Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)

$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x \plus{} y^2)(x^2 \minus{} y)/(xy)$ ? $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17$

Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$.

Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn? $ \textbf{(A)}\ 0.75 \qquad \textbf{(B)}\ 0.8 \qquad \textbf{(C)}\ 1.35 \qquad \textbf{(D)}\ 1.5 \qquad \textbf{(E)}\ 3$

Consider a table with three rows and eleven columns. There are zeroes prefilled in the cell of the first row and the first column and in the cell of the second row and the last column. Determine the least real number $\alpha$ such that the table can be filled with non-negative numbers and the following conditions hold simultaneously: (1) the sum of numbers in every column is one, (2) the sum of every two neighboring numbers in the first row is at most one, (3) the sum of every two neighboring numbers in the second row is at most one, (4) the sum of every two neighboring numbers in the third row is at most $\alpha$.

A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, then $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, $\angle ESF=35^{\circ}$. Let $T$ be an intersection of $BE$ and $CF$. Find $\angle BTC$.

Let $P(x)$ be a polynomial with integer coefficients that has at least one rational root. Let $n$ be a positive integer. Alan and Allan are playing a game. First, Alan writes down $n$ integers at $n$ different locations on a board. Then Allan may make moves of the following kind: choose a position that has integer $a$ written, then choose a different position that has integer $b$ written, then at the first position erase $a$ and in its place write $a+P(b)$. After any nonnegative number of moves, Allan may choose to end the game. Once Allan ends the game, his score is the number of times the mode (most common element) of the integers on the board appears. Find, in terms of $P(x)$ and $n$, the maximum score Allan can guarantee. [i]Henrick Rabinovitz[/i]

A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.) Given this information, find all possible values for the number of elements of $S$.

What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$

The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ . .

Ken has a six sided die. He rolls the die, and if the result is not even, he rolls the die one more time. Find the probability that he ends up with an even number. [i]Proposed by Gabriel Wu[/i]

$ABCD$ is a tetrahedron such that $BA\perp AC$, $DB\perp (ABC)$ and $AC\neq BD$. Denote by $O$ the midpoint of $AB$ and $K$ the foot of the perpendicular from $O$ to $DC$. Prove that $$\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$$ if and only if $2AC\cdot BD=AB^2$. [i]Vietnam Olympiad[/i]

Rectangle $ ABCD$, pictured below, shares $50\%$ of its area with square $ EFGH$. Square $ EFGH$ shares $20\%$ of its area with rectangle $ ABCD$. What is $ \frac{AB}{AD}$? [asy]unitsize(5mm); defaultpen(linewidth(0.8pt)+fontsize(10pt)); pair A=(0,3), B=(8,3), C=(8,2), D=(0,2), Ep=(0,4), F=(4,4), G=(4,0), H=(0,0); fill(shift(0,2)*xscale(4)*unitsquare,grey); draw(Ep--F--G--H--cycle); draw(A--B--C--D); label("$A$",A,W); label("$B$",B,E); label("$C$",C,E); label("$D$",D,W); label("$E$",Ep,NW); label("$F$",F,NE); label("$G$",G,SE); label("$H$",H,SW);[/asy]$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 10$

Let $A, B \in \mathbb{N}$. Consider a sequence $x_1, x_2, \ldots$ such that for all $n\geq 2$, $$x_{n+1}=A \cdot \gcd(x_n, x_{n-1})+B. $$ Show that the sequence attains only finitely many distinct values.

Let $a>0$, $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that $$ \lim_{n \to \infty} S_n = a-1.$$