Horizontal line $m$ passes the center of circle $\odot O$. Line $l\perp m$, $l$ and $m$ intersect at $M$, and $M$ is on the right side of $O$. Three points $A,B,C$ ($B$ is in the middle) lie on line $l$, which are outside the circle, above line $m$. $AP,BQ,CR$ are tangent to $\odot O$ at $P,Q,R$. Prove: [b](a)[/b] If $l$ is tangent to $\odot O$, then $AB\cdot CR+BC\cdot AP=AC\cdot BQ$. [b](b)[/b] If $l$ and $\odot O$ intersect, then $AB\cdot CR+BC\cdot AP<AC\cdot BQ$. [b](c)[/b] If $l$ and $\odot O$ are apart, then $AB\cdot CR+BC\cdot AP>AC\cdot BQ$.

An unequal acute-angled triangle $ABC$ with an orthocenter $H$ is given, $M$ is the midpoint of side $BC$. Points $K$ and $L$ lie on a line passing through $H$ and perpendicular to $AM$ such a $KB$ and $LC$ perpendicular to $BC$. Point $N$ lies on the line $HM$, and the lines $AN$ and $AH$ are symmetric with respect to the line $AM$. Prove that a circle with a diameter $AN$ touches two circles: centered at $K$ and with a radius $KB$ and with a center $L$ and radius $LC$.

[u]Round 1[/u] [b]p1.[/b] How many distinct ways are there to scramble the letters in $EXETER$? [b]p2.[/b] Given that $\frac{x - y}{x - z}= 3$, find $\frac{x - z}{y - z}$. [b]p3.[/b] When written in base $10$, $9^9 =\overline{ABC420DEF}.$ Find the remainder when $A + B + C + D + E + F$ is divided by $9$. [u]Round 2[/u] [b]p4.[/b] How many positive integers, when expressed in base $7$, have exactly $3$ digits, but don't contain the digit $3$? [b]p5.[/b] Pentagon $JAMES$ is such that its internal angles satisfy $\angle J = \angle A = \angle M = 90^o$ and $\angle E = \angle S$. If $JA = AM = 4$ and $ME = 2$, what is the area of $JAMES$? [b]p6.[/b] Let $x$ be a real number such that $x = \frac{1+\sqrt{x}}{2}$ . What is the sum of all possible values of $x$? [u]Round 3[/u] [b]p7.[/b] Farmer James sends his favorite chickens, Hen Hao and PEAcock, to compete at the Fermi Estimation All Star Tournament (FEAST). The first problem at the FEAST requires the chickens to estimate the number of boarding students at Eggs-Eater Academy given the number of dorms $D$ and the average number of students per dorm $A$. Hen Hao rounds both $D$ and $A$ down to the nearest multiple of $10$ and multiplies them, getting an estimate of $1200$ students. PEAcock rounds both $D$ and $A$ up to the nearest multiple of $10$ and multiplies them, getting an estimate of $N$ students. What is the maximum possible value of $N$? [b]p8.[/b] Farmer James has decided to prepare a large bowl of egg drop soup for the Festival of Eggs-Eater Annual Soup Tasting (FEAST). To flavor the soup, Hen Hao drops eggs into it. Hen Hao drops $1$ egg into the soup in the first hour, $2$ eggs into the soup in the second hour, and so on, dropping $k$ eggs into the soup in the $k$th hour. Find the smallest positive integer $n$ so that after exactly n hours, Farmer James finds that the number of eggs dropped in his egg drop soup is a multiple of $200$. [b]p9.[/b] Farmer James decides to FEAST on Hen Hao. First, he cuts Hen Hao into $2018$ pieces. Then, he eats $1346$ pieces every day, and then splits each of the remaining pieces into three smaller pieces. How many days will it take Farmer James to eat Hen Hao? (If there are fewer than $1346$ pieces remaining, then Farmer James will just eat all of the pieces.) [u]Round 4[/u] [b]p10.[/b] Farmer James has three baskets, and each basket has one magical egg. Every minute, each magical egg disappears from its basket, and reappears with probability $\frac12$ in each of the other two baskets. Find the probability that after three minutes, Farmer James has all his eggs in one basket. [b]p11.[/b] Find the value of $\frac{4 \cdot 7}{\sqrt{4 +\sqrt7} +\sqrt{4 -\sqrt7}}$. [b]p12.[/b] Two circles, with radius $6$ and radius $8$, are externally tangent to each other. Two more circles, of radius $7$, are placed on either side of this configuration, so that they are both externally tangent to both of the original two circles. Out of these $4$ circles, what is the maximum distance between any two centers? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949222p26406222]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let be three nonzero rational numbers $ a,b,c $ under the relation $ (a+b)(b+c)(c+a)=a^2b^2c^2. $ Show that the expression $ \sqrt[3]{3+1/a^3+1/b^3+1/c^3} $ is rational. [i]Ion Bursuc[/i]

Let $n \geq 2$ and $m$ be positive integers. $m$ ballot boxes are placed in a line. Two players $A$ and $B$ play by turns, beginning with $A$, in the following manner. Each turn, $A$ chooses two boxes and places a ballot in each of them. Afterwards, $B$ chooses one of the boxes, and removes every ballot from it. $A$ wins if after some turn of $B$, there exists a box containing $n$ ballots. For each $n$, find the minimum value of $m$ such that $A$ can guarantee a win independently of how $B$ plays.

Let $x^{(n)}=x(x-1)\cdots (x-n+1)$ for $n$ a positive integer and let $x^{(0)}=1.$ Prove that $$(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.$$

Let $T_n$ be the number of non-congruent triangles with positive area and integer side lengths summing to $n$. Prove that $T_{2022}=T_{2019}$. [i]Proposed by Constantinos Papachristoforou[/i]

A vertical line divides the triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ in the $xy\text{-plane}$ into two regions of equal area. The equation of the line is $x=$ $\textbf {(A) } 2.5 \qquad \textbf {(B) } 3.0 \qquad \textbf {(C) } 3.5 \qquad \textbf {(D) } 4.0\qquad \textbf {(E) } 4.5$

Let $ABC$ be a triangle with $\angle A = 90^o$ and let $P$ be a point on the hypotenuse $BC$. Prove that $$\frac{AB^2}{PC}+\frac{AC^2}{PB} \ge \frac{BC^3}{PA^2 + PB \cdot PC}$$

Solve the system of equations $$\begin{cases} \dfrac{x+y}{xy}+\dfrac{xy}{x+y}= a+ \dfrac{1}{a}\\ \\\dfrac{x-y}{xy}+\dfrac{xy}{x-y}= c+ \dfrac{1}{c}\end{cases}$$

Consider two geometric sequences $16$, $a_1$, $a_2$, $ . . .$ and $56$, $b_1$, $b_2$, $. . . $ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.

If $n$ is positive integer and $p, q, r$ are primes solve the system: $pqr=n$ and $(p+1)(q+1)r=n+138$

Given a positive integer $n \geq 2$. Let $a_{ij}$ $(1 \leq i,j \leq n)$ be $n^2$ non-negative reals and their sum is $1$. For $1\leq i \leq n$, define $R_i=max_{1\leq k \leq n}(a_{ik})$. For $1\leq j \leq n$, define $C_j=min_{1\leq k \leq n}(a_{kj})$ Find the maximum value of $C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)$

If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded: $a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.

For a rhombus with side length of 5, length of one of its diagonal is not larger than $6$, length of the other diagonal is not smaller than $6$, then the maximum value of the sum of the two diagonals is $\text{(A)}10\sqrt{2}\qquad\text{(B)}14\qquad\text{(C)}5\sqrt{6}\qquad\text{(D)}12$

Let $ f(x)\equal{}5x^{13}\plus{}13x^5\plus{}9ax$. Find the least positive integer $ a$ such that $ 65$ divides $ f(x)$ for every integer $ x$.

A real number is written on each square of a $2024 \times 2024$ chessboard. It is given that the sum of all real numbers on the board is $2024$. Then, the board is covered by $1 \times 2$ or $2\times 1$ dominos such that there isn't any square that is covered by two different dominoes. For each domino, Aslı deletes $2$ numbers covered by it and writes $0$ on one of the squares and the sum of the two numbers on the other square. Find the maximum number $k$ such that after Aslı finishes her moves, there exists a column or row where the sum of all the numbers on it is at least $k$ regardless of how dominos were replaced and the real numbers were written initially.

Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square. N. Sedrakian , I.Voronovich

Fede chooses $50$ distinct integers from the set $\{1, 2, 3, \ldots, 100\}$ such that their sum equals $2900$. Determine the minimum number of even numbers that can be among the $50$ numbers chosen by Fede.

Let $n \geq 2$ be an integer. Define $x_i =1$ or $-1$ for every $i=1,2,3,\cdots, n$. Call an operation [i]adhesion[/i], if it changes the string $(x_1,x_2,\cdots,x_n)$ to $(x_1x_2, x_2x_3, \cdots ,x_{n-1}x_n, x_nx_1)$ . Find all integers $n \geq 2$ such that the string $(x_1,x_2,\cdots, x_n)$ changes to $(1,1,\cdots,1)$ after finitely [i]adhesion[/i] operations.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xy) = f(x)f(y) + f(f(x + y))$$ holds for all $x, y \in \mathbb{R}$.

Let $VABCD$ be a regular pyramid, having the square base $ABCD$. Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$. Prove that $4AM=3AC$. [i]Mircea Fianu[/i]

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]

Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:\\ • For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.\\ • For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.

The permutation $\sigma$ consisting of four words $A,B,C,D$ has $f_{AB}(\sigma)$, the sum of the number of $B$ placed rightside of every $A$. We can define $f_{BC}(\sigma)$,$f_{CD}(\sigma)$,$f_{DA}(\sigma)$ as the same way too. For example, $\sigma=ACBDBACDCBAD$, $f_{AB}(\sigma)=3+1+0=4$, $f_{BC}(\sigma)=4$,$f_{CD}(\sigma)=6$, $f_{DA}(\sigma)=3$ Find the maximal value of $f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)$, when $\sigma$ consists of $2020$ letters for each $A,B,C,D$