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 $BF = EF$

Let $x,y,z$ be non-zero real numbers. Suppose $\alpha, \beta, \gamma$ are complex numbers such that $|\alpha|=|\beta|=|\gamma|=1$. If $x+y+z=0=\alpha x+\beta y+\gamma z$, then prove that $\alpha =\beta =\gamma$.

Let $ABC$ be a triangle such that $AB > AC$, with a circumcircle $\omega$. Draw the tangents to $\omega$ at $B$ and $C$ and these intersect at $P$. The perpendicular to $AP$ through $A$ cuts $BC$ at $R$. Let $S$ be a point on the segment $PR$ such that $PS = PC$. (a) Prove that the lines $CS$ and $AR$ intersect on $\omega$. (b) Let $M$ be the midpoint of $BC$ and $Q$ be the point of intersection of $CS$ and $AR$. Circle $\omega$ and the circumcircle of $\vartriangle AMP$ intersect at a point $J$ ($J \ne A$), prove that $P$, $J$ and $Q$ are collinear.

Prove the equality:\\ $\tan (\frac{3\pi}{7})-4\sin (\frac{\pi}{7})= \sqrt{7}$ .

Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90 degrees left turn after every $ \ell$ kilometer driving from start, Rob makes a 90 degrees right turn after every $ r$ kilometer driving from start, where $ \ell$ and $ r$ are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair ($ \ell$, $ r$) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?

Jim starts with a positive integer $ n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $ n=55$, then his sequence contains $ 5$ numbers: \begin{align*} &55\\ 55-7^2=&\ 6\\ 6-2^2=&\ 2\\ 2-1^2=&\ 1\\ 1-1^2=&\ 0 \end{align*}Let $ N$ be the smallest number for which Jim's sequence has 8 numbers. What is the units digit of $ N$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$

Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1}+f_{n-2}$. For each $i$, $1 \le i \le 200$, we calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. What is the sum of the distinct values of $g_i$?

Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$