Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]

In triangle $ABC$, $AB = AC$. Point $D$ is the midpoint of side $BC$. Point $E$ lies outside the triangle $ABC$ such that $CE \perp AB$ and $BE = BD$. Let $M$ be the midpoint of segment $BE$. Point $F$ lies on the minor arc $\widehat{AD}$ of the circumcircle of triangle $ABD$ such that $MF \perp BE$. Prove that $ED \perp FD.$ [asy] defaultpen(fontsize(10)); size(6cm); pair A = (3,10), B = (0,0), C = (6,0), D = (3,0), E = intersectionpoints( Circle(B, 3), C--(C+100*dir(B--A)*dir(90)) )[1], M = midpoint(B--E), F = intersectionpoints(M--(M+50*dir(E--B)*dir(90)), circumcircle(A,B,D))[0]; dot(A^^B^^C^^D^^E^^M^^F); draw(B--C--A--B--E--D--F--M^^circumcircle(A,B,D)); pair point = extension(M,F,A,D); pair[] p={A,B,C,D,E,F,M}; string s = "A,B,C,D,E,F,M"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} d[4] = -50; string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

$a_{1}=1$ and for all $n \geq 1$, \[ (a_{n+1}-2a_{n})\cdot \left (a_{n+1} - \dfrac{1}{a_{n}+2} \right )=0.\] If $a_{k}=1$, which of the following can be equal to $k$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.

There are two given different polynomials $P(x),Q(x)$ with real coefficients such that $P(Q(x))=Q(P(x))$. Prove that $\forall n\in \mathbb{Z_{+}}$ polynomial: \[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\] is divisible by $P(x)-Q(x)$.

In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\). For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\). Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]

Observe the set $S =\{(x, y) \in Z^2 : |x| \le 5$ and $-10 \le y\le 0\}$. Find the number of points $P$ in $S$ such that there exists a tangent line from $P$ to the parabola $y = x^2 + 1$ that can be written in the form $y = mx + b$, where $m$ and $b$ are integers.

Circles $O_1$ and $O_2$ interest at two points $ B$ and $ C,$ and $ BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $ C$ and intersecting circle $O_2$ at another point $ A.$ We join $ AB$ to intersect circle $O_1$ at point $ E,$ then join $ CE$ and extend it to intersect circle $O_2$ at point $ F.$ Assume $ H$ is an arbitrary point on line segment $ AF.$ We join $ HE$ and extend it to intersect circle $O_1$ at point $ G,$ and then join $ BG$ and extend it to intersect the extend line of $ AC$ at point $ D.$ Prove that \[ \frac{AH}{HF} = \frac{AC}{CD}.\]

A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of $2$, $4$, and $6$), and the other die has only odd numbers (two each of $1$, $3$, and $5$). The pair of dice is rolled. What is the probability that the sum of the numbers on top of the two dice is $7$? $ \textbf{(A)}\ \dfrac{1}{6} \qquad\textbf{(B)}\ \dfrac{1}{5} \qquad\textbf{(C)}\ \dfrac{1}{4} \qquad\textbf{(D)}\ \dfrac{1}{3} \qquad\textbf{(E)}\ \dfrac{1}{2} $

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

A natural number is a [i]palindrome[/i] when one obtains the same number when writing its digits in reverse order. For example, $481184$, $131$ and $2$ are palindromes. Determine all pairs $(m,n)$ of positive integers such that $\underbrace{111\ldots 1}_{m\ {\rm ones}}\times\underbrace{111\ldots 1}_{n\ {\rm ones}}$ is a palindrome.

Archimedes planned to count all of the prime numbers between $2$ and $1000$ using the Sieve of Eratosthenes as follows: (a) List the integers from $2$ to $1000$. (b) Circle the smallest number in the list and call this $p$. (c) Cross out all multiples of $p$ in the list except for $p$ itself. (d) Let $p$ be the smallest number remaining that is neither circled nor crossed out. Circle $p$. (e) Repeat steps $(c)$ and $(d)$ until each number is either circled or crossed out. At the end of this process, the circled numbers are prime and the crossed out numbers are composite. Unfortunately, while crossing out the multiples of $2$, Archimedes accidentally crossed out two odd primes in addition to crossing out all the even numbers (besides $2$). Otherwise, he executed the algorithm correctly. If the number of circled numbers remaining when Archimedes finished equals the number of primes from $2$ to $1000$ (including $2$), then what is the largest possible prime that Archimedes accidentally crossed out?

Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?

We say that a positive integer $n$ is nice if $\frac{4}{n}$ cannot be written as $\frac{1}{x}+\frac{1}{xy}+\frac{1}{z}$ for any positive integers $x, y, z$. Let us denote by $ a_n$ the number of nice numbers smaller than $n$. Prove that the sequence $\frac{n}{a_n}$ is not bounded.

Let $P$ be a variable point on the circumference of a quarter-circle with radii $OA, OB$ and $\angle AOB = 90^\circ$. H is the projection of $P$ on $OA$. Find the locus of the incenter of the right-angled triangle $HPO.$

Two medians of a triangle with unequal sides are $ 3$ inches and $ 6$ inches. Its area is $ 3 \sqrt{15}$ square inches. The length of the third median in inches, is: $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt{3} \qquad \textbf{(C)}\ 3 \sqrt{6} \qquad \textbf{(D)}\ 6 \sqrt{3} \qquad \textbf{(E)}\ 6 \sqrt{6}$