Let $F_1 = 0$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$. Compute $$\sum_{n=1}^\infty \frac{\sum_{n=1}^\infty F_i}{3^n}.$$

Let $a,b,c$ be positive real numbers that satisfy $a+b+c=abc$. Prove that $$\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.$$

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

Let $b$ be a positive integer such that 2032 has 3 digits when expressed in base $b$. Define the function $S_k(n)$ as the sum of the digits of the base $k$ representation of $n$. Given that $S_b(2032)+S_{b^2}(2032) = 14$, find $b$. [i]Proposed by Anthony Yang[/i]

There are $30$ bags and there are $100$ similar coins in each bag (coins in each bag are similar, coins of different bags can be different). The weight of each coin is an one digit number in grams. We have a digital scale which can weigh at most $999$ grams in each weighing. Using this scale, we want to find the weight of coins of each bag. [b](a)[/b] Show that this operation is possible by $10$ times of weighing, and [b](b)[/b] It's not possible by $9$ times of weighing.

Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that the sequence $S_n=a_1+a_2+\ldots+a_n$ is upperbounded and lowerbounded and find its limit as $n\to\infty$.

Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.

Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.

Consider this histogram of the scores for $81$ students taking a test: [asy] unitsize(12); draw((0,0)--(26,0)); draw((1,1)--(25,1)); draw((3,2)--(25,2)); draw((5,3)--(23,3)); draw((5,4)--(21,4)); draw((7,5)--(21,5)); draw((9,6)--(21,6)); draw((11,7)--(19,7)); draw((11,8)--(19,8)); draw((11,9)--(19,9)); draw((11,10)--(19,10)); draw((13,11)--(19,11)); draw((13,12)--(19,12)); draw((13,13)--(17,13)); draw((13,14)--(17,14)); draw((15,15)--(17,15)); draw((15,16)--(17,16)); draw((1,0)--(1,1)); draw((3,0)--(3,2)); draw((5,0)--(5,4)); draw((7,0)--(7,5)); draw((9,0)--(9,6)); draw((11,0)--(11,10)); draw((13,0)--(13,14)); draw((15,0)--(15,16)); draw((17,0)--(17,16)); draw((19,0)--(19,12)); draw((21,0)--(21,6)); draw((23,0)--(23,3)); draw((25,0)--(25,2)); for (int a = 1; a < 13; ++a) { draw((2*a,-.25)--(2*a,.25)); } label("$40$",(2,-.25),S); label("$45$",(4,-.25),S); label("$50$",(6,-.25),S); label("$55$",(8,-.25),S); label("$60$",(10,-.25),S); label("$65$",(12,-.25),S); label("$70$",(14,-.25),S); label("$75$",(16,-.25),S); label("$80$",(18,-.25),S); label("$85$",(20,-.25),S); label("$90$",(22,-.25),S); label("$95$",(24,-.25),S); label("$1$",(2,1),N); label("$2$",(4,2),N); label("$4$",(6,4),N); label("$5$",(8,5),N); label("$6$",(10,6),N); label("$10$",(12,10),N); label("$14$",(14,14),N); label("$16$",(16,16),N); label("$12$",(18,12),N); label("$6$",(20,6),N); label("$3$",(22,3),N); label("$2$",(24,2),N); label("Number",(4,8),N); label("of Students",(4,7),N); label("$\textbf{STUDENT TEST SCORES}$",(14,18),N); [/asy] The median is in the interval labeled $\text{(A)}\ 60 \qquad \text{(B)}\ 65 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 80$

Find all functions $f : \mathbb{R^{+}}\to \mathbb{R^{+}}$ such that $$f(x)^2=f(xy)+f(x+f(y))-1$$ for all $x, y\in \mathbb{R^{+}}$

The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new sequence add to $836$. Find the sum of the first, last, and middle terms of the original sequence.

[asy] size(150); dotfactor=4; draw(circle((0,0),4)); draw(circle((10,-6),3)); pair O,A,P,Q; O = (0,0); A = (10,-6); P = (-.55, -4.12); Q = (10.7, -2.86); dot("$O$", O, NE); dot("$O'$", A, SW); dot("$P$", P, SW); dot("$Q$", Q, NE); draw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle); draw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle); draw(P--Q--cycle); //Credit to happiface for the diagram[/asy] In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is $\textbf{(A) }\text{the average of the lengths of the internal and external common tangents}\qquad$ $\textbf{(B) }\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}\qquad$ $\textbf{(C) }\text{always equal to the length of an external common tangent}\qquad$ $\textbf{(D) }\text{greater than the length of an external common tangent}\qquad$ $\textbf{(E) }\text{the geometric mean of the lengths of the internal and external common tangents}$

Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Given a set of $100$ different numbers such that if each number in the set is replaced by the sum of the others, the same set will be obtained. Prove that the product of numbers in a set is positive.

Consider a circle $ \Gamma(O,R)$ and a point $ P$ found in the interior of this circle. Consider a chord $ AB$ of $ \Gamma$ that passes through $ P$. Suppose that the tangents to $ \Gamma$ at the points $ A$ and $ B$ intersect at $ Q$. Let $ M\in QA$ and $ N\in QB$ s.t. $ PM\perp QA$ and $ PN\perp QB$. Prove that the value of $ \frac {1}{PN} \plus{} \frac {1}{PM}$ doesn't depend of choosing the chord $ AB$.

There are $99$ space stations. Each pair of space stations is connected by a tunnel. There are $99$ two-way main tunnels, and all the other tunnels are strictly one-way tunnels. A group of $4$ space stations is called [i]connected[/i] if one can reach each station in the group from every other station in the group without using any tunnels other than the $6$ tunnels which connect them. Determine the maximum number of connected groups.

Find all solutions of $\frac{a}{x}=\frac{x-a}{a}$ for $x$.

Let $a$, $b$ and $c$ be numbers with $0 < a < b < c$. Which of the following is impossible? $\textbf{(A)}\ a+c<b \qquad \textbf{(B)}\ a\cdot b<c \qquad \textbf{(C)}\ a+b<c \qquad \textbf{(D)}\ a\cdot c<b \qquad \textbf{(E)}\ \frac{b}{c}=a$

What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?

Let $a$ and $ b$ be real numbers with $0\le a, b\le 1$. Prove that $$\frac{a}{b + 1}+\frac{b}{a + 1}\le 1$$ When does equality holds? (K. Czakler, GRG 21, Vienna)

Given a 4×4 grid of points, the points at two opposite corners are denoted $A$ and $D$. We need to choose two other points $ B$ and $C$ such that the six pairwise distances of these four points are all distinct. (a) How many such quadruples of points are there? (b) How many such quadruples of points are non-congruent? (c) If each point is assigned a pair of coordinates $(x_i,y_i)$, prove that the sum of the expressions $|x_i-x_j |+|y_i-y_j|$ over all six pairs of points in a quadruple is constant.

Maria buys computer disks at a price of 4 for 5 dollars and sells them at a price of 3 for 5 dollars. How many computer disks must she sell in order to make a profit of 100 dolars? $ \text{(A)}\ 100\qquad\text{(B)}\ 120\qquad\text{(C)}\ 200\qquad\text{(D)}\ 240\qquad\text{(E)}\ 1200 $

Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.

Find all triples $ (x,y,z)$ of real numbers which satisfy the simultaneous equations \[ x \equal{} y^3 \plus{} y \minus{} 8\] \[y \equal{} z^3 \plus{} z \minus{} 8\] \[ z \equal{} x^3 \plus{} x \minus{} 8.\]

Suppose $x,y,z$ are positive numbers such that $x+y+z=1$. Prove that (a) $xy+yz+xz\ge 9xyz$; (b) $xy+yz+xz<\frac{1}{4}+3xyz$;