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 $13^4+16^5-172^2$.
In $\triangle{ABC}$, $\angle{BAC} = 40^\circ$ and $\angle{ABC} = 60^\circ$. Points $D$ and $E$ are on sides $AC$ and $AB$, respectively, such that $\angle{DBC}=40^\circ$ and $\angle{ECB}=70^\circ$. Let $F$ be the intersection point of $BD$ and $CE$. Show that $AF\perp BC$.
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5 .$ Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C.$ Given $EC=1,$ find the radius of the circle
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 $CE$.
During the annual frog jumping contest at the county fair, the height of the frog's jump, in feet, is given by $$f(x)= -
\frac{1}{3}x^2+\frac{4}{3}x\:$$ What was the maximum height reached by the frog?
Let $x, y,$ and $z$ be some real numbers such that: $x+2y-z=6$ and $x-y+2z=3$. Find the minimal value of $x^2 + y^2 + z^2$.
Let $x$ be a negative real number. Find the maximum value of $y=x+\frac{4}{x} +2007$.
Let real numbers $a$ and $b$ satisfy $a^2 + ab + b^2 = 1$. Find the range of $a^2 - ab + b^2$.
A farmer wants to fence in 60,000 square feet of land in a rectangular plot along a straight highway. The fence he plans to use along the highway costs \$2 per foot, while the fence for the other three sides costs \$1 per foot. How much of each type of fence will he have to buy in order to keep expenses to a minimum? What is the minimum expense?
Prove the Maximum Area of a Triangle with Fixed Perimeter is Equilateral
Use the Arithmetic Mean-Geometric Mean Inequality to find the maximum volume of a box made from a 25 by 25 square sheet of cardboard by removing a small square from each corner and folding up the sides to form a lidless box.
In $\triangle{ABC}$ show that $$\tan nA + \tan nB + \tan nC = \tan nA \tan nB \tan nC$$ where $n$ is an integer.
See the attachments
In $\triangle{ABC}$, if $A:B:C=4:2:1$, prove $$\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$$
In acute $\triangle{ABC}$, $\angle{ACB}=2\angle{ABC}$. Let $D$ be a point on $BC$ such that $\angle{ABC}=2\angle{BAD}$. Show that $$\frac{1}{BD}=\frac{1}{AB}+\frac{1}{AC}$$
As shown.
Let $O$ be a point inside a convex pentagon, as shown, such that $\angle{1} = \angle{2}, \angle{3} = \angle{4}, \angle{5} = \angle{6},$ and $\angle{7} = \angle{8}$. Show that either $\angle{9} = \angle{10}$ or $\angle{9} + \angle{10} = 180^\circ$ holds.
