In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

A square and an equilateral triangle have the same perimeter. Let $ A$ be the area of the circle circumscribed about the square and $ B$ be the area of the circle circumscribed about the triangle. Find $ A/B$. $ \textbf{(A)}\ \frac{9}{16} \qquad \textbf{(B)}\ \frac{3}{4} \qquad \textbf{(C)}\ \frac{27}{32} \qquad \textbf{(D)}\ \frac{3\sqrt{6}}{8} \qquad \textbf{(E)}\ 1$

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.

If the digit $1$ is placed after a two digit number whose tens' digit is $t$, and units' digit is $u$, the new number is: $\textbf{(A)}\ 10t+u+1 \qquad \textbf{(B)}\ 100t+10u+1 \qquad \textbf{(C)}\ 100t+10u+1\qquad \textbf{(D)}\ t+u+1 \qquad \textbf{(E)}\ \text{None of these answers}$

A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$ Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible. (E. Barabanov)

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

From a square board $11$ squares long and $11$ squares wide, the central square is removed. Prove that the remaining $120$ squares cannot be covered by $15$ strips each $8$ units long and one unit wide.

How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers $1, 2,..., 6$?

A whole number is written on each square of a $3\times 2021$ board. If the number written in each square is greater than or equal to at least two of the numbers written in the neighboring squares, how many different numbers written on the board can there be at most? Note: Two squares are neighbors when they have a common side.

We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations \[ A = ABA \quad \text{ and } \quad B=BAB. \] a) Prove that any square matrix has at least a pseudo-inverse. b) For which matrix $A$ is the pseudo-inverse unique? [i]Marius Cavachi[/i]

Determine the planar finite configurations $C$ consisting of at least $3$ points, satisfying the following conditions; if $x$ and $y$ are distinct points of $C$, there exist $z\in C$ such that $xyz$ are three vertices of equilateral triangles

Prove that no pair of different positive integers $(m, n)$ exist, such that $\frac{4m^{2}n^{2}-1}{(m^{2}-n^2)^{2}}$ is an integer.

The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\] Show that $ a_{2008} < 5$.

9.7 Is there an infinite arithmetic sequence $\{a_n\}\subset \mathbb N$ s.t. $a_n+...+a_{n+9}\mid a_n...a_{n+9}$ for all $n$? ([i]V. Senderov[/i])

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$?

Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.

Solve the equation $(x^3-1000)^{1/2}=(x^2+100)^{1/3}$

At Olympic High School, $\frac25$ of the freshmen and $\frac45$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true? $\text{(A)}$ There are five times as many sophomores as freshmen. $\text{(B)}$ There are twice as many sophomores as freshmen. $\text{(C)}$ There are as many freshmen as sophomores. $\text{(D)}$ There are twice as many freshmen as sophomores. $\text{(E)}$ There are five times as many freshmen as sophomores.

Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.

Chris has a bag with 4 black socks and 6 red socks (so there are $10$ socks in total). Timothy reaches into the bag and grabs two socks [i]without replacement[/i]. Find the probability that he will not grab two red socks. [i]Proposed by Chris Tong[/i]

Find all quadruples $(x,y,z,w)$ of integers satisfying the sys­tem of equations $$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$

Jay has a $24\times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e. it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once, such that after each toggle he waits for one minute before the next toggle. Each light uses no energy while off and 1 kiloJoule of energy per minute while on. To express his creativity, Jay chooses to toggle the rows and columns in a random order. Compute the expected value of the total amount of energy in kiloJoules which has been expended by all the lights after all $48$ toggles. [i]Proposed by James Lin

How many even integers between 4000 and 7000 have four different digits?

Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.