For each integer $m$, consider the polynomial \[ P_m(x)=x^4-(2m+4)x^2+(m-2)^2. \] For what values of $m$ is $P_m(x)$ the product of two non-consant polynomials with integer coefficients?

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$? \[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3 \]

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?

The discriminant of the equation $ x^2\plus{}2x\sqrt{3}\plus{}3\equal{}0$ is zero. Hence, its roots are: $ \textbf{(A)}\ \text{real and equal} \qquad \textbf{(B)}\ \text{rational and equal} \qquad \textbf{(C)}\ \text{rational and unequal} \\ \textbf{(D)}\ \text{irrational and unequal} \qquad \textbf{(E)}\ \text{imaginary}$

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

Let $f(x) = x^2 - 2x$. A set of real numbers $S$ is [i]valid[/i] if it satisfies the following: $\bullet$ If $x \in S$, then $f(x) \in S$. $\bullet$ If $x \in S$ and $\underbrace{f(f(\dots f}_{k\ f\text{'s}}(x)\dots )) = x$ for some integer $k$, then $f(x) = x$. Compute the number of 7-element valid sets. [i]Proposed by Lewis Chen[/i]

Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993? [asy] int x=101, y=3*floor(x/4); draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x)); int i; for(i=y-2; i<y+4; i=i+1) { dot(dir(360*i/x)); } label("3", dir(360*(y-2)/x), dir(360*(y-2)/x)); label("2", dir(360*(y+1)/x), dir(360*(y+1)/x)); label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]

If $x$ is a real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is: $\text{(A)} \ x\le -2~\text{or}~x\ge3 \qquad \text{(B)} \ x\le 2~\text{or}~x\ge3 \qquad \text{(C)} \ x\le -3 ~\text{or}~x\ge 2$ $\text{(D)} \ -3\le x \le 2\qquad \text{(E)} \ \-2\le x \le 3$

A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $ R$ is the ratio of the lesser part to the greater part, then the value of \[ R^{[R^{(R^2\plus{}R^{\minus{}1})}\plus{}R^{\minus{}1}]}\plus{}R^{\minus{}1}\] is $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2R \qquad \textbf{(C)}\ R^{\minus{}1} \qquad \textbf{(D)}\ 2\plus{}R^{\minus{}1} \qquad \textbf{(E)}\ 2\plus{}R$

For how many ordered pairs $(x,y)$ of integers is it true that $0<x<y<10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y?$

If in applying the quadratic formula to a quadratic equation \[ f(x)\equiv ax^2 \plus{} bx \plus{} c \equal{} 0, \] it happens that $ c \equal{} \frac {b^2}{4a}$, then the graph of $ y \equal{} f(x)$ will certainly: $ \textbf{(A)}\ \text{have a maximum} \qquad\textbf{(B)}\ \text{have a minimum} \qquad\textbf{(C)}\ \text{be tangent to the x \minus{} axis} \\ \qquad\textbf{(D)}\ \text{be tangent to the y \minus{} axis} \qquad\textbf{(E)}\ \text{lie in one quadrant only}$

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

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$

Prove that for infinitely many pairs $(a,b)$ of integers the equation \[x^{2012} = ax + b\] has among its solutions two distinct real numbers whose product is 1.

For how many integers $n$ between 1 and 100 does $x^2+x-n$ factor into the product of two linear factors with integer coefficients? $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 9 \qquad \text{(E)} \ 10$

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".)

Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$. Find the $ x$-coordinate of the point $ X$ on the $ x$-axis maximizing $ \angle AXB$.

In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$. [i]Proposed by Eugene Chen [/i]

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r,$ where $m, n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0.$ Find $m+n+r.$

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

The complex numbers $z$ and $w$ satisfy the system \begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*} Find the smallest possible value of $|zw|^2$.

Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized? [i]Proposed by Lewis Chen[/i]

Find positive integer $n$ so that $\tfrac{80-6\sqrt{n}}{n}$ is the reciprocal of $\tfrac{80+6\sqrt{n}}{n}.$