What is the largest possible height of a right cylinder with radius $3$ that can fit in a cube with side length $12$?

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

Solve the system $\begin{cases} x^2y^2+|xy|=\frac{4}{9}\\ xy+1=x+y^2\\ \end{cases}$

Let $x$, $y$, and $z$ be positive real numbers satisfying $x^2+y^2+z^2=1$. Prove that \[x^2yz+xy^2z+xyz^2\le\frac{1}{3}.\]

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

On a faded piece of paper it is possible to read the following: \[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\] Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term $a$? We assume that all polynomials in the statement have only integer coefficients.

Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? ${{ \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72}\qquad\textbf{(E)}\ 80} $

Let $O$ be the center of an equilateral triangle $ABC$ of area $1/\pi$. As shown in the diagram below, a circle centered at $O$ meets the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect each of the triangle's sides. Compute the total area of all six shaded regions. [asy] unitsize(90); pair A = dir(0); pair B = dir(120); pair C = dir(240); draw(A -- B -- C -- cycle); pair D = (2*A + B)/3; pair E = (A + 2*B)/3; pair F = (2*B + C)/3; pair G = (B + 2*C)/3; pair H = (2*C + A)/3; pair I = (C + 2*A)/3; draw(E -- F); draw(G -- H); draw(I -- D); draw(D -- G); draw(E -- H); draw(F -- I); pair O = (0, 0); real r = 1/sqrt(3); draw(circle(O, r)); fill(O -- D -- E -- cycle, gray); fill(O -- F -- G -- cycle, gray); fill(O -- H -- I -- cycle, gray); fill(arc(O, r, -30, 30) -- cycle, gray); fill(arc(0, r, 90, 150) -- cycle, gray); fill(arc(0, r, 210, 270) -- cycle, gray); label("$A$", A, A); label("$B$", B, B); label("$C$", C, C); label("$D$", D, unit(D)); label("$E$", E, unit(E)); label("$F$", F, unit(F)); label("$G$", G, unit(G)); label("$H$", H, unit(H)); label("$I$", I, unit(I)); label("$O$", O, C); [/asy]

Find the greatest prime divisor of $29! + 33!$.

Find the ratio of two numbers if the ratio of their arithmetic mean to their geometric mean is $25 : 24$

Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c, 2^c, \cdots , n^c$ are integers. Prove that $c$ is an integer.

Let $(a, b, c, d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}$. For how many such quadruples is it true that $a\cdot d-b\cdot c$ is odd$?$ (For example, $(0, 3, 1, 1)$ is one such quadruple, because $0\cdot 1-3\cdot 1=-3$ is odd.) $\textbf{(A) } 48 \qquad \textbf{(B) } 64 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 128 \qquad \textbf{(E) } 192$

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

$32$ competitors participate in a tournament. No two of them are equal and in a one against one match the better always wins. Show that the gold, silver, and bronze medal winners can be found in $39$ matches.

Let H be a complex separable Hilbert space and denote $B(H)$ the algebra of bounded linear operators on H. Find all *-subalgebras C of $B(H)$ for which for all $A \in B(H)$ and $T \in C$ there exists $S \in C$ that $$TA-AT^{\ast} = TS-ST^{\ast}$$ note: *-algebra is also known as involutive algebra.

A [i]regular spatial pentagon[/i] consists of five points $P_1,P_2,P_3,P_4$ and $P_5$ in $\mathbb{R}^3$ such that $|P_iP_{i+1}|=|P_jP_{j+1}|$ and $\angle P_{i-1}P_iP_{i+1}=\angle P_{j-1}P_jP_{j+1}$ for all $1\leq i,\leq 5$, where $P_0=P_5$ and $P_{6}=P_{1}$. A regular spatial pentagon is [i]planar[/i] if there is a plane passing through all five points $P_1,P_2,P_3,P_4$ and $P_5$. Show that every regular spatial pentagon is planar.

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

All sides of the convex pentagon $ ABCDE$ are of equal length, and $ \angle A \equal{} \angle B \equal{} 90^{\circ}$. What is the degree measure of $ \angle E$? $ \textbf{(A)}\ 90 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$

Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$

Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true: i. Either each of the three cards has a different shape or all three of the card have the same shape. ii. Either each of the three cards has a different color or all three of the cards have the same color. iii. Either each of the three cards has a different shade or all three of the cards have the same shade. How many different complementary three-card sets are there?

Let $R > 0$, be an integer, and let $n(R)$ be the number um triples $(x, y, z) \in \mathbb{Z}^3$ such that $2x^2+3y^2+5z^2 = R$. What is the value of $\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?

Given a list $A$, let $f(A) = [A[0] + A[1], A[0] - A[1]]$. Alef makes two programs to compute $f(f(...(f(A))))$, where the function is composed $n$ times: \begin{tabular}{l|l} 1: \textbf{FUNCTION} $T_1(A, n)$ & 1: \textbf{FUNCTION} $T_2(A, n)$ \\ 2: $\quad$ \textbf{IF} $n = 0$ & 2: $\quad$ \textbf{IF} $n = 0$ \\ 3: $\quad$ $\quad$ \textbf{RETURN} $A$ & 3: $\quad$ $\quad$ \textbf{RETURN} $A$ \\ 4: $\quad$ \textbf{ELSE} & 4: $\quad$ \textbf{ELSE} \\ 5: $\quad$ $\quad$ \textbf{RETURN} $[T_1(A, n - 1)[0] + T_1(A, n - 1)[1],$ & 5: $\quad$ $\quad$ $B \leftarrow T_2(A, n - 1)$ \\ $\quad$ $\quad$ $\quad$ $T_1(A, n - 1)[0] - T_1(A, n - 1)[1]]$ & 6: $\quad$ $\quad$ \textbf{RETURN} $[B[0] + B[1], B[0] - B[1]]$ \\ \end{tabular} Each time $T_1$ or $T_2$ is called, Alef has to pay one dollar. How much money does he save by calling $T_2([13, 37], 4)$ instead of $T_1([13, 37], 4)$?

In the triangle $ABC$ with $| AB | = c, | BC | = a, | CA | = b$ the relations hold simultaneously $$a \ge max \{ b, c, \sqrt{bc}\}, \sqrt{(a - b) (a + c)} + \sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc}$$ Prove that the triangle $ABC$ is isosceles.

On a magical island there are lions, wolves and goats. Wolves can eat goats while lions can eat both wolves and goats. But if a lion eats a wolf, the lion becomes a goat. Likewise if a wolf eats a goat, the wolf becomes a lion. And if a lion eats a goat, the lion becomes a wolf. Initially on the island there are $17$ goats, $55$ wolves and $6$ lions. If they start eating until they no longer possible to eat more, what is the maximum number of animals that they can stay alive?

Let $a,b$ and $c$ be real numbers such that $| (a-b) (b-c) (c-a) | = 1$. Find the smallest value of the expression $| a | + | b | + | c |$. (K.Satylhanov )