Points $D$ and $E$ are taken on sides $BC$ and $CA$ of a triangle $ BD\equal{}AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of $\angle ACB$ intersects $AD$ and $BE$ at $Q$ and $R$ respectively. Prove that $ \frac{PQ}{PR}\equal{}\frac{AD}{BE}$.

Let $k$ be a positive integer and let $n$ be the smallest number with exactly $k$ divisors. Given $n$ is a cube, is it possible that $k$ is divisible by a prime factor of the form $3j+2$?

Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]

Determine all pairs $(a,b)$ of positive integers such that $(a+b)^3-2a^3-2b^3$ is a power of two.

For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ satisfying the two following properties: [b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and [b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.

A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is $\textbf{(A) }\frac{\sqrt{3}}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{2}{\sqrt{3}}\qquad\textbf{(D) }\frac{3}{2}\qquad\textbf{(E) }\sqrt{3}$

$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .

Let $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p$ be real numbers with $p >- 1$. Prove that $$\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0$$

An ellipse has foci $A$ and $B$ and has the property that there is some point $C$ on the ellipse such that the area of the circle passing through $A$, $B$, and, $C$ is equal to the area of the ellipse. Let $e$ be the largest possible eccentricity of the ellipse. One may write $e^2$ as $\frac{a+\sqrt{b}}{c}$ , where $a, b$, and $c$ are integers such that $a$ and $c$ are relatively prime, and b is not divisible by the square of any prime. Find $a^2 + b^2 + c^2$.

Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30$

In an $n \times n$ square array of $1\times1$ cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.

If $M=\{z\in\mathbb{C}|z=\frac{t}{1+t}+\text{i}\frac{1+t}{t},t\in\mathbb{R},t\neq0,t\neq-1\}$, $N=\{z\in\mathbb{C}|z=\sqrt2[\cos(\arcsin t)+\text{i}\cos(\arccos t)],t\in\mathbb{R},|t|\leq1\}$, then $|M\cap N|$ is $\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}4$

Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls and 463 balls, or one can choose 360 red balls and 540 blue balls.

Given any positive integer, we can write the integer in base $12$ and add together the digits of its base $12$ representation. We perform this operation on the number $ 7^{6^{5^{4^{3^{2^{1}}}}}}$ repeatedly until a single base $12$ digit remains. Find this digit.

Three vertices $KLM$ of the rhombus (diamond) $KLMN$ lays on the sides $[AB], [BC]$ and $[CD]$ of the given unit square. Find the area of the set of all the possible vertices $N$.

Let $ABC$ be a triangle, and let $BE, CF$ be the altitudes. Let $\ell$ be a line passing through $A$. Suppose $\ell$ intersect $BE$ at $P$, and $\ell$ intersect $CF$ at $Q$. Prove that: i) If $\ell$ is the $A$-median, then circles $(APF)$ and $(AQE)$ are tangent. ii) If $\ell$ is the inner $A$-angle bisector, suppose $(APF)$ intersect $(AQE)$ again at $R$, then $AR$ is perpendicular to $\ell$.

[u]Algebra Round[/u] [b]p1.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$. [b]p2.[/b] $f(x) = x^2 + bx + c$ is a function with the property that the $x$-coordinate of the vertex is equal to the positive difference of the two roots of $f(x)$. Given that $c = 48$, compute $b$. [b]p3.[/b] Suppose we have a function $f(x)$ such that $f(x)^2 = f(x - 5)f(x + 5)$ for all integers $x$. Given that $f(1) = 1$ and $f(16) = 8$, what is $f(2016)$? [b]p4.[/b] Suppose that we have the following set of equations $$\log_2 x + \log_3 x + \log_4 x = 20$$ $$\log_4 y + \log_9 y + \log_{16} y = 16$$ Compute $\log_x y$. [b]p5.[/b] Let $\{a_n\}$ be the arithmetic sequence defined as $a_n = 2(n - 1) + 6$ for all $n \ge 1$. Compute $$\sum^{\infty}_{i=1} \frac{1}{a_ia_{i+2}}.$$ [b]p6.[/b] Let $a, b, c, d, e, f$ be non-negative real numbers. Suppose that $a + b + c + d + e + f = 1$ and $ad + be + cf \ge \frac{1}{18} $. Find the maximum value of $ab + bc + cd + de + ef + fa$. [b]p7.[/b] Let f be a continuous real-valued function defined on the positive real numbers. Determine all $f$ such that for all positive real $x, y$ we have $f(xy) = xf(y) + yf(x)$ and $f(2016) = 1$. [b]p8.[/b] Find the maximum of the following expression: $$21 cos \theta + 18 sin \theta sin \phi + 14 sin \theta cos \phi $$ [b]p9.[/b] $a, b, c, d$ satisfy the following system of equations $$ab + c + d = 13$$ $$bc + d + a = 27$$ $$cd + a + b = 30$$ $$da + b + c = 17.$$ Compute the value of $a + b + c + d$. [b]p10.[/b] Define a sequence of numbers $a_{n+1} = \frac{(2+\sqrt3)a_n+1}{(2+\sqrt3)-a_{n}}$ for $n > 0$, and suppose that $a_1 = 2$. What is $a_{2016}$? [u]Algebra Tiebreakers[/u] [b]Tie 1.[/b] Mark takes a two digit number $x$ and forms another two digit number by reversing the digits of $x$. He then sums the two values, obtaining a value which is divisible by $13$. Compute the smallest possible value of $x$. [b]Tie 2.[/b] Let $p(x) = x^4 - 10x^3 + cx^2 - 10x + 1$, where $c$ is a real number. Given that $p(x)$ has at least one real root, what is the maximum value of $c$? [b]Tie 3.[/b] $x$ satisfies the equation $(1 + i)x^3 + 8ix^2 + (-8 + 8i)x + 36 = 0$. Compute the largest possible value of $|x|$. PS. You should use hide for answers.

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

Two functions of $x$ are differentiable and not identically zero. Find an example of two such functions having the property that the derivative of their quotient is the quotient of their derivatives.

If a person A is taller or heavier than another peoson B, then we note that A is [i]not worse than[/i] B. In 100 persons, if someone is [i]not worse than[/i] other 99 people, we call him [i]excellent boy[/i]. What's the maximum value of the number of [i]excellent boys[/i]? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}50\qquad\text{(D)}100$

The number of functions $g:\mathbb{R}^4\to\mathbb{R}$ such that, $\forall a,b,c,d,e,f\in\mathbb{R}$ : (i) $g(1,0,0,1)=1$ (ii) $g(ea,b,ec,d)=eg(a,b,c,d)$ (iii) $g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)$ (iv) $g(a,b,c,d)+g(b,a,d,c)=0$ is : (A)$1$ (B)$0$ (C)$\text{infinitely many}$ (D)$\text{None of these}$ [Hide=Hint(given in question)] Think of matrices[/hide]

Let $f(x)=\frac{9^x}{9^x + 3}$. Evaluate $\sum_{i=1}^{1995}{f\left(\frac{i}{1996}\right)}$.

Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.

For $n = 1,2,3,...$. $a_n$ is defined by: $$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$ Prove that for every $n$ holds that $$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$

Consider $D$ the midpoint of the base $[BC]$ of the isosceles triangle ABC in which $\angle BAC < 90^o$. On the perpendicular from $B$ on the line $BC$ consider the point $E$ such that $\angle EAB= \angle BAC$, and on the line passing though $C$ parallel to the line $AB$ we consider the point $F$ such that $F$ and $D$ are on different side of the line $AC$ and $\angle FAC = \angle CAD$. Prove that $AE = CF$ and $BF = EF$