Let $ E$ be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let $ f$ be the function of area of a triangle. Determine the set of values $ f(E)$ of $ f.$

Wu starts out with exactly one coin. Wu flips every coin he has [i]at once[/i] after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a, b$ are positive integers such that $a$ is odd. Find $a+b$. [i]Proposed by Bradley Guo[/i]

Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\] $\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$ $\textbf{(B) }\text{if and only if }a=b^2\qquad$ $\textbf{(C) }\text{if and only if }b=a^2\qquad$ $\textbf{(D) }\text{if and only if }x=ab\qquad$ $ \textbf{(E) }\text{for none of these}$

Let $(p_1,p_2,..., p_n)$ be a random permutation of the set $\{1,2,...,n)$. If $n$ is odd, prove that the product $(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n)$ is an even number. @below fixed.

The eight points below are the vertices and the midpoints of the sides of a square. We would like to draw a number of circles through the points, in such a way that each pair of points lie on (at least) one of the circles. Determine the smallest number of circles needed to do this. [asy] unitsize(1 cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((0,1)); dot((2,1)); dot((0,2)); dot((1,2)); dot((2,2)); [/asy]

Let $ (a_{n})_{n\ge 1} $ be a sequence such that: $ a_{1}=1; a_{n+1}=\frac{n}{a_{n}+1}.$ Find $ [a_{2008}] $

Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighborhood picnic? $\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}6\dfrac23 \qquad \textbf{(C)}\hspace{.05in}7\dfrac12 \qquad \textbf{(D)}\hspace{.05in}8 \qquad \textbf{(E)}\hspace{.05in}9 $

For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that: \[ \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC\]

Find all non negative integers $(a, b)$ such that $$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$

What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?

How many ordered triples $(a, b, c)$ of integers with $-15 \le a, b, c \le 15$ are there such that the three equations $ax + by = c$, $bx + cy = a$, and $cx + ay = b$ correspond to lines that are distinct and concurrent?

In a plane three points $P,Q,R,$ not on a line, are given. Let $k, l, m$ be positive numbers. Construct a triangle $ABC$ whose sides pass through $P, Q,$ and $R$ such that $P$ divides the segment $AB$ in the ratio $1 : k$, $Q$ divides the segment $BC$ in the ratio $1 : l$, and $R$ divides the segment $CA$ in the ratio $1 : m.$

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? $\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$

Each angle of a rectangle is trisected. The intersections of the pairs of trisectors adjacent to the same side always form: $ \textbf{(A)}\ \text{a square} \qquad\textbf{(B)}\ \text{a rectangle} \qquad\textbf{(C)}\ \text{a parallelogram with unequal sides} \\ \textbf{(D)}\ \text{a rhombus} \qquad\textbf{(E)}\ \text{a quadrilateral with no special properties}$

Is it possible to partition the set $S=\{1,2,\ldots,21\}$ into subsets that in each of these subsets the largest number is equal to the sum of the other numbers?

Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$? [i]Proposed by Evan Chen[/i]

In a $44\times 44$ board, some of the cells are blue, and the rest are red. No blue cells borders another blue cell on the side. The red cells, on the other hand, form a connected component (one may get from any red cell to any other red cell only by traversing edge-adjacent red cells). Prove that less than one third of the cells on the board are blue. [i]Proposed by B. Frenkin[/i]

Determine all integer solutions of the equation $x^n+y^n=1994$ where $n\geq 2$

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

There are two very strict laws in the country of Militaria. (i) Anyone who is shorter than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $r$ from him) is freed from the military service. (ii) Anyone who is taller than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $R$ from him) is allowed to serve in the police. A nice thing is that each man $X$ may choose his own (possibly different) positive numbers $r = r(X)$ and $R = R(X)$. Can it happen that $90\%$ (or more) of the men in Militaria are free from the army and, at the same time, $90\%$ (or more) of the men in Militaria are allowed to serve in the police? (The places of living of the men are fixed points in the plane.) (N Konstantinov)

Ted's favorite number is equal to \[1\cdot\binom{2007}1+2\cdot\binom{2007}2+3\cdot\binom{2007}3+\cdots+2007\cdot\binom{2007}{2007}.\] Find the remainder when Ted's favorite number is divided by $25$. $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ \textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\ \textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\ \textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\ \textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\ \textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\ \textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\ \textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23\\\\ \textbf{(Y) }24 \end{array}$

Consider a convex solid $ K$ in space and two parallel planes $ \epsilon _1$ and $ \epsilon _2$ on the distance $ 1$ tangent to $ K$. A plane $ \epsilon$ between $ \epsilon _1$ and $ \epsilon _2$ is on the distance $ d_1$ from $ \epsilon _1$. Find all $ d_1$ such that the part of $ K$ between $ \epsilon _1$ and $ \epsilon$ always has a volume not exceeding half the volume of $ K$.

Triangle $ABC$ is right-angled and isosceles with a right angle at the vertex $C$. On rays $CB$ on vertex $B$ is selected point F, on rays $BA$ on vertex $A$ is selected point G so that $AG = BF.$ The ray $GD$ is drawn so that it intersects with ray $AC$ at point $D$ with $\angle FGD = 45^o$. Find $\angle FDG$. (Bogdan Rublev)

Triangle $ABC$ is an scalene triangle. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.

We call a set $ A$ a good set if it has the following properties: 1. $ A$ consists circles in plane. 2. No two element of $ A$ intersect. Let $ A,B$ be two good sets. We say $ A,B$ are equivalent if we can reach from $ A$ to $ B$ by moving circles in $ A$, making them bigger or smaller in such a way that during these operations each circle does not intersect with other circles. Let $ a_{n}$ be the number of inequivalent good subsets with $ n$ elements. For example $ a_{1}\equal{} 1,a_{2}\equal{} 2,a_{3}\equal{} 4,a_{4}\equal{} 9$. [img]http://i5.tinypic.com/4r0x81v.png[/img] If there exist $ a,b$ such that $ Aa^{n}\leq a_{n}\leq Bb^{n}$, we say growth ratio of $ a_{n}$ is larger than $ a$ and is smaller than $ b$. a) Prove that growth ratio of $ a_{n}$ is larger than 2 and is smaller than 4. b) Find better bounds for upper and lower growth ratio of $ a_{n}$.