Let $\omega=\cos\frac{\pi}{5}+\text{i}\sin\frac{\pi}{5}$, which equation has roots $\omega,\omega^3,\omega^7,\omega^9$? $\text{(A)}x^4+x^3+x^2+x+1=0\qquad\text{(B)}x^4-x^3+x^2-x+1=0$ $\text{(C)}x^4-x^3-x^2+x+1=0\qquad\text{(D)}x^4+x^3+x^2-x+1=0$

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\tfrac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of his points against the other nine of the ten). What was the total number of players in the tournament?

What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$. [i]Proposed by Levi Iszler[/i]

For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]

In concurrent computing, two processes may have their steps interwoven in an unknown order, as long as the steps of each process occur in order. Consider the following two processes: \begin{tabular}{c|cc} Process & $A$ & $B$\\ \hline Step 1 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 2 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$\\ Step 3 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 4 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$ \end{tabular} One such interweaving is $A1$, $B1$, $A2$, $B2$, $A3$, $B3$, $B4$, $A4$, but $A1$, $A3$, $A2$, $A4$, $B1$, $B2$, $B3$, $B4$ is not since the steps of $A$ do not occur in order. We run $A$ and $B$ concurrently with $x$ initially valued at $6$. Find the minimal possible value of $x$ among all interweavings.

Let $n>1$ be a positive integer. Find the number of binary strings $(a_1, a_2, \ldots, a_n)$, such that the number of indices $1\leq i \leq n-1$ such that $a_i=a_{i+1}=0$ is equal to the number of indices $1 \leq i \leq n-1$, such that $a_i=a_{i+1}=1$.

Let $x$ and $y$ be integers from $-10$ to $10$, inclusive, with $xy \ne1$. Compute the number of ordered pairs $(x, y) $ such that $$\left| \frac{x + y}{1 - xy} \right|\le 1.$$

Let $ \triangle ABC $ be an acute triangle with circumcenter $ O $ and centroid $ G .$ Let $ D $ be the midpoint of $ BC $ and $ E\in \odot (BC) $ be a point inside $ \triangle ABC $ such that $ AE \perp BC . $ Let $ F=EG \cap OD $ and $ K, L $ be the point lie on $ BC $ such that $ FK \parallel OB, FL \parallel OC . $ Let $ M \in AB $ be a point such that $ MK \perp BC $ and $ N \in AC $ be a point such that $ NL \perp BC . $ Let $ \omega $ be a circle tangent to $ OB, OC $ at $ B, C, $ respectively $ . $ Prove that $ \odot (AMN) $ is tangent to $ \omega $

Show that there exist no integers $a$ and $b$ such that $a^3+a^2b+ab^2+b^3=2001$.

In a village with a population of $1000$, two hundred people have been infected by a disease. A diagnostic test can be done to check whether a person is infected, but the result could be erroneous. That is, there is a $5\%$ probability that the test result of an infected person shows that they are not infected and a $5\%$ probability that the test result of a healthy person shows that they are infected. We randomly choose someone from the population of this village and take the diagnostic test from him. What is the probability that the test result declares that person is infected?

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is: $\textbf{(A)}\ \dfrac{16}{9} \qquad \textbf{(B)}\ \dfrac{16}{3}\qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{4}{3} \qquad \textbf{(E)}\ -\dfrac{4}{3}$

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.\] Show that $a_{n}$ is a square if and only if $n=2^{k}+k-2$ for some $k \in \mathbb{N}$.

9. Let [i]k[/i] be a positive integer and let $x_0, x_1, x_2, \cdots$ be an infinite sequence defined by the relationship $$x_0 = 0$$ $$x_1 = 1$$ $$x_{n+1} = kx_n +x_{n-1}$$ For all [i]n[/i] $\ge$ 1 (a) For the special case [i]k[/i] = 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2 (b) For the general case of integers [i]k[/i] $\ge$ 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2

Given is the square $ABCD$. Point $E$ lies on the diagonal $AC$, where $AE> EC$. On the side $AB$, a different point from $B$ has been selected for which $EF = DE$. Prove that $\angle DEF = 90^o$.

A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon? $ \textbf{(A)}\ \sqrt 3\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ \sqrt 6\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 6$

On a certain math exam, $ 10 \%$ of the students got 70 points, $ 25 \%$ got 80 points, $ 20 \%$ got 85 points, $ 15 \%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

Let $\left \{ x_n \right \} _{n\ge 1}$ and $\left \{ y_n \right \} _{n\ge 1}$ be two infinite sequences of integers. Prove that there exists an infinite sequence of integers $\left \{ z_n \right \} _{n\ge 1}$ such that for any positive integer \( n \), the following holds: \[ \sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right). \]

Eight players, Ann, Bob, Con, Dot, Eve, Fay, Guy and Hal compete in a chess tournament. No pair plays together more than once and there is no group of five people in which each one plays against all of the other four. (a) Write down an arrangement for a tournament of $24$ games satisfying these conditions. (b) Show that it is impossible to have a tournament of more than $24$ games satisfying these conditions.

Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and \[2^ap^b=(p+2)^c-1.\]

(D.Shnol) Two opposite angles of a convex quadrilateral with perpendicular diagonals are equal. Prove that a circle can be inscribed in this quadrilateral.

What is the smallest result that can be obtained from the following process? *Choose three different numbers from the set $\{3,5,7,11,13,17\}$. *Add two of these numbers. *Multiply their sum by the third number. $\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56$