ARML

back to index

3110   
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 $a+b+c$.

3111   
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 $\tan{A}$.

3112   
Compute $$\frac{\sqrt[4]{1}\cdot\sqrt[4]{3}\cdot\sqrt[4]{5}\cdots\sqrt[4]{2015}}{\sqrt[4]{2}\cdot\sqrt[4]{4}\cdot\sqrt[4]{6}\cdots\sqrt[4]{2016}}$$

3113   
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 $i, 1\le i < 6$.

3114   
Compute the least possible non-zero value of $A^2+B^2+C^2$ such that $A, B,$ and $C$ are integers satisfying $A\log16+B\log18+C\log24=0$.

3115   
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 $\triangle{LEO}$.

3116   
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.

3117   
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 $(A, B, C)$.

3118   
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.

3119   
Find the largest of three prime divisors of $13^4+16^5-172^2$.

3122   
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 $CE$.

back to index