Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m\ne0$, $n\ne0$, then the sum of the roots is: $\textbf{(A)}\ -\dfrac{1}{2} \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{Undetermined} $

The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.

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

A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

If $ \left|BC\right| \equal{} a$, $ \left|AC\right| \equal{} b$, $ \left|AB\right| \equal{} c$, $ 3\angle A \plus{} \angle B \equal{} 180{}^\circ$ and $ 3a \equal{} 2c$, then find $ b$ in terms of $ a$. $\textbf{(A)}\ \frac {3a}{2} \qquad\textbf{(B)}\ \frac {5a}{4} \qquad\textbf{(C)}\ a\sqrt {2} \qquad\textbf{(D)}\ a\sqrt {3} \qquad\textbf{(E)}\ \frac {2a\sqrt {3} }{3}$

Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.

A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.

Let $P(x)$ be a quadratic polynomial satisfying the following conditions: [list] [*] $P(x)$ has leading coefficient $1$. [*] $P(x)$ has nonnegative integer roots that are at most $2022$. [*] the set of the roots of $P(x)$ is a subset of the set of the roots of $P(P(x))$. [/list] Let $S$ be the set of all such possible $P(x)$, and let $Q(x)$ be the polynomial obtained upon summing all the elements of $S$. Find the sum of the roots of $Q(x)$.

The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; marker m=marker(scale(5)*d,Fill); path f1=(0,0); path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ (-2,2)--(-3,3); draw(f1,m); draw(shift(5,0)*f2,m); draw(shift(5,0)*g2); draw(shift(12,0)*f3,m); draw(shift(12,0)*g3); draw(shift(21,0)*f4,m); draw(shift(21,0)*g4); label("$F_1$",(0,-4)); label("$F_2$",(5,-4)); label("$F_3$",(12,-4)); label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$

If $ a > 1$, then the sum of the real solutions of \[\sqrt{a \minus{} \sqrt{a \plus{} x}} \equal{} x\] is equal to $ \textbf{(A)}\ \sqrt{a} \minus{} 1\qquad \textbf{(B)}\ \frac{\sqrt{a} \minus{} 1}{2}\qquad \textbf{(C)}\ \sqrt{a \minus{} 1}\qquad \textbf{(D)}\ \frac{\sqrt{a \minus{} 1}}{2}\qquad \textbf{(E)}\ \frac{\sqrt{4a \minus{} 3} \minus{} 1}{2}$

Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$? $ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $

Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots. Determine the maximum number of elements in $S(a, b, c)$.

How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

Find all the positive integers $x,y,z$ satisfiing : $x^{2}+y^{2}+z^{2}=2xyz$

Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. [i]Alex Zhu.[/i]

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]

Find all pairs of integers $ (x,y)$, such that \[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0 \]

Given such positive real numbers $a, b$ and $c$, that the system of equations: $ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $ has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.