In triangle $ABC$, $AB=3\sqrt{30}-\sqrt{10}$, $BC=12$, and $CA=3\sqrt{30}+\sqrt{10}$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AC$. Denote $l$ as the line passing through the circumcenter $O$ and orthocenter $H$ of $ABC$, and let $E$ and $F$ be the feet of the perpendiculars from $B$ and $C$ to $l$, respectively. Let $l'$ be the reflection of $l$ in $BC$ such that $l'$ intersects lines $AE$ and $AF$ at $P$ and $Q$, respectively. Let lines $BP$ and $CQ$ intersect at $K$. $X$, $Y$, and $Z$ are the reflections of $K$ over the perpendicular bisectors of sides $BC$, $CA$, and $AB$, respectively, and $R$ and $S$ are the midpoints of $XY$ and $XZ$, respectively. If lines $MR$ and $NS$ intersect at $T$, then the length of $OT$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $100p+q$. [i]Proposed by Vincent Huang and James Lin[/i]

A trapezium has parallel sides of length equal to $a$ and $b$ ($a <b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel lines intersect at a point that is a vertex of two triangles that have as sides the parallel sides of the trapezium. Express the areas of the triangles as functions of $a,b$ and $h$.

Given triangle $ABC$ with circumcircle $\Gamma$, let $D$, $E$, and $F$ be the midpoints of sides $BC$, $CA$, and $AB$, respectively, and let the lines $AD$, $BE$, and $CF$ intersect $\Gamma$ again at points $J$, $K$, and $L$, respectively. Show that the area of triangle $JKL$ is at least that of triangle $ABC$.

Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the number in each group will be the same. In how many ways can this be done? $\textbf{(A) }0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$

For which $n > 0$ it is possible to mark several different points and several different circles on the plane in such a way that: — exactly $n$ marked circles pass through each marked point; — exactly $n$ marked points lie on each marked circle; — the center of each marked circle is marked?

Two circles intersect at the points $A$ and $B$. Tangent lines drawn to both of the circles at the point $A$ intersect the circles at the points $M$ and $N$. The lines $BM$ and $BN$ intersect the circles once more at the points $P$ and $Q$ respectively. Prove that the segments $MP$ and $NQ$ are equal. (I Nagel)

Let $\Delta ABC$ be an acute triangle and $H$ its orthocenter. $B_1$ and $C_1$ are the feet of heights from $B$ and $C$, $M$ is the midpoint of $AH$. Point $K$ is on the segment $B_1C_1$, but isn't on line $AH$. Line $AK$ intersects the lines $MB_1$ and $MC_1$ in $E$ and $F$, the lines $BE$ and $CF$ intersect at $N$. Prove that $K$ is the orthocenter of $\Delta NBC$.

$2048$ frogs are sitting in a circle and each have a $\$1$ bill. After each minute, each frog will independently give away each of their $\$1$ bills to either the closest frog to their left or the closest frog to their right with equal probability. If a frog has $\$0$ at the end of any given minute, then they will not give any money but may receive money. The expected number of frogs to have at least $\$1$ after $3$ minutes can be denoted as a common fraction in the form $\tfrac{a}{b}.$ Find $a+b.$

Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$

$50$ gangsters simultaneously shoot at each other, and each shoots at the nearest gangster (if there are several of them, then at one of them) and kills him. Find the smallest possible number of people killed.

A sequence of integers $\{f(n)\}$ for $n=0,1,2,\ldots$ is defined as follows: $f(0)=0$ and for $n>0$, $$\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,\\&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,\\&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}$$Derive an explicit formula for $f(n)$ when $n\equiv0\pmod6$, showing all necessary details in your derivation.

For a convex polygon with $n$ edges $F$, if all its diagonals have the equal length, then $\text{(A)}F\in \{\text{quadrilaterals}\}$ $\text{(B)}F\in \{\text{pentagons}\}$ $\text{(C)}F\in \{\text{pentagons}\} \cup\{\text{quadrilaterals}\}$ $\text{(D)}F\in \{\text{convex polygons that have all edges' length equal}\} \cup\{\text{convex polygons that have all inner angles equal}\}$

John ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by $ 50\%$. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was: $\textbf{(A)}\ 4:1 \qquad \textbf{(B)}\ 2:1 \qquad \textbf{(C)}\ 1:4 \qquad \textbf{(D)}\ 1:2 \qquad \textbf{(E)}\ 1:8$

Find all functions $g : R \to R$, for which there exists a strictly increasing function $f : R \to R$ such that $f(x + y) = f(x)g(y) + f(y)$.

Let $n \ge 2$ be an integer, and let $x_1, x_2, \ldots, x_n$ be reals for which: (a) $x_j > -1$ for $j = 1, 2, \ldots, n$ and (b) $x_1 + x_2 + \ldots + x_n = n.$ Prove that $$ \sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2} $$ and determine when does the equality occur.

Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$. We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$. Compute $a + b$.

Minivan and Megavan play a game. For a positive integer $n$, Minivan selects a sequence of integers $a_1,a_2,\ldots,a_n$. An operation on $a_1,a_2,\ldots,a_n$ means selecting an $a_i$ and increasing it by $1$. Minivan and Megavan take turns, with Minivan going first. On Minivan's turn, he performs at most $2025$ operations, and he may choose the same integer repeatedly. On Megavan's turn, he performs exactly $1$ operation instead. Megavan wins if at any point in the game, including in the middle of Minivan's operations, two numbers in the sequence are equal. [i](Proposed by Ho Janson)[/i]

Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$

Let $n{}$ be a natural number. The pairwise distinct nonzero integers $a_1,a_2,\ldots,a_n$ have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by $a_1a_2\cdots a_n$ for any integer $k{}.$ Find the largest possible value of $a_n.$ [i]Proposed by F. Petrov and K. Sukhov[/i]

[b]p1.[/b] How many perfect squares are in the set: $\{1, 2, 4, 9, 10, 16, 17, 25, 36, 49\}$? [b]p2.[/b] If $a \spadesuit b = a^b - ab - 5$, what is the value of $2 \spadesuit 11$? [b]p3.[/b] Joe can catch $20$ fish in $5$ hours. Jill can catch $35$ fish in $7$ hours. If they work together, and the number of days it takes them to catch $900$ fish is represented by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, what is $m + n$? Assume that they work at a constant rate without taking breaks and that there are an infinite number of fish to catch. [b]p4.[/b] What is the units digit of $187^{10}$? [b]p5.[/b] What is the largest number of regions we can create by drawing $4$ lines in a plane? [b]p6.[/b] A regular hexagon is inscribed in a circle. If the area of the circle is $2025\pi$, given that the area of the hexagon can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any number other than $1$, find $a + b + c$. [b]p7.[/b] Find the number of trailing zeroes in the product $3! \cdot 5! \cdot 719!$. [b]p8.[/b] How many ordered triples $(x, y, z)$ of odd positive integers satisfy $x + y + z = 37$? [b]p9.[/b] Let $N$ be a number with $2021$ digits that has a remainder of $1$ when divided by $9$. $S(N)$ is the sum of the digits of $N$. What is the value of $S(S(S(S(N))))$? [b]p10.[/b] Ayana rolls a standard die $10$ times. If the probability that the sum of the $10$ die is divisible by $6$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$, what is $m + n$? [b]p11.[/b] In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. The inscribed circle touches the side $BC$ at point $D$. The line $AI$ intersects side $BC$ at point $K$ given that $I$ is the incenter of triangle $ABC$. What is the area of the triangle $KID$? [b]p12.[/b] Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$. [b]p13.[/b] In tetrahedron $ABCD$, $AB=6$, $BC=8$, $CA=10$, and $DA$, $DB$, $DC=20$. If the volume of $ABCD$ is $a\sqrt{b}$ where $a$, $b$ are positive integers and in simplified radical form, what is $a + b$? [b]p14.[/b] A $2021$-digit number starts with the four digits $2021$ and the rest of the digits are randomly chosen from the set $0$,$1$,$2$,$3$,$4$,$5$,$6$. If the probability that the number is divisible by $14$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. what is $m + n$? [b]p15.[/b] Let $ABCD$ be a cyclic quadrilateral with circumcenter $O_1$ and circumradius $20$, Let the intersection of $AC$ and $BD$ be $E$. Let the circumcenter of $\vartriangle EDC$ be $O_2$. Given that the circumradius of 4EDC is $13$; $O_1O_2 = 11$, $BE = 11 \sqrt2$, find $O_1E^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first). The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies? Proposed by Denis Afrizonov

Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$. [i]Proposed by Viktor Simjanoski[/i]

The sum of two numbers is $ S$. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? $ \textbf{(A)} \ 2S \plus{} 3 \qquad \textbf{(B)} \ 3S \plus{} 2 \qquad \textbf{(C)} \ 3S \plus{} 6 \qquad \textbf{(D)} \ 2S \plus{} 6 \qquad \textbf{(E)} \ 2S \plus{} 12$

Prove that if $ a > 0 $, $ b > 0 $, $ c > 0 $, then the inequality holds $$ ab (a + b) + bc (b + c) + ca (c + a) \geq 6abc.$$

Compute the largest prime factor of $357!+358!+359!+360!$. [i]Proposed by Luke Robitaille