Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$. (For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)

Solve the equation $$ x + \sqrt{x^2-9} = \frac{2(x+3)}{(x-3)^2}$$

An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.) $ \displaystyle \textbf{(A)} \ \frac {1}{2} \sqrt {3} \qquad \textbf{(B)} \ 1 \qquad \textbf{(C)} \ \sqrt {2} \qquad \textbf{(D)} \ \frac {3}{2} \qquad \textbf{(E)} \ 2$

There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

Find all possible triples of integers $a, b, c$ satisfying $a+b-c = 1$ and $a^2+b^2-c^2 =-1$.

Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$. [B. Kovacs]

Let $ABCD$ be a quadrilateral with a incircle $\omega$. Let $I$ be the center of $\omega$, suppose that the lines $AD$ and $BC$ intersect at $Q$ and the lines $AB$ and $CD$ intersect at $P$ with $B$ is in the segment $AP$ and $D$ is in the segment $AQ$. Let $X$ and $Y$ the incenters of $\triangle PBD$ and $\triangle QBD$ respectively. Let $R$ be the intersection of $PY$ and $QX$. Prove that the line $IR$ is perpendicular to $BD$.

Can we find an (infinite) sequence of disks in the Euclidean plane such that: $(1)$ their centers have no (finite) limit point in the plane; $(2)$ the total area of the disks is finite; and $(3)$ every line in the plane intersects at least one of the disks?

Two circles with centers $ S_{1}$ and $ S_{2}$ are externally tangent at point $ K$. These circles are also internally tangent to circle $ S$ at points $ A_{1}$ and $ A_{2}$, respectively. Denote by $ P$one of the intersection points of $ S$ and common tangent to $ S_{1}$ and $ S_{2}$ at $ K$.Line $ PA_{1}$ intersects $ S_{1}$ at $ B_{1}$ while $ PA_{2}$ intersects $ S_{2}$ at $ B_{2}$. Prove that $ B_{1}B_{2}$ is common tangent of circles $ S_{1}$ and $ S_{2}$.

Find all functions $f : \mathbb R \to \mathbb R$ such that \[f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.\]

What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction.

Let $a$, $b$ and $n$ be positive integers, satisfying that $bn$ divides $an-a+1$. Let $\alpha=a/b$. Prove that, when the numbers $\lfloor\alpha\rfloor,\lfloor2\alpha\rfloor,\dots,\lfloor(n-1)\alpha\rfloor$ are divided by $n$, the residues are $1,2,\dots,n-1$, in some order.

There exists infinitely many positive integers such that $a^3 + 1990b^3 = c^4$.

Let $F_1$ be an arbitrary convex quadrilateral. For $k\ge2$, $F_k$ is obtained by cutting $F_{k-1}$ into two pieces along one of its diagonals, flipping one piece over, and the glueing them back together along the same diagonal. What is the maximum number of non-congruent quadrilaterals in the sequence $\{F_k\}$?

Let $A,B,C,...,Z$ be $26$ nonzero real numbers. Suppose that $T=TNYWR$. Compute the smallest possible value of \[ \left\lceil A^2+B^2+\cdots+Z^2\right\rceil . \] (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.) [i]Proposed by Luke Robitaille[/i]

[u]Round 1[/u] [b]p1.[/b] Six pirates – Captain Jack and his five crewmen – sit in a circle to split a treasure of $99$ gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The five crewmen will then vote on Jack's decision. Each is greedy and will vote “aye” only if he gets more coins than each of his two neighbors. If a majority vote “aye”, Jack's decision is accepted. Otherwise Jack is thrown overboard and gets nothing. What is the most coins Captain Jack can take for himself and survive? [b]p2[/b]. Rose and Bella take turns painting cells red and blue on an infinite piece of graph paper. On Rose's turn, she picks any blank cell and paints it red. Bella, on her turn, picks any blank cell and paints it blue. Bella wins if the paper has four blue cells arranged as corners of a square of any size with sides parallel to the grid lines. Rose goes first. Show that she cannot prevent Bella from winning. [img]https://cdn.artofproblemsolving.com/attachments/d/6/722eaebed21a01fe43bdd0dedd56ab3faef1b5.png[/img] [b]p3.[/b] A $25\times 25$ checkerboard is cut along the gridlines into some number of smaller square boards. Show that the total length of the cuts is divisible by $4$. For example, the cuts shown on the picture have total length $16$, which is divisible by $4$. [img]https://cdn.artofproblemsolving.com/attachments/c/1/e152130e48b804fe9db807ef4f5cd2cbad4947.png[/img] [b]p4.[/b] Each robot in the Martian Army is equipped with a battery that lasts some number of hours. For any two robots, one's battery lasts at least three times as long as the other's. A robot works until its battery is depleted, then recharges its battery until it is full, then goes back to work, and so on. A battery that lasts $N$ hours takes exactly $N$ hours to recharge. Prove that there will be a moment in time when all the robots are recharging (so you can invade the planet). [b]p5.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png [/img] [u]Round 2[/u] [b]p6.[/b] The sum of $2015$ rational numbers is an integer. The product of every pair of them is also an integer. Prove that they are all integers. (A rational number is one that can be written as $m/n$, where $m$ and $n$ are integers and $n\ne 0$.) [b]p7.[/b] An $N \times N$ table is filled with integers such that numbers in cells that share a side differ by at most $1$. Prove that there is some number that appears in the table at least $N$ times. For example, in the $5 \times 5$ table below the numbers $1$ and $2$ appear at least $5$ times. [img]https://cdn.artofproblemsolving.com/attachments/3/8/fda513bcfbe6834d88fb8ca0bfcdb504d8b859.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A positive integer is called extra-even if all of its digits are even. Compute the number of positive integers $n$ less than or equal to $2022$ such that both $n$ and $2n$ are both extra-even.

Petya gave Vasya a number puzzle. Petya chose a digit $X$ and said, "I am thinking of a three digit number that is divisible by 11. The hundreds digit is $X$ and the tens digit is 3. Find the units digit." Vasya was excited because he knew how to solve this problem, but then realized that the problem Petya gave did not have an answer. What digit $X$ did Petya chose? [i]Ray Li.[/i]

Determine the smallest real number $C$ such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} \] holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$.

Determine the real numbers $x$, $y$, $z > 0$ for which $xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$

Let $ABC$ be a triangle, $T$ its centroid and $S$ its incenter. Prove that the following conditions are equivalent: (1) line $TS$ is parallel to one side of triangle $ABC$, (2) one of the sides of triangle $ABC$ is equal to the half-sum of the other two sides.

Let $n\ge2$ be a positive integer. The numbers $1,2,..., n^2$ are consecutively placed into squares of an $n\times n$, so the first row contains $1,2,...,n$ from left to right, the second row contains $n+1,n+2,...,2n$ from left to right, and so on. The [i]magic square value[/i] of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of $\frac{n^2 + 1}{2}$. Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.) [i]Proposed by Ray Li[/i]

Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations $x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$. Find the maximum possible value of $x + 1000y$.

$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $. Find the maximum value of \[ (a-2bc)(b-2ca)(c-2ab) \]