The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face? $\text{\textbf{(A) }red}\qquad \text{\textbf{(B) } white}\qquad\text{\textbf{(C) }green}\qquad\text{\textbf{(D) }brown }\qquad\text{\textbf{(E)} purple}$ [asy] unitsize(2 cm); pair x, y, z, trans; int i; x = dir(-5); y = (0.6,0.5); z = (0,1); trans = (2,0); for (i = 0; i <= 2; ++i) { draw(shift(i*trans)*((0,0)--x--(x + y)--(x + y + z)--(y + z)--z--cycle)); draw(shift(i*trans)*((x + z)--x)); draw(shift(i*trans)*((x + z)--(x + y + z))); draw(shift(i*trans)*((x + z)--z)); } label(rotate(-3)*"$R$", (x + z)/2); label(rotate(-5)*slant(0.5)*"$B$", ((x + z) + (y + z))/2); label(rotate(35)*slant(0.5)*"$G$", ((x + z) + (x + y))/2); label(rotate(-3)*"$W$", (x + z)/2 + trans); label(rotate(50)*slant(-1)*"$B$", ((x + z) + (y + z))/2 + trans); label(rotate(35)*slant(0.5)*"$R$", ((x + z) + (x + y))/2 + trans); label(rotate(-3)*"$P$", (x + z)/2 + 2*trans); label(rotate(-5)*slant(0.5)*"$R$", ((x + z) + (y + z))/2 + 2*trans); label(rotate(-85)*slant(-1)*"$G$", ((x + z) + (x + y))/2 + 2*trans); [/asy]

Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$. Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.

A positive integer $n$ is called $special$ if there exist integers $a > 1$ and $b > 1$ such that $n=a^b + b$. Is there a set of $2014$ consecutive positive integers that contains exactly $2012$ $special$ numbers?

You have an infinite stack of T-shaped tetrominoes (composed of four squares of side length 1), and an n × n board. You are allowed to place some tetrominoes on the board, possibly rotated, as long as no two tetrominoes overlap and no tetrominoes extend off the board. For which values of n can you cover the entire board?

Determine all triples $(k, m, n) $ of positive integers satisfying $$k!+m!=k!n!.$$

Consider the roots of the polynomial $x^{2013}-2^{2013}=0$. Some of these roots also satisfy $x^k-2^k=0$, for some integer $k<2013$. What is the product of this subset of roots?

Let $n=(q + 2)q^{2021}$ where $q=10^9+7$. For every $k<=n$ and prime $p|n$, define $f_{p,k}(n)$ =$ v_{p}$$ (\binom{n}{k}) $ ($v_{p}$$(i)$ is the highest power of $p$ that divides $i$). Let $m$ be the maximum possible (over all $k$) value of the expression $\prod_{p\text{,prime,} p|n} f_{p,k}$. Find the sum of the digits of $m$.

Juli has a deck of $54$ cards and proposes the following game to Bruno. Juli places the cards in a row, some face-up and others face-down. Bruno can repeatedly perform the following move: select a card and flip it along with its two neighbors (turning face-up cards face-down, and vice versa for face-down cards). Bruno wins if, through this process, he manages to turn all the cards face up. Otherwise, Juli wins. Determine which player has a winning strategy and explain it. [b]Note:[/b] When Bruno selects the first or the last card in the row, he flips only two cards. In all other cases, he flips three cards.

Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$

There are $200$ boxes on the table. In the beginning, each of the boxes contains a positive integer (the integers are not necessarily distinct). Every minute, Alice makes one move. A move consists of the following. First, she picks a box $X$ which contains a number $c$ such that $c = a + b$ for some numbers $a$ and $b$ which are contained in some other boxes. Then she picks a positive integer $k > 1$. Finally, she removes $c$ from $X$ and replaces it with $kc$. If she cannot make any mobes, she stops. Prove that no matter how Alice makes her moves, she won't be able to make infinitely many moves.

Determine the last two digits of the number $2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}$ , written in decimal notation.

The numbers $3$, $4$ ,$...$, $2019$ are written on a blackboard. David and Edgardo play alternately, starting with David. On their turn, each player must erase a number from the board and write two positive integers whose sum is equal to the number just deleted. The winner is the one who makes all the numbers on the board equal. Determine who has a winning strategy and describe it.

In $\triangle ABC$, $AB = 360$, $BC = 507$, and $CA = 780$. Let $M$ be the midpoint of $\overline{CA}$, and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC$. Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}$. Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E$. The ratio $DE: EF$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

If $x \neq 0$, $\frac x{2} = y^2$ and $\frac{x}{4} = 4y$, then $x$ equals $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 128 $

Are there integers $x,y$ and $z$ which satisfy the equation $$z^2=(x+1)(y^2-1)+n$$ when (a) $n=2006$ (b) $n=2007$?

A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch? Note: $1$ mile= $5280$ feet [asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,1)--(5.5,1)); D((-1.5,0)--(5.5,0),dashed); D((-1.5,-1)--(5.5,-1)); [/asy] $\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$

Let $x_1, x_2, ..., x_n$ be positive numbers, with $n \ge 2$. Prove that $$\left(x_1+\frac{1}{x_1}\right)\left(x_2+\frac{1}{x_2}\right)...\left(x_n+\frac{1}{x_n}\right)\ge \left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)...\left(x_{n-1}+\frac{1}{x_n}\right)\left(x_n+\frac{1}{x_1}\right)$$

Percy buys $3$ apples for $6$ dollars, $4$ pears for $16$ dollars, and $1$ watermelon for $5$ dollars. Assuming the rates stay the same, how much would it cost to buy $10$ apples, $3$ pears, and $2$ watermelons? $\textbf{(A) } 38\qquad\textbf{(B) } 39\qquad\textbf{(C) } 40\qquad\textbf{(D) } 41\qquad\textbf{(E) } 42$

In an acute-angled triangle $ABC$ with orthocenter $H$, the line $AH$ cuts $BC$ at point $A_1$. Let $\Gamma$ be a circle centered on side $AB$ tangent to $AA_1$ at point $H$. Prove that $\Gamma$ is tangent to the circumscribed circle of triangle $AMA_1$, where $M$ is the midpoint of $AC$.

Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.

Let $ n$ be an integer, $ n\geq 2$, and the integers $ a_1,a_2,\ldots,a_n$, such that $ 0 < a_k\leq k$, for all $ k \equal{} 1,2,\ldots,n$. Knowing that the number $ a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n$ is even, prove that there exists a choosing of the signs $ \plus{}$, respectively $ \minus{}$, such that \[ a_1 \pm a_2 \pm \cdots \pm a_n\equal{} 0. \]

Toward the end of a game of Fish, the $2$ through $7$ of spades, inclusive, remain in the hands of three distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known that DBR either has more than one card or has an even-numbered spade, or both, in how many ways can the players’ hands be distributed?

Determine the least real number $L$ such that $$\dfrac{1}{a}+\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}\leqslant L$$ for all quadruples $(a,b,c,d)$ of integers satisfying $1<a<b<c<d$.

Let $ a\ge 2 $ be a natural number. Show that the following relations are equivalent: $ \text{(i)} \ a $ is the hypothenuse of a right triangle whose sides are natural numbers. $ \text{(ii)}\quad $ there exists a natural number $ d $ for which the polynoms $ X^2-aX\pm d $ have integer roots.

$p(x)$ is a polynomial of degree $n$ with every coefficient $0 $ or $\pm1$, and $p(x)$ is divisible by $ (x - 1)^k$ for some integer $ k > 0$. $q$ is a prime such that $\frac{q}{\ln q}< \frac{k}{\ln n+1}$. Show that the complex $q$-th roots of unity must be roots of $ p(x). $