Box A contains $3$ black and $4$ blue marbles. Box B has $7$ black and $1$ blue, whereas Box C has $2$ black, $3$ blue and $1$ green marble. I close my eyes and pick two marbles from $2$ different boxes. If it turns out that I get $1$ black and $1$ blue marble, what is the probability that the black marble is from box A and the blue one is from C?

Outside triangle $ ABC $ equilateral triangles $ BMC $, $ CNA $, and $ APB $ are constructed. Prove that the centers $ S $, $ T $, $ U $ of these triangles form an equilateral triangle.

$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.

At a given plane with $2,000$ lines, all those with an odd number of different points of intersection with intersecting lines. a) Can there be an odd number of red lines if in the plane given there are no parallel lines? b) Can there be an odd number of red lines if none of any 3 given lines intersect at one point?

[b]1.[/b] Consider in the plane a set $H$ of pairwise disjoint circles of radius 1 such that, for infinitely many positive integers $n$, the circle $k_n$ with centre at the origin and of radius $n$ contains at least $cn^2$ elements of the set $H$. Prove that there exists a straight line which intersects infinitely many of the circles of $H$. Show further that if we require only that the circles $k_n$ contain o(n²) elements of $H$, the proposition will be false. [b](G. 5)[/b]

The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon? [asy] size(130); pair A, B, C, D, E, F, G, H, I, J, K, L; A = dir(120); B = dir(60); C = dir(0); D = dir(-60); E = dir(-120); F = dir(180); draw(A--B--C--D--E--F--cycle); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); G = midpoint(A--B); H = midpoint(B--C); I = midpoint(C--D); J = midpoint(D--E); K = midpoint(E--F); L = midpoint(F--A); draw(G--H--I--J--K--L--cycle); label("$A$", A, dir(120)); label("$B$", B, dir(60)); label("$C$", C, dir(0)); label("$D$", D, dir(-60)); label("$E$", E, dir(-120)); label("$F$", F, dir(180)); [/asy] $\textbf{(A)}\ \displaystyle \frac{1}{2} \qquad \textbf{(B)}\ \displaystyle \frac{\sqrt 3}{3} \qquad \textbf{(C)}\ \displaystyle \frac{2}{3} \qquad \textbf{(D)}\ \displaystyle \frac{3}{4} \qquad \textbf{(E)}\ \displaystyle \frac{\sqrt 3}{2}$

Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

How many rearrangements of the letters of "$HMMTHMMT$" do not contain the substring "$HMMT$"? (For instance, one such arrangement is $HMMHMTMT$.)

The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown. 1. AUSSG 9981 2. USSGA 9819 3. SSGAU 8199 etc. If the pattern continues in this way, what number will appear in front of GAUSS 1998? $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$

Jimmy’s garden has right angled triangle shape that lies on island of circular shape in such a way that the corners of the triangle are on the shore of the island. When he made fences along the garden he realized that the length of the shortest side is $36$ meter shorter than the longest side, and third side required $48$ meter long fence. In the middle of the garden he built a house of circular shape that has the largest possible size. Jimmy measured the distance between the center of his house and the center of the island. What is the square of this distance?

There are several coins in a row, and the [i]allowed move[/i] is to remove exactly one coin from the row, which can either be the first or the last. In the initial distribution, there are $n$ coins with not necessarily equal values. Ana and María alternate turns. Ana starts, making two moves, then María makes one move, then Ana makes two moves, and so on until no coins remain: Ana makes two moves and María makes one. (Only in the last move can Ana take one coin if only one coin is left.) Ana's goal is to ensure she takes at least $\dfrac{2}{3}$ of the total value of the coins. Determine if Ana can achieve her goal with certainty if [list=a] [*]$n=2013$ [*]$n=2014$ [/list] If the answer is yes, provide a strategy to achieve it; if the answer is no, give a specific sequence of coins and explain how María prevents Ana from achieving her goal.

A circle of radius $2$ is inscribed in an isosceles trapezoid with the area of $28$. Find the length of the side of the trapezoid.

A non-self intersecting hexagon $RANDOM$ is formed by assigning the labels $R, A, N, D, O, M$ in some order to the points $$(0,0), (10,0), (10,10), (0,10), (3,4), (6,2).$$ Let $a_{\text{max}}$ be the greatest possible area of $RANDOM$ and $a_{\text{min}}$ the least possible area of $RANDOM.$ Find $a_{\text{max}}-a_{\text{min}}.$

Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

A triangle $ABC$ is given. Points $E,F,G$ are arbitrarily selected on the sides $AB,BC,CA$, respectively, such that $AF\perp EG$ and the quadrilateral $AEFG$ is cyclic. Find the locus of the intersection point of $AF$ and $EG$.

Find all integers $ n\ge 2$ with the following property: There exists three distinct primes $p,q,r$ such that whenever $ a_1,a_2,a_3,\cdots,a_n$ are $ n$ distinct positive integers with the property that at least one of $ p,q,r$ divides $ a_j - a_k \ \forall 1\le j\le k\le n$, one of $ p,q,r$ divides all of these differences.

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$