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 )

Let $ABC$ be a triangle with $AB=7, AC=9, BC=10$, circumcenter $O$, circumradius $R$, and circumcircle $\omega$. Let the tangents to $\omega$ at $B,C$ meet at $X$. A variable line $\ell$ passes through $O$. Let $A_1$ be the projection of $X$ onto $\ell$ and $A_2$ be the reflection of $A_1$ over $O$. Suppose that there exist two points $Y,Z$ on $\ell$ such that $\angle YAB+\angle YBC+\angle YCA=\angle ZAB+\angle ZBC+\angle ZCA=90^{\circ}$, where all angles are directed, and furthermore that $O$ lies inside segment $YZ$ with $OY*OZ=R^2$. Then there are several possible values for the sine of the angle at which the angle bisector of $\angle AA_2O$ meets $BC$. If the product of these values can be expressed in the form $\frac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $b$ squarefree and $a,c$ coprime, determine $a+b+c$. [i]Proposed by Vincent Huang

Let $f(x)=\frac{ax+b}{cx+d}$ $F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$) Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$, show that $F_n(x)=x$ (for any valid x).

Let $T=\text{TNFTPP}$. Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y|\leq T-500$ and $|y|\leq T-500$. Find the area of region $R$.

Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.

Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\vartriangle ABC$ out of paper and then folding $\vartriangle AFD$ up along $FD$, folding $\vartriangle BED$ up along $DE$, and folding $\vartriangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$ , where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$.

Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$