Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]

Let $ q \equal{} 2p\plus{}1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.

Let $b \geq 2$ be a positive integer. (a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits. (b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits. (c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even. (d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.

If $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$, the larger root minus the smaller root is: $ \textbf{(A)}\ -2+3\sqrt{3}\qquad\textbf{(B)}\ 2-\sqrt{3}\qquad\textbf{(C)}\ 6+3\sqrt{3}\qquad\textbf{(D)}\ 6-3\sqrt{3}\qquad\textbf{(E)}\ 3\sqrt{3}+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 $p + q$.

Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

Solve the following system for real $x,y,z$ \[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]

Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots. a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly. b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$

Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

The polynomial $ f(x) \equal{} x^{4} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ has real coefficients, and $ f(2i) \equal{} f(2 \plus{} i) \equal{} 0.$ What is $ a \plus{} b \plus{} c \plus{} d?$ $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 16$

The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.

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]

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

Find all positive integers $n$ such that $4^{n}+2007$ is a perfect square.

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$

Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$. Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.

If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{none of these} $

Let $p\in \mathbb P,p>3$. Calcute: a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$ b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$

Show that the equation $x^2-7y^2 = 1$ has infinitely many solutions in natural numbers.

The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?

Laura and Daniel play with quadratic polynomials. First Laura says a nonzero real number $r$. Then Daniel says a nonzero real number $s$, and then again Laura says another nonzero real number $t$. Finally. Daniel writes the polynomial $P(x) = ax^2 + bx + c$ where $a,b$, and $c$ are $r,s$, and $t$ in some order Daniel chooses. Laura wins if the equation $P(x) = 0$ has two different real solutions, and Daniel wins otherwise. Determine who has a winning strategy and describe that strategy.

A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.

For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which are mutually distinct and form an arithmetic progression?

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.

$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$