Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i ]$ ($i=1,2,3$). If the distribution of each random number is uniform with respect to the length of the interval it is chosen from, determine the expected value of the smallest number chosen.

In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters? $\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$

Let $k$ be a positive integer, let $z_1,z_2, \ldots, z_k \in \mathbb{C}$ be distinct and let $u_1,u_2,\ldots,u_k \in \mathbb{C}$ be such that the set $\big\{a_n=u_1z_1^n+u_2z_2^n+\ldots+u_kz_k^n : n \in \mathbb{Z}_{>0} \big\}$ is finite. Prove that there exists a positive integer $p$ such that $a_n=a_{n+p},$ for any positive integer $n.$

Given an acute triangle $ABC$, $D$ is a point on $BC$. A circle with diameter $BD$ intersects line $AB,AD$ at $X,P$ respectively (different from $B,D$).The circle with diameter $CD$ intersects $AC,AD$ at $Y,Q$ respectively (different from $C,D$). Draw two lines through $A$ perpendicular to $PX,QY$, the feet are $M,N$ respectively.Prove that $\triangle AMN$ is similar to $\triangle ABC$ if and only if $AD$ passes through the circumcenter of $\triangle ABC$.

How many $7$-digit positive integers are there such that the number remains same when its digits are reversed and is multiple of $11$? $ \textbf{(A)}\ 900 \qquad\textbf{(B)}\ 854 \qquad\textbf{(C)}\ 818 \qquad\textbf{(D)}\ 726 \qquad\textbf{(E)}\ \text{None} $

Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$ be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$, $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if \[\frac{BC}{CD}=\frac{EF}{FA}.\]

Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$

Let $\triangle ABC$ be a right triangle with $\angle C=90^{\circ}$. Two squares $S_1$ and $S_2$ are inscribed in the triangle $ABC$ such that $S_1$ and $ABC$ share a common vertex $C$ and $S_2$ has one of its sides on $AB$. Suppose that $\text{Area}(S_1)=1+\text{Area}(S_2)=441$, then calculate $AC+BC$ (A)$400$ (B)$420$ (C)$441$ (D)$462$

Does there exist a finite sequence of integers $c_1,c_2,\ldots ,c_n$ such that all the numbers $a+c_1,a+c_2,\ldots ,a+c_n$ are primes for more than one but not infinitely many different integers $a$?

An equilateral triangle $\bigtriangleup ABC$ is split into $4$ triangles with equal area; three congruent triangles $\bigtriangleup ABX,\bigtriangleup BCY, \bigtriangleup CAZ$, and a smaller equilateral triangle $\bigtriangleup XYZ$, as shown. Prove that the points $X, Y, Z$ lie on the incircle of triangle $\bigtriangleup ABC$. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Let $P$ be a point inside triangle $ABC$ such that $\angle CPA = 90^o$ and $\angle CBP = \angle CAP$. Prove that $\angle P XY = 90^o$, where $X$ and $Y$ are the midpoints of $AB$ and $AC$ respectively.

Find all functions $f:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, such that $$xf(1+xf(y))=f(f(x)+f(y))$$ for all positive reals $x, y$.

Two lines $m$ and $n$ intersect in a unique point $P$. A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$, but not at the same time. Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.

Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.

Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$

Given that $\{a_n\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$

How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

How many real numbers $x$ are there such that $\sqrt{ x + 1 - 4\sqrt{x-3}} + \sqrt{ x + 6 - 6\sqrt{x-3}} = 1$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

$50$ silver coins ordered by weight and $51$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupying the $51$st place in weight among all $101$ coins) using $7$ weighings? (A. Andjans)

[asy] for (int i=0; i<9; ++i) { draw(dir(10+40*i)--dir(50+40*i)); } draw(dir(50) -- dir(90)); label("$a$", dir(50) -- dir(90), N); draw(dir(10) -- dir(90)); label("$b$", dir(10) -- dir(90), SW); draw(dir(-70) -- dir(90)); label("$d$", dir(-70) -- dir(90), E); //Credit to MSTang for the diagram[/asy] If $a,b,$ and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then $\textbf{(A) }d=a+b\qquad\textbf{(B) }d^2=a^2+b^2\qquad\textbf{(C) }d^2=a^2+ab+b^2\qquad$ $\textbf{(D) }b=\frac{a+d}{2}\qquad \textbf{(E) }b^2=ad$

let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$ find all values of $\frac{1}{x}+\frac{1}{y}$

Let $1=d_1<d_2<\ldots<d_{2m}=n$ be the divisors of a positive integer $n$, where $n$ is not a perfect square. Consider the determinant $$D=\begin{vmatrix}n+d_1&n&\ldots&n\\n&n+d_2&\ldots&n\\\ldots&\ldots&&\ldots\\n&n&\ldots&n+d_{2m}\end{vmatrix}.$$ (a) Prove that $n^m$ divides $D$. (b) Prove that $1+d_1+d_2+\ldots+d_{2m}$ divides $D$.

There are $100$ points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?

For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. (Once a sapling is turned into a perfectly-shaped apple tree, it will stay a perfectly-shaped apple tree.) The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Yannick Yao[/i]