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.

Bisectrices $AA_1$ and $BB_1$ of triangle $ABC$ meet in $I$. Segments $A_1I$ and $B_1I$ are the bases of isosceles triangles with opposite vertices $A_2$ and $B_2$ lying on line $AB$. It is known that line $CI$ bisects segment $A_2B_2$. Is it true that triangle $ABC$ is isosceles?

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that \[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]

A regular hexagon of area $H$ is inscribed in a convex polygon of area $P$. Show that $P \leq \frac{3}{2}H$. When does the equality occur?

Below is pictured a regular seven-pointed star. Find the measure of angle $a$ in radians. [asy] size(150); draw(unitcircle, white); pair A = dir(180/7); pair B = dir(540/7); pair C = dir(900/7); pair D = dir(180); pair E = dir(-900/7); pair F = dir(-540/7); pair G = dir (-180/7); draw(A--D); draw(B--E); draw(C--F); draw(D--G); draw(E--A); draw(F--B); draw(G--C); label((-0.1,0.5), "$a$"); [/asy]

Suppose that $n$ polygons of area $s = (n - 1)^2$ are placed on a polygon of area $S = \frac{n(n - 1)^2}{2}$. Prove that there exist two of the $n$ smaller polygons whose intersection has the area at least $1$.

There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.

Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]

For $n > 1$, let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$. Find the maximum value of $\frac{a_n}{n}$.

There is a unique choice of positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime and the infinite sums \[ \sum_{n=0}^{\infty} \left(\left(\frac{a - b\sqrt{c}}{10}\right)^{n-10}\cdot\prod_{k=0}^{9} (n - k)\right) \quad \text{and} \quad \sum_{n=0}^{\infty} \left((a - b\sqrt{c})^{n+1}\cdot\prod_{k=0}^{9} (n - k)\right) \] are equal (i.e., converging to the same finite value). Compute $a + b + c$.

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.

Let $ABC$ be a triangle with altitudes $AD, BE, CF$ and orthocenter $H$. The perpendicular bisector of $HD$ meets $EF$ at $P$ and $N$ is the center of the nine-point circle. Let $L$ be a point on the circumcircle of $ABC$ such that $\angle PLN=90^{\circ}$ and $A, L$ are in distinct sides of the line $PN$. Show that $ANDL$ is cyclic.

Find all the odd positive integers $n$ such that there are $n$ odd integers $x_1, x_2,..., x_n$ such that $$x_1^2+x_2^2+...+x_n^2=n^4$$

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

Find the sum of the squares of the values $x$ that satisfy $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$.

$ (6?3)+4-(2-1) = 5. $ To make this statement true, the question mark between the 6 and the 3 should be replaced by $ \text{(A)}\div\qquad\text{(B)}\ \times\qquad\text{(C)}+\qquad\text{(D)}\ -\qquad\text{(E)}\ \text{None of these} $

Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$. Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.

Let $ n $ be a natural number. Find all integer numbers that can be written as $$ \frac{1}{a_1} +\frac{2}{a_2} +\cdots +\frac{n}{a_n} , $$ where $ a_1,a_2,...,a_n $ are natural numbers.

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

Let $y_{1}, y_{2}, y_{3},\ldots$ be a sequence such that $y_{1}=1$ and, for $k>0,$ is defined by the relationship: \[y_{2k}=\begin{cases}2y_{k}& \text{if}~k~ \text{is even}\\ 2y_{k}+1 & \text{if}~k~ \text{is odd}\end{cases}\]\[y_{2k+1}=\begin{cases}2y_{k}& \text{if}~k~ \text{is odd}\\ 2y_{k}+1 & \text{if}~k~ \text{is even}\end{cases}\]Show that the sequence takes on every positive integer value exactly once.

Let $z\neq0$ be a complex number satisfying $z^2=z+i|z|$. ($|z|$ denotes the length between the origin and $z$ in the complex plane.) Find $z\cdot\overline{z}$, where $\overline{z}=a-bi$ is the complex conjugate of $z=a+bi$.

There were ten points $X_1, \ldots , X_{10}$ on a line in this particular order. Pete constructed an isosceles triangle on each segment $X_1X_2, X_2X_3,\ldots, X_9X_{10}$ as a base with the angle $\alpha{}$ at its apex. It so happened that all the apexes of those triangles lie on a common semicircle with diameter $X_1X_{10}$. Find $\alpha{}$. [i]Egor Bakaev[/i]

Determine the polygons with $n$ sides $(n \ge 4)$, not necessarily convex, which satisfy the property that the reflection of every vertex of polygon with respect to every diagonal of the polygon does not fall outside the polygon. [b]Note:[/b] Each segment joining two non-neighboring vertices of the polygon is a diagonal. The reflection is considered with respect to the support line of the diagonal.

13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, .. 13$. Their scores are $s_1,s_2, ... s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, ... w_5$. Each score $s_i$ is equal to the sums of the weights of all problems solved by student $i$. On the other hand, each weight $w_j$ is assigned to be $\frac{1}{\sum_ {s_i} }$, where the sum is over all the scores of students who solved problem $j$. (If nobody solved a problem, the score doesn't matter). If the largest possible average score of the students can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ is square-free, find $a+b$. [i]Proposed by Jeff Lin[/i]