for $\begin{tabular}{ccc} X & X & X \\ & Y & X \\ + & & X \\ \hline \end{tabular}$ has the form $\text{(A)}\ XXY \qquad \text{(B)}\ XYZ \qquad \text{(C)}\ YYX \qquad \text{(D)}\ YYZ \qquad \text{(E)}\ ZZY$

There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$. Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.

A triangle $ABC$ is given. The segment connecting the points where the excircles touch $AB$ and $AC$ meets the bisector of angle $C$ at $X$. The segment connecting the points where the excircles touch $BC$ and $AC$ meets the bisector of angle $A$ at $Y$. Prove that the midpoint of $XY$ is equidistant from $A$ and $C$.

Evaluate $1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + ... + 2018 - 2019$.

The base $AD$ of a trapezoid $ABCD$ is twice greater than the base $BC$, and the angle $C$ equals one and a half of the angle $A$. The diagonal $AC$ divides angle $C$ into two angles. Which of them is greater?

For an integer $n\ge 3$ and a permutation $\sigma=(p_{1},p_{2},\cdots ,p_{n})$ of $\{1,2,\cdots , n\}$, we say $p_{l}$ is a $landmark$ point if $2\le l\le n-1$ and $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$. For example , for $n=7$,\\ the permutation $(2,7,6,4,5,1,3)$ has four landmark points: $p_{2}=7$, $p_{4}=4$, $p_{5}=5$ and $p_{6}=1$. For a given $n\ge 3$ , let $L(n)$ denote the number of permutations of $\{1,2,\cdots ,n\}$ with exactly one landmark point. Find the maximum $n\ge 3$ for which $L(n)$ is a perfect square.

Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.

The LHS Math Team wants to play Among Us. There are so many people who want to play that they are going to form several games. Each game has at most 10 people. People are $\textit{happy}$ if they are in a game that has at least 8 people in it. What is the largest possible number of people who would like to play Among Us such that it is impossible to make everyone $\textit{happy}$? [i]Proposed by Sammy Charney[/i]

$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that: \[f(m)+f(n) \ | \ m+n .\]

Karlson and Smidge divide a cake in a shape of a square in the following way. First, Karlson places a candle on the cake (chooses some interior point). Then Smidge makes a straight cut from the candle to the boundary in the direction of his choice. Then Karlson makes a straight cut from the candle to the boundary in the direction perpendicular to Smidge's cut. As a result, the cake is split into two pieces; Smidge gets the smaller one. Smidge wants to get a piece which is no less than a quarter of the cake. Can Karlson prevent Smidge from getting the piece of that size?

Let’s consider the real numbers $a, b, c$ satisfying the condition \[21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.\] Find the minimal value of the expression \[P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\]

Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $. We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.

Let $ABC$ be an acute-angled triangle, $O$ be its circumcenter, $BM$ be a median, and $BH$ be an altitude. Circles $AOB$ and $BHC$ meet for the second time at point $E$, and circles $AHB$ and $BOC$ meet at point $F$. Prove that $ME = MF$.

Solve the equation: $$x=1978 - \dfrac{1977}{1978 - \dfrac{1977}{\frac{...}{...\dfrac{1977}{1978 -\dfrac{1977}{x}}}}}{}$$

The natural numbers from 1 to 100 are arranged on a circle with the characteristic that each number is either larger as their two neighbours or smaller than their two neighbours. A pair of neighbouring numbers is called "good", if you cancel such a pair, the above property remains still valid. What is the smallest possible number of good pairs?

The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$. How is the tenth vertex labeled? [i]Proposed by A. Golovanov[/i]

Alice starts at the top of Pascal’s triangle. Every move, she moves one layer below, choosing either the left or the right with equal probability. After making $6$ moves, what is the expected sum of the values she visited, including the starting and ending values? For example, in the path shown below, the sum of the values Alice visited is $1 + 1 + 1 + 3 + 6 + 10 + 20 = 42$. [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMC84L2JjZDhiYjkzNjgyMTczMGQ0ZWIzZjE4NDVkOWIxODQxYzQxODdlLnBuZw==&rn=QS5wbmc=[/img][/center]

$f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation \[f(x)f(y)-af(xy)=x+y\] , for every real numbers $x,y$. Find all possible real values of $a$.

[b]p1.[/b] If $$ \begin{cases} a^2 + b^2 + c^2 = 1000 \\ (a + b + c)^2 = 100 \\ ab + bc = 10 \end{cases}$$ what is $ac$? [b]p2.[/b] If a and b are real numbers such that $a \ne 0$ and the numbers $1$, $a + b$, and $a$ are, in some order, the numbers $0$, $\frac{b}{a}$ , and $b$, what is $b - a$? [b]p3.[/b] Of the first $120$ natural numbers, how many are divisible by at least one of $3$, $4$, $5$, $12$, $15$, $20$, and $60$? [b]p4.[/b] For positive real numbers $a$, let $p_a$ and $q_a$ be the maximum and minimum values, respectively, of $\log_a(x)$ for $a \le x \le 2a$. If $p_a - q_a = \frac12$ , what is $a$? [b]p5.[/b] Let $ABC$ be an acute triangle and let $a$, $b$, and $c$ be the sides opposite the vertices $A$, $B$, and $C$, respectively. If $a = 2b \sin A$, what is the measure of angle $B$? [b]p6.[/b] How many ordered triples $(x, y, z)$ of positive integers satisfy the equation $$x^3 + 2y^3 + 4z^3 = 9?$$ [b]p7.[/b] Joe has invented a robot that travels along the sides of a regular octagon. The robot starts at a vertex of the octagon and every minute chooses one of two directions (clockwise or counterclockwise) with equal probability and moves to the next vertex in that direction. What is the probability that after $8$ minutes the robot is directly opposite the vertex it started from? [b]p8.[/b] Find the nonnegative integer $n$ such that when $$\left(x^2 -\frac{1}{x}\right)^n$$ is completely expanded the constant coefficient is $15$. [b]p9.[/b] For each positive integer $k$, let $$f_k(x) = \frac{kx + 9}{x + 3}.$$ Compute $$f_1 \circ f_2\circ ... \circ f_{13}(2).$$ [b]p10.[/b] Exactly one of the following five integers cannot be written in the form $x^2 + y^2 + 5z^2$, where $x$, $y$, and $z$ are integers. Which one is it? $$2003, 2004, 2005, 2006, 2007$$ [b]p11.[/b] Suppose that two circles $C_1$ and $C_2$ intersect at two distinct points $M$ and $N$. Suppose that $P$ is a point on the line $MN$ that is outside of both $C_1$ and $C_2$. Let $A$ and $B$ be the two distinct points on $C_1$ such that AP and BP are each tangent to $C_1$ and $B$ is inside $C_2$. Similarly, let $D$ and $E$ be the two distinct points on $C_2$ such that $DP$ and $EP$ are each tangent to $C_2$ and $D$ is inside $C_1$. If $AB = \frac{5\sqrt2}{2}$ , $AD = 2$, $BD = 2$, $EB = 1$, and $ED =\sqrt2$, find $AE$. [b]p12.[/b] How many ordered pairs $(x, y)$ of positive integers satisfy the following equation? $$\sqrt{x} +\sqrt{y} =\sqrt{2007}.$$ [b]p13.[/b] The sides $BC$, $CA$, and $CB$ of triangle $ABC$ have midpoints $K$, $L$, and $M$, respectively. If $$AB^2 + BC^2 + CA^2 = 200,$$ what is $AK^2 + BL^2 + CM^2$? [b]p14.[/b] Let $x$ and $y$ be real numbers that satisfy: $$x + \frac{4}{x}= y +\frac{4}{y}=\frac{20}{xy}.$$ Compute the maximum value of $|x - y|$. [b]p15.[/b] $30$ math meet teams receive different scores which are then shuffled around to lend an aura of mystery to the grading. What is the probability that no team receives their own score? Express your answer as a decimal accurate to the nearest hundredth. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the coordinate plane, a point $A$ is chosen on the line $y =\frac32 x$ in the first quadrant. Two perpendicular lines $\ell_1$ and $\ell_2$ intersect at A where $\ell_1$ has slope $m > 1$. Let $\ell_1$ intersect the $ x$-axis at $B$, and $\ell_2$ intersects the $ x$ and $y$ axes at $C$ and $D$, respectively. Suppose that line $BD$ has slope $-m$ and $BD = 2$. Compute the length of $CD$.