The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.

Find $\displaystyle \sum_{k=0}^{49}(-1)^k\binom{99}{2k}$, where $\binom{n}{j}=\frac{n!}{j!(n-j)!}$. $ \textbf{(A)}\ -2^{50} \qquad\textbf{(B)}\ -2^{49} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2^{49} \qquad\textbf{(E)}\ 2^{50} $

A [i]binary string[/i] is a sequence, each of whose terms is $0$ or $1$. A set $\mathcal{B}$ of binary strings is defined inductively according to the following rules. [list] [*]The binary string $1$ is in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ odd, then both $s_1,s_2,\dotsc ,s_n,0$ and $0,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ even, then both $s_1,s_2,\dotsc ,s_n,1$ and $1,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*] [*]No other binary strings are in $\mathcal{B}$.[/*] [/list] For each positive integer $n$, let $b_n$ be the number of binary strings in $\mathcal{B}$ of length $n$. [list=a] [*]Prove that there exist constants $c_1,c_2>0$ and $1.6<\lambda_1,\lambda_2<1.9$ such that $c_1\lambda_1^n<b_n<c_2\lambda_2^n$ for all positive integer $n$.[/*] [*]Determine $\liminf_{n\to \infty} {\sqrt[n]{b_n}}$ and $\limsup_{n\to \infty} {\sqrt[n]{b_n}}$[/*] [/list] [i]Note: The problem is open in the sense that no solution is currently known to part (b).[/i]

A square with sides of length $1$ cm is given. There are many different ways to cut the square into four rectangles. Let $S$ be the sum of the four rectangles’ perimeters. Describe all possible values of $S$ with justification.

In a circle $O$, there are six points, $ A$, $ B$, $C$, $D$, $E$, $F$ in a counterclockwise order such that $BD \perp CF$ , and $CF$, $BE$, $AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE \parallel MN$.

Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant rate, how many hours would it have taken Alex to rake the entire lawn by himself?

Charlotte writes a test consisting of 100 questions, where the answer to each question is either TRUE or FALSE. Charlotte’s teacher announces that for every five consecutive questions on the test, the answers to exactly three of them are TRUE. Just before the test starts, the teacher whispers to Charlotte that the answers to the first and last questions are both FALSE. (a) Determine the number of questions for which the correct answer is TRUE. (b) What is the correct answer to the sixth question on the test? (c) Explain how Charlotte can correctly answer all 100 questions on the test.

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent: (i) there is a numerical interval without any values of $f(x)$ , (ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ . (A Kanel)

It can be shown that there exists a unique polynomial $P$ in two variables such that for all positive integers $m$ and $n,$ $$P(m,n)=\sum_{i=1}^m\sum_{i=1}^n (i+j)^7.$$ Compute $P(3,-3).$

Let $a_1,b_1,c_1$ are three lines each two of them are mutually crossed and aren't parallel to some plane. The lines $a_2,b_2,c_2$ intersect the lines $a_1,b_1,c_1$ at the points $a_2$ in $A$, $C_2$, $B_1$; $b_2$ in $C_1$, $B$, $A_2$; $c_2$ in $B_2$, $A_1$, $C$ respectively in such a way that $A$ is the perpendicular bisector of $B_1C_2$, $B$ is the perpendicular bisector of $C_1A_2$ and $C$ is the perpendicular bisector of $A_1B_2$. Prove that: (a) $A$ is the perpendicular bisector of $B_2C_1$, $B$ is the perpendicular bisector of $C_2A_1$ and $C$ is the perpendicular bisector of $A_2B_1$; (b) triangles $A_1B_1C_1$ and $A_2B_2C_2$ are the same.

The cube $nxnxn$ consists of $n^3$ unit cubes $1x1x1$, and at least one of these unit cubes is black. Show that we can always cut the cube in $2$ parallelepiped pieces so that each piece contains exactly one black 1x1 square .

A point $P{}$ is considered on the incircle of the triangle $ABC$. We draw the tangent segments from $P{}$ to the three excircles of $ABC$. Prove that from the obtained three tangent segments it is possible to make a right triangle if and only if the point $P{}$ lies on one of the lines connecting two of the midpoints of the sides of $ABC$.

There are two circles with a common center $ O $ and a point $ A $. Construct a circle with center $ A $ intersecting the given circles at points $ M $ and $ N $ such that the line $ MN $ passes through point $ O $.

Real numbers $x_1,x_2,...x_{2004},y_1,y_2,...y_{2004}$ differ from $1$ and are such that $x_ky_k=1$ for every $k=1,2,...,2004$. Calculate the sum $$S=\frac{1}{1-x_1^3}+\frac{1}{1-x_2^3}+...+\frac{1}{1-x_{2004}^3}+\frac{1}{1-y_1^3}+\frac{1}{1-y_2^3}+...+\frac{1}{1-y_{2004}^3}$$

The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born? $\textbf{(A) }1979\qquad\textbf{(B) }1980\qquad\textbf{(C) }1981\qquad\textbf{(D) }1982\qquad \textbf{(E) }1983$

Choose a four-digit number (none of them zero) and, starting with it, build a list of $21$ different numbers, each with four digits, that satisfies the following rule: after writing each new number in the list, all the averages are calculated Between two digits of that number, those averages that do not give a whole number are discarded, and with the rest a four-digit number is formed that will occupy the next place in the list. For example, if $2946$ was written in the list, the next one can be $3333$ or $3434$ or $5345$ or any other number armed with the figures $3$, $4$ or $5$.

Evaluate $$\prod_{k=1}^{14} cos \big(\frac{k\pi}{15}\big)$$

Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$

Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.

Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$ Is $N$ even or odd? Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.

One day, Ivan was imprisoned by an evil king. The evil king said : "If you can correctly determine the polynomial that I'm thinking of, you'll be free. If after $2014$ tries, you can't guess it, you'll be executed." Ivan answered : "Are there any clues?" The evil king replied : "I can tell you that the polynomial has real coefficients and is monic. Furthermore, all roots are positive real numbers." That night, a kind wizard, told him the polynomial. The conversation was heard by the king who was visiting Ivan. He killed the wizard. The next day, Ivan forgot the polynomial, except that the coefficients of $x^{2013}$ is $2014$, and that the constant term is $1$. Can Ivan guarantee freedom? And if so, in how many tries? (Assume that Ivan is very unlucky so any random guess fails.)

Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares? [i]David Yang.[/i]

$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.