A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$ for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$

The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.

Find all polynomials $P$ with real coefficients having no repeated roots, such that for any complex number $z$, the equation $zP(z) = 1$ holds if and only if $P(z-1)P(z + 1) = 0$. Remark: Remember that the roots of a polynomial are not necessarily real numbers.

Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows: $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$. Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.

Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\tfrac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137\le W\le 1061$ is $\tfrac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.

If $r+s+t=3$, $r^2+s^2+t^2=1$, and $r^3+s^3+t^3=3$, compute $rst$.

For positive integers $n, p$ with $n \geq p$, define real number $K_{n, p}$ as follows: $K_{n, 0} = \frac{1}{n+1}$ and $K_{n, p} = K_{n-1, p-1} -K_{n, p-1}$ for $1 \leq p \leq n.$ (i) Define $S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots$ . Find $\lim_{n \to \infty} S_n.$ (ii) Find $T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots$.

Determine all integers $n\ge 2$ with the following property: the number $2^k\cdot n-1$ is prime for all $k\in\{2,3,\ldots,n\}$.

Let $NAS$ be a triangle such that $NA=NS=5$ and $AS=6$. Let $D$ be the foot of the altitude from $N$ to $AS$ and $E$ the foot of the altitude from $A$ to $NS$. Point $X$ lies on line $DE$ outside the triangle such that $XA=\tfrac{18}{5}$. Find $XS$.

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

For how many primes $p$, $|p^4-86|$ is also prime? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$. Let $m$ be a positive integer. Express $$\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m$$in the form $(-1)^ma^{f(m)}g(m)!$, where $a$ is an integer and $f$ and $g$ are polynomials.

A telephone number has the form $ ABC \minus{} DEF \minus{} GHIJ$, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, $ A > B > C$, $ D > E > F$, and $ G > H > I > J$. Furthermore, $ D$, $ E$, and $ F$ are consecutive even digits; $ G$, $ H$, $ I$, and $ J$ are consecutive odd digits; and $ A \plus{} B \plus{} C \equal{} 9$. Find $ A$. $ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8$

Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$ $\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$

In the plane there is a square $ Q $ and a point $ P $. The point $ K $ runs through the perimeter of the square $ Q $. Find the locus of the vertex $ M $ of the equilateral triangle $ KPM $.

A set $M$ of natural numbers is called a [i]spectrum[/i] if there is a first-order language $L$ and a sentence $\phi$ over $L$ such that: $$M = \{ n \mid \phi \text{ has a model containing exactly $n$ elements}\}$$ For example, consider a sentence $\phi = \exists e . (\forall x. x = e)$ in a first order language with no relation symbol, no function symbol, and no constant symbol. The formula $\phi$ only admits a model containing exactly 1 element. Therefore, the set $\{1\}$ is a spectrum.\\ Show that: [list=1] [*] Every finite subset of $\mathbb{N} \setminus \{0\} $ is a spectrum [/*] [*] The set of even numbers, i.e., $\{2k \mid k \in \mathbb{N}\}$ is a spectrum [/*] [*] For any fixed $m \geq 1$, the set of numbers greater than $0$ that are divisible by $m$, i.e., $\{m\cdot k \mid k \in \mathbb{N}\}$ is a spectrum[/*] [/list]