Let the complex number $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\tfrac{1}{2}.$

$ABCD$ is a quadrilateral and $P,Q$ are the midpoints of $CD, AB, AP, DQ$ meet at $X$ and $BP, CQ$ meet at $Y$. Prove that $A[ADX]+A[BCY] = A[PXOY]$.

$2n-1$ distinct positive real numbers with sum $S $ are given. Prove that there are at least $\binom {2n-2}{n-1}$ different ways to choose $n $ numbers among them such that their sum is at least $\frac {S}{2}$. [i]Proposed by Amirhossein Gorzi[/i]

Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?

Find the sum of all perfect squares that divide 2016.

Prove that every positive integer can be represented in the form $$3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k}$$ where $m_1 > m_2 > \dots > m_k \geq 0$ and $0 \leq n_1 < n_2 < \dots < n_k$ are integers.

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

The sequence $a_0,a_1,a_2,...$ is defined such that $a_0 = 2+ \sqrt3$, $a_1 =\sqrt{5-2\sqrt5}$, and $$a_n a_{n-1}a_{n-2} - a_n + a_{n-1} + a_{n-2} = 0.$$ Find the least positive integer $n$ such that $a_n = 1$.

Formulate a criterion for the conguence of triangles by two medians and an altitude.

Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers

Find one complex value of $x$ that satisfies the equation $\sqrt{3}x^7+x^4+2=0$.

Meghana writes two (not necessarily distinct) primes $q$ and $r$ in base $10$ next to each other on a blackboard, resulting in the concatenation of $q$ and $r$ (for example, if $q = 13$ and $r = 5$, the number on the blackboard is now $135$). She notices that three more than the resulting number is the square of a prime $p$. Find all possible values of $p$.

Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$, and meets $BC$ at $M$. What is the nearest integer to $100 \frac{AM}{CM}$?

A natural number $n$ is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting $k$ from $n$ and the larger by adding $l$ to $n.$ Prove that $n-kl$ is a perfect square.

Suppose that positive integers $m,n,k$ satisfy the equations $$m^2+1=2n^2, 2m^2+1=11k^2.$$ Find the residue when $n$ is divided by $17$.

Let $ n$ be a nonzero positive integer. A set of persons is called a $ n$-balanced set if in any subset of $ 3$ persons there exists at least two which know each other and in each subset of $ n$ persons there are two which don't know each other. Prove that a $ n$-balanced set has at most $ (n \minus{} 1)(n \plus{} 2)/2$ persons.

Given a triangle $ABC$, let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that \[\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}. \]

For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$. Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.

The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $ \$3.15$. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $ \$4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to $ \textbf{(A)}\ \$1.70 \qquad \textbf{(B)}\ \$1.65 \qquad \textbf{(C)}\ \$1.20 \qquad \textbf{(D)}\ \$1.05 \qquad \textbf{(E)}\ \$0.95$

A triangle $\vartriangle ABC$ has $\angle B = 90^o$ . A circle is tangent to $AB$ at $B$ and also tangent to $AC$. Another circle is tangent to the first circle as well as the two sides $AB$ and $AC$. Suppose that $AB =\sqrt3$ and $BC = 3$. What is the radius of the second circle?

A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$. Find the minimum possible value of $$BC^6+BD^6-AC^6-AD^6.$$

The points $A_1,A_...,A_n$ lie on a circle with radius 1. The points $B_1,B_2,…,B_n$ are such that $B_i B_j<A_i A_j$ for $i\neq j$. Is it always true that the points $B_1,B_2,...,B_n$ lie on a circle with radius lesser than 1?

Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

Construct a triangle knowing an angle, the ratio of the sides that form it and the radius of the inscribed circle.