Let triangle ABC with $ AB=c$ $AC=b$ $BC=a$ $R$ circumradius, $p$ half peremetr of $ABC$. I f $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$ then find all value of $cosA$.

Suppose $s,k,t\in \mathbb N$. We've colored each natural number with one of the $k$ colors, such that each color is used infinitely many times. We want to choose a subset $\mathcal A$ of $\mathbb N$ such that it has $t$ disjoint monochromatic $s$-element subsets. What is the minimum number of elements of $A$? [i]Proposed by Navid Adham[/i]

In the circumcircle of triange $\triangle ABC,$ $AA'$ is a diameter. We draw lines $l'$ and $l$ from $A'$ parallel with Internal and external bisector of the vertex $A$. $l'$ Cut out $AB , BC$ at $B_1$ and $B_2$. $l$ Cut out $AC , BC$ at $C_1$ and $C_2$. Prove that the circumcircles of $\triangle ABC$ $\triangle CC_1C_2$ and $\triangle BB_1B_2$ have a common point. (20 points)

A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985. [asy] size(200); pair A=(0,1), B=(1,1), C=(1,0), D=origin; draw(A--B--C--D--A--(1,1/6)); draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1)); pair point=( 0.5 , 0.5 ); //label("$A$", A, dir(point--A)); //label("$B$", B, dir(point--B)); //label("$C$", C, dir(point--C)); //label("$D$", D, dir(point--D)); label("$1/n$", (11/12,1), N, fontsize(9));[/asy]

Let $ABC$ be a triangle. Let $ r$ denote the inradius of $\vartriangle ABC$. Let $r_a$ denote the $A$-exradius of $\vartriangle ABC$. Note that the $A$-excircle of $\vartriangle ABC$ is the circle that is tangent to segment $BC$, the extension of ray $AB$ beyond $ B$ and the extension of $AC$ beyond $C$. The $A$-exradius is the radius of the $A$-excircle. Define $ r_b$ and $ r_c$ analogously. Prove that $$\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$$

The real numbers $a, b, c$ are such that $a^2 + b^2 = 2c^2$, and also such that $a \ne b, c \ne -a, c \ne -b$. Show that \[\frac{(a+b+2c)(2a^2-b^2-c^2)}{(a-b)(a+c)(b+c)}\] is an integer.

In a row of $2009$ weights, the weight of each weight is an integer grams and does not exceed $1$ kg. The weights of any two adjacent weights differ by exactly $1$ g, and the total weight of all weights in grams is an even number. Prove that weights can be separated into two piles, the sums of the weights in which are equal.

Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $. Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]

For given positive integer $n$ find all quartets $(x_1,x_2,x_3,x_4)$ such that $x_1^2+x_2^2+x_3^2+x_4^2=4^n$

On a certain planet, the alien inhabitants are born without any arms, legs, or noses. Every year, on their birthday, each alien randomly grows either an arm, a leg, or a nose, with equal probability for each. After its sixth birthday, the probability that an alien will have at least $2$ arms, at least $2$ legs, and at least $1$ nose on the day is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $n$ be a positive integer, and let $a_0,\,a_1,\dots,\,a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that \[ \sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}. \] [i]Note:[/i] $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$.

The equation $ 3y^2 \plus{} y \plus{} 4 \equal{} 2(6x^2 \plus{} y \plus{} 2)$ where $ y \equal{} 2x$ is satisfied by: $ \textbf{(A)}\ \text{no value of }x \qquad\textbf{(B)}\ \text{all values of }x \qquad\textbf{(C)}\ x \equal{} 0\text{ only}$ $ \textbf{(D)}\ \text{all integral values of }x\text{ only} \qquad\textbf{(E)}\ \text{all rational values of }x\text{ only}$

Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

A positive number $ x$ has the property that $ x\%$ of $ x$ is $ 4$. What is $ x$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 40$

Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.

[b]p1.[/b] Fleming has a list of 8 mutually distinct integers between $90$ to $99$, inclusive. Suppose that the list has median $94$, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads. [b]p2.[/b] Find the number of ordered pairs $(x,y)$ of three digit base-$10$ positive integers such that $x-y$ is a positive integer, and there are no borrows in the subtraction $x-y$. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows. $$\begin{tabular}{ccccc} & 4 & 7 & 2 \\ - & 1 & 9 & 1\\ \hline & 2 & 8 & 1 \\ \end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc} & 3 & 7 & 9 \\ - & 2 & 6 & 3\\ \hline & 1 & 1 & 6 \\ \end{tabular}$$ [b]p3.[/b] Evaluate $$1 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021$$ [b]p4.[/b] Find the number of ordered pairs of integers $(a,b)$ such that $$\frac{ab+a+b}{a^2+b^2+1}$$ is an integer. [b]p5.[/b] Lin Lin has a $4\times 4$ chessboard in which every square is initially empty. Every minute, she chooses a random square $C$ on the chessboard, and places a pawn in $C$ if it is empty. Then, regardless of whether $C$ was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from $C$. The expected number of minutes before the entire chessboard is occupied with pawns equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Find $m+n$. A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally. [b]p6.[/b] Let $P(x) = x^5-3x^4+2x^3-6x^2+7x+3$ and $a_1,...,a_5$ be the roots of$ P(x)$. Compute $$\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).$$ [b]p7.[/b] Rectangle $AXCY$ with a longer length of $11$ and square $ABCD$ share the same diagonal $\overline{AC}$. Assume $B$,$X$ lie on the same side of $\overline{AC}$ such that triangle$ BXC$ and square $ABCD$ are non-overlapping. The maximum area of $BXC$ across all such configurations equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Compute $m+n$. [b]p8.[/b] Earl the electron is currently at $(0,0)$ on the Cartesian plane and trying to reach his house at point $(4,4)$. Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line $y=x$. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house. Earl visits a chronologically ordered sequence of distinct points $(0,0)$, $...$, $(4,4)$ due to his choice of actions. This is called an [i]Earl-path[/i]. How many possible such [i]Earl-paths[/i] are there? [b]p9.[/b] Let $P(x)$ be a degree-$2022$ polynomial with leading coefficient $1$ and roots $\cos \left( \frac{2\pi k}{2023} \right)$ for $k = 1$ , $...$,$2022$ (note $P(x)$ may have repeated roots). If $P(1) =\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then find the remainder when $m+n$ is divided by $100$. [b]p10.[/b] A randomly shuffled standard deck of cards has $52$ cards, $13$ of each of the four suits. There are $4$ Aces and $4$ Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p11.[/b] The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The [i]angle of incidence[/i] is marked by the shaded angle; the[i] angle of reflection[/i] is marked by the unshaded angle. [img]https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png[/img] The sides of a unit square $ABCD$ are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant $q$ degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from $A$ strikes $\overline{CD}$ at $W_1$ such that $2DW_1 =CW_1$, reflects off of $\overline{CD}$ and then strikes $\overline{BC}$ at $W_2$ such that $2CW_2 = BW_2$, reflects off of $\overline{BC}$, etc. To this end, denote $W_i$ the $i$-th point at which the light beam strikes $ABCD$. As $i$ grows large, the area of $W_iW_{i+1}W_{i+2}W_{i+3}$ approaches $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] For any positive integer $m$, define $\phi (m)$ the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime. Find the smallest positive integer $N$ such that $\sqrt{ \phi (n) }\ge 22$ for any integer $n \ge N$. [b]p13.[/b] Let $n$ be a fixed positive integer, and let $\{a_k\}$ and $\{b_k\}$ be sequences defined recursively by $$a_1 = b_1 = n^{-1}$$ $$a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1$$ $$b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1$$ When $n = 2021$, then $a_{2021} +b_{2021} = m \cdot 2017^2$ for some positive integer $m$. Find the remainder when $m$ is divided by $2017$. [b]p14.[/b] Consider the quadratic polynomial $g(x) = x^2 +x+1020100$. A positive odd integer $n$ is called $g$-[i]friendly[/i] if and only if there exists an integer $m$ such that $n$ divides $2 \cdot g(m)+2021$. Find the number of $g$-[i]friendly[/i] positive odd integers less than $100$. [b]p15.[/b] Let $ABC$ be a triangle with $AB < AC$, inscribed in a circle with radius $1$ and center $O$. Let $H$ be the intersection of the altitudes of $ABC$. Let lines $\overline{OH}$, $\overline{BC}$ intersect at $T$. Suppose there is a circle passing through $B$, $H$, $O$, $C$. Given $\cos (\angle ABC-\angle BCA) = \frac{11}{32}$ , then $TO = \frac{m\sqrt{p}}{n}$ for relatively prime positive integers $m$,$n$ and squarefree positive integer $p$. Find $m+n+ p$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].