#### ARML

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Given that $a, b,$ and $c$ are positive integers such that $a^b\cdot b^c$ is a multiple of 2016. Compute the least possible value of $a+b+c$.

Triangle $ABC$ is isosceles. An ant begins at $A$, walks exactly halfway along the perimeter of $\triangle{ABC}$, and then returns directly $A$, cutting through the interior of the triangle. The ant's path surround exactly 90% of the area of $\triangle{ABC}$. Compute the maximum value of $\tan{A}$.

Compute $$\frac{\sqrt[4]{1}\cdot\sqrt[4]{3}\cdot\sqrt[4]{5}\cdots\sqrt[4]{2015}}{\sqrt[4]{2}\cdot\sqrt[4]{4}\cdot\sqrt[4]{6}\cdots\sqrt[4]{2016}}$$

Compute the number of permutations $x_1, \cdots, x_6$ of integers $1, \cdots, 6$ such that $x_{i+1}\le 2x_i$ for all $i, 1\le i < 6$.

Compute the least possible non-zero value of $A^2+B^2+C^2$ such that $A, B,$ and $C$ are integers satisfying $A\log16+B\log18+C\log24=0$.

In $\triangle{LEO}$, point $J$ lies on $\overline{LO}$ such that $\overline{JE}\perp\overline{EO}$, and point $S$ lies on $\overline{LE}$ such that $\overline{JS}\perp\overline{LE}$. Given that $JS=9, EO=20,$ and $JO+SE=37$, compute the perimeter of $\triangle{LEO}$.

Compute the least possible area of a non-degenerate right triangle with sides of lengths $\sin{x}$, $\cos{x}$ and $\tan{x}$ where $x$ is a real number.

Let $P(x)$ be the polynomial $x^3 + Ax^2 +Bx+C$ for some constants $A, B,$ and $C$. There exists constant $D$ and $E$ such that for all $x$, $P(x+1)=x^3 + Dx^2 + 54x +37$ and $P(x+2)=x^3 + 26x + Ex+115$. Compute the ordered triple $(A, B, C)$.

An $n$-sided die has the integers between $1$ and $n$ (inclusive) on its faces. All values on the faces of this die are equally likely to be rolled. An $8$-sides side, a $12$-sided die, and a $20$-sided die are rolled. Compute the probability that one of the values rolled equal to the sum of the other two values rolled.

Find the largest of three prime divisors of $13^4+16^5-172^2$.

Chords $AB$ and $CD$ of a given circle are perpendicular to each other and intersect at a right angle at point $E$. If $BE=16$, $DE=4$, and $AD=5$, find $CE$.