Let $x$ and $y$ be relatively prime integers. Show that $x^2+xy+y^2$ and $x^2+3xy+y^2$ are relatively prime.

Ari and Beri play a game using a deck of $2020$ cards with exactly one card with each number from $1$ to $2020$. Ari gets a card with a number $a$ and removes it from the deck. Beri sees the card, chooses another card from the deck with a number $b$ and removes it from the deck. Then Beri writes on the board exactly one of the trinomials $x^2-ax+b$ or $x^2-bx+a$ from his choice. This process continues until no cards are left on the deck. If at the end of the game every trinomial written on the board has integer solutions, Beri wins. Otherwise, Ari wins. Prove that Beri can always win, no matter how Ari plays.

Let $ x_1, x_2, \cdots, x_{25} $ real numbers such that $ 0 \le x_i \le i (i=1, 2, \cdots, 25) $. Find the maximum value of \[x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 ) \]

Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.

Is there a set $S$ of positive integers such that a number is in $S$ if and only if it is the sum of two distinct members of $S$ or a sum of two distinct positive integers not in $S$?

Which of the following describes the graph of the equation $ (x \plus{} y)^2 \equal{} x^2 \plus{} y^2$? $ \textbf{(A)}\text{ the empty set}\qquad \textbf{(B)}\text{ one point}\qquad \textbf{(C)}\text{ two lines}$ $\textbf{(D)}\text{ a circle}\qquad \textbf{(E)}\text{ the entire plane}$

Let $[a, b] = ab - a - b$. Shaq sees the numbers $2, 3, \dots , 101$ written on a blackboard. Let $V$ be the largest number that Shaq can obtain by repeatedly choosing two numbers $a, b$ on the board and replacing them with $[a, b]$ until there is only one number left. Suppose $N$ is the integer with $N!$ nearest to $V$. Find the nearest integer to $10^6 \cdot \tfrac{|V-N!|}{N!}$.

Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$ [hide= original wording] before finding a typo .. Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $1,0,0,4.$ Posts 2 and 3 reply to this wording [/hide]

Let $ABC$ be a triangle with $AB=2\sqrt6, BC=5, CA=\sqrt{26}$, midpoint $M$ of $BC$, circumcircle $\Omega$, and orthocenter $H$. Let $BH$ intersect $AC$ at $E$ and $CH$ intersect $AB$ at $F$. Let $R$ be the midpoint of $EF$ and let $N$ be the midpoint of $AH$. Let $AR$ intersect the circumcircle of $AHM$ again at $L$. Let the circumcircle of $ANL$ intersect $\Omega$ and the circumcircle of $BNC$ at $J$ and $O$, respectively. Let circles $AHM$ and $JMO$ intersect again at $U$, and let $AU$ intersect the circumcircle of $AHC$ again at $V \neq A$. The square of the length of $CV$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Michael Ren[/i]

Determine the numbers $x,y$, with $x$ integer and $y$ rational, for which equality holds: $$5(x^2+xy+y^2) = 7(x+2y)$$

Let $ C $ denote the complex unit circle centered at the origin. [b]a)[/b] Prove that $ \left( |z+1|-\sqrt 2 \right)\cdot \left( |z-1|-\sqrt 2 \right)\le 0,\quad\forall z\in C. $ [b]b)[/b] Prove that for any twelve numbers from $ C, $ namely $ z_1,\ldots ,z_{12} , $ there exist another twelve numbers $ \varepsilon_1,\ldots ,\varepsilon_{12}\in\{-1,1\} $ such that $$ \sum_{k=1}^{12} \left| z_k+\varepsilon_k \right| <17. $$

Let $ABC$ be atriangle with $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.

Determine all functions $f : R \to R$ satisfying $f(f(x) - y) \le xf(x) + f(y)$ for all real numbers $x, y$.

In the triangle $ABC,\angle A=\frac{\pi}{3},D,M$ are points on the line $AC$ and $E,N$ are points on the line $AB$ such that $DN$ and $EM$ are the perpendicular bisectors of $AC$ and $AB$ respectively. Let $L$ be the midpoint of $MN$. Prove that $\angle EDL=\angle ELD$

Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.

For all non-negative real numbers $x,y,z$ with $x \geq y$, prove the inequality $$\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.$$