Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by \[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , \] where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$). a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$. [i]Sugested by Calin Popescu.[/i]

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.

Express $\sqrt{2 + \sqrt{3}}$ in the form $\frac{a + \sqrt{b}}{\sqrt{c}}$, where $a$ is a positive integer and $b$ and $c$ are square-free positive integers.

Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$? $ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$ $\textbf{(B)}\ \text{The discriminant is 9}\qquad$ $\textbf{(C)}\ \text{The roots are imaginary}\qquad$ $\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$ $\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $

Find the sum of the absolute values of the roots of $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$.

Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url] [i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]

In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis

$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ? $\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$

Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system: \[a_1^2 + a_1 - 1 = a_2\] \[ a_2^2 + a_2 - 1 = a_3\] \[\hspace*{3.3em} \vdots \] \[a_{n}^2 + a_n - 1 = a_1\]

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

For what values of $ m$ does the equation $ x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0$ has two real roots, both of them greater than $ \minus{}1$.

What is the smallest integral value of $k$ such that \[ 2x(kx-4)-x^2+6=0 \] has no real roots? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

The roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ will be reciprocal if: $ \textbf{(A)}\ a \equal{} b \qquad\textbf{(B)}\ a \equal{} bc \qquad\textbf{(C)}\ c \equal{} a \qquad\textbf{(D)}\ c \equal{} b \qquad\textbf{(E)}\ c \equal{} ab$

Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$ Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

Let $b$ be a real number randomly sepected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.

Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$ $ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

Let $ x,y$ be positive integers with $ y>3$ and $ x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2).$ Prove that: $ x^2\plus{}y^4\equal{}1994.$

Let $ p(x) $ be a quadratic polynomial for which : \[ |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} \] Prove that: \[ \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]\]

Triangle $ ABC$ has a right angle at $ B$, $ AB \equal{} 1$, and $ BC \equal{} 2$. The bisector of $ \angle BAC$ meets $ \overline{BC}$ at $ D$. What is $ BD$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,1), B=(0,0), C=(2,0); pair D=extension(A,bisectorpoint(B,A,C),B,C); pair[] ds={A,B,C,D}; dot(ds); draw(A--B--C--A--D); label("$1$",midpoint(A--B),W); label("$B$",B,SW); label("$D$",D,S); label("$C$",C,SE); label("$A$",A,NW); draw(rightanglemark(C,B,A,2));[/asy]$ \textbf{(A)}\ \frac {\sqrt3 \minus{} 1}{2} \qquad \textbf{(B)}\ \frac {\sqrt5 \minus{} 1}{2} \qquad \textbf{(C)}\ \frac {\sqrt5 \plus{} 1}{2} \qquad \textbf{(D)}\ \frac {\sqrt6 \plus{} \sqrt2}{2}$ $ \textbf{(E)}\ 2\sqrt3 \minus{} 1$