For reals $x,y$, find the maximum of A. $ A=\frac{-x^2-y^2-2xy+30x+30y+75}{3x^2-12xy+12y^2+12} $

Let $R$ be a convex region symmetrical about the origin with area greater than $4$. Show that $R$ must contain a lattice point different from the origin.

Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying \begin{align*} 2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4 \\ 2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3 \\ 2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2 \\ 2a_4+a_5 &= 6 + a_1 \end{align*} Compute $a_1+a_2+a_3+a_4+a_5$. [i]Proposed by Evan Chen[/i]

Let $ABC$ be an acute triangle, with $AB <AC$. Let $M$ be the midpoint of $AB$, $H$ the foot of the altitude from $A$, and $Q$ be point on side $AC$ such that $\angle ABQ = \angle BCA$. Show that the circumcircles of the triangles $ABQ$ and $BHM$ are tangent.

The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then the number of possible coverings is even. [i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]

On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

On an $8\times 8$ chessboard, place a stick on each edge of each grid (on a common edge of two grid only one stick will be placed). What is the minimum number of sticks to be deleted so that the remaining sticks do not form any rectangle?

Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.

Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers $ 29$, $ 23$, $ 21$, and $ 17$ are obtained. One of the original integers is: $ \textbf{(A)}\ 19 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 29 \qquad \textbf{(E)}\ 17$

Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.

A straight line $\ell$ and a point $A$ outside of it are given on the plane. Find the locus of the vertices $C$ of the equilateral triangle $ABC$, the vertex $B$ of which lies on the straight line $\ell$.

A basketball team's players were successful on $50\%$ of their two-point shots and $40\%$ of their three-point shots, which resulted in $54$ points. They attempted $50\%$ more two-point shots than three-point shots. How many three-point shots did they attempt? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }30 $

Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.

A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$. Prove that $\ell \le \sin \frac{2\pi}{5}$ .

On the Alphamegacentavra planet there are $2023$ cities, some of which are connected by non-directed flights. It turned out that among any $4$ cities one can find two with no flight between them. Find the maximum number of triples of cities such that between any two of them there is a flight.

The sequence of polynomials $ \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty}$ is defined inductively by $ P_0(x) \equal{} 0$ and $ P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}$.

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

$T$ is a given triangle with vertices $P_1,P_2,P_3$. Consider an arbitrary subdivision of $T$ into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex $V$ of the subtriangles there is assigned a number $n(V)$ according to the following rules: $(\text{i})$ If $V$ = $P_i$, then $n(V) = i$. $(\text{ii})$ If $V$ lies on the side $P_i P_j$ of $T$, then $n(V) = i$ or $j$. $(\text{iii})$ If $V$ lies inside the triangle $T$, then $n(V)$ is any of the numbers $1,2,3$. Prove that there exists at least one subtriangle whose vertices are numbered $1, 2, 3$.

Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that $$P(x)^3+Q(x)^3=x^{12}+1.$$

Let $a,b,n$ be positive integers. Prove that $n!$ divides \[b^{n-1}a(a+b)(a+2b)...(a+(n-1)b)\]

Let $n$ be a positive integer. Consider an $n\times n$ grid of unit squares. How many ways are there to partition the horizontal and vertical unit segments of the grid into $n(n + 1)$ pairs so that the following properties are satisfied? (i) Each pair consists of a horizontal segment and a vertical segment that share a common endpoint, and no segment is in more than one pair. (ii) No two pairs of the partition contain four segments that all share the same endpoint. (Pictured below is an example of a valid partition for $n = 2$.) [asy] import graph; size(2.6cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((-3,4)--(-3,2)); draw((-3,4)--(-1,4)); draw((-1,4)--(-1,2)); draw((-3,2)--(-1,2)); draw((-3,3)--(-1,3)); draw((-2,4)--(-2,2)); draw((-2.8,4)--(-2,4), linewidth(2)); draw((-3,3.8)--(-3,3), linewidth(2)); draw((-1.8,4)--(-1,4), linewidth(2)); draw((-2,4)--(-2,3.2), linewidth(2)); draw((-3,3)--(-2.2,3), linewidth(2)); draw((-3,2.8)--(-3,2), linewidth(2)); draw((-3,2)--(-2.2,2), linewidth(2)); draw((-2,3)--(-2,2.2), linewidth(2)); draw((-1,2)--(-1.8,2), linewidth(2)); draw((-1,4)--(-1,3.2), linewidth(2)); draw((-2,3)--(-1.2,3), linewidth(2)); draw((-1,2.8)--(-1,2), linewidth(2)); dot((-3,2),dotstyle); dot((-1,4),dotstyle); dot((-1,2),dotstyle); dot((-3,3),dotstyle); dot((-2,4),dotstyle); dot((-2,3),dotstyle);[/asy]

Evaluate $\dbinom{6}{0}+\dbinom{6}{1}+\dbinom{6}{4}+\dbinom{6}{3}+\dbinom{6}{4}+\dbinom{6}{5}+\dbinom{6}{6}$ [i]Proposed by Jonathan Liu[/i] [hide=Solution] [i]Solution.[/i] $\boxed{64}$ We have that $\dbinom{6}{4}=\dbinom{6}{2}$, so $\displaystyle\sum_{n=0}^{6} \dbinom{6}{n}=2^6=\boxed{64}.$ [/hide]

A positive real number $k$, a triangle $ABC$ with circumcircle $\omega$, and a point $M$ on its side $AB$ are fixed. The point $P$ moves along $\omega$, and $Q$ on segment $CP$ is such that $CQ : QP = k$. The line through $P$, parallel to $CM$, intersects the line $MQ$ at point $N$. Prove that $N$ lies on a constant circle, independent of the choice of $P$.

Let $\overline{AB}$ be a line segment with length $10$. Let $P$ be a point on this segment with $AP = 2$. Let $\omega_1$ and $\omega_2$ be the circles with diameters $\overline{AP}$ and $\overline{P B}$, respectively. Let $XY$ be a line externally tangent to $\omega_1$ and $\omega_2$ at distinct points $X$ and $Y$ , respectively. Compute the area of $\vartriangle XP Y$ .

Let $v \in \mathbb{R}^2$ a vector of length 1 and $A$ a $2 \times 2$ matrix with real entries such that: (i) The vectors $A v, A^2 v$ y $A^3 v$ are also of length 1. (ii) The vector $A^2 v$ isn't equal to $\pm v$ nor to $\pm A v$. Prove that $A^t A=I_2$.