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$

Solve the equation [b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $ [b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $ [i]Cristinel Mortici[/i]

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$

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

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?

Let $ f_m: \mathbb R \to \mathbb R$, $ f_m(x)\equal{}(m^2\plus{}m\plus{}1)x^2\minus{}2(m^2\plus{}1)x\plus{}m^2\minus{}m\plus{}1,$ where $ m \in \mathbb R$. 1) Find the fixed common point of all this parabolas. 2) Find $ m$ such that the distance from that fixed point to $ Oy$ is minimal.

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $ f(x) \equal{} ax^{2} \plus{} bx \plus{} c$ are equal. Their common value must also be which of the following? $ \textbf{(A)}\ \text{the coefficient of }x^{2}\qquad \textbf{(B)}\ \text{the coefficient of }x$ $ \textbf{(C)}\ \text{the y \minus{} intercept of the graph of }y \equal{} f(x)$ $ \textbf{(D)}\ \text{one of the x \minus{} intercepts of the graph of }y \equal{} f(x)$ $ \textbf{(E)}\ \text{the mean of the x \minus{} intercepts of the graph of }y \equal{} f(x)$

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]

To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be: $ \textbf{(A)}\ \text{both rational} \qquad\textbf{(B)}\ \text{both real but not rational} \qquad\textbf{(C)}\ \text{both not real}\qquad$ $\textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not real} $

Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.

The sum of the roots of the equation $ 4x^2\plus{}5\minus{}8x\equal{}0$ is equal to: $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ -5 \qquad \textbf{(C)}\ -\dfrac{5}{4} \qquad \textbf{(D)}\ -2 \qquad \textbf{(E)}\ \text{None of these}$

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]

Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$.

Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]

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 $m+n+r.$

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

Two men set out at the same time to walk towards each other from $ M$ and $ N$, $ 72$ miles apart. The first man walks at the rate of $ 4$ mph. The second man walks $ 2$ miles the first hour, $ 2\frac {1}{2}$ miles the second hour, $ 3$ miles the third hour, and so on in arithmetic progression. Then the men will meet: $ \textbf{(A)}\ \text{in 7 hours} \qquad \textbf{(B)}\ \text{in }{8\frac {1}{4}}\text{ hours}\qquad \textbf{(C)}\ \text{nearer }{M}\text{ than }{N}\qquad \\ \textbf{(D)}\ \text{nearer }{N}\text{ than }{M}\qquad \textbf{(E)}\ \text{midway between }{M}\text{ and }{N}$

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.

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

Find $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$, and the digits $\displaystyle a_1,a_2,\ldots,a_n$ such that \[ \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n . \]

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 $m+n$.

Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$