Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$: \begin{align*} x^2+py+q&=0,\\ y^2+px+q&=0. \end{align*} Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.

Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.

A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$? $ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$

Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $ 400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? $ \textbf{(A)}\ 26 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 34$

Find all triples $(a, b, c)$ of real numbers such that $$ a^2 + ab + c = 0, $$ $$b^2 + bc + a = 0, $$ $$c^2 + ca + b = 0.$$

If the sum of all the angles except one of a convex polygon is $ 2190^{\circ}$, then the number of sides of the polygon must be $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 21$

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

The difference of the roots of $ x^2 \minus{} 7x \minus{} 9 \equal{} 0$ is: $ \textbf{(A)}\ \plus{} 7 \qquad\textbf{(B)}\ \plus{} \frac {7}{2} \qquad\textbf{(C)}\ \plus{} 9 \qquad\textbf{(D)}\ 2\sqrt {85} \qquad\textbf{(E)}\ \sqrt {85}$

The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.

Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that (i) The numbers $a,b,c$ are all positive. (ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.

A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.

Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.

For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.

Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.

Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

For what values of $a$ does the system of equations \[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?

If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals $\textbf{(A) }-1\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }-1\text{ or }2\qquad \textbf{(E) }-1\text{ or }-2$

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.

Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.