Initially, there are $n$ red boxes numbered with the numbers $1,2,\dots ,n$ and $n$ white boxes numbered with the numbers $1,2,\dots ,n$ on the table. At every move, we choose $2$ different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is $6$ more than the number of balls from the white one or the number of balls from the white one is $16$ more than the number of balls from the red one. With that given information find all possible values of $n$

Find the smallest positive integer $n$ such that $32^n = 167x + 2$ for some integer $x$

Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?

$4n$ first grade students at Songkhla Primary School, including $2n$ boys and $2n$ girls, participate in a taekwondo tournament where every pair of students compete against each other exactly once. The tournament is scored as follows: $\bullet$ In a match between two boys or between two girls, a win is worth $3$ points, a draw $1$ point, and a loss $0$ points. $\bullet$ In a math between a boy and a girl, if the boy wins, he receives $2$ points, else he receives $0$ points. If the girl wins, she receives $3$ points, if she draws, she receives $2$ points, and if she loses, she receives $0$ points. After the tournament, the total score of each student is calculated. Let $P$ be the number of matches ending in a draw, and let $Q$ be the total number of matches. Suppose that the maximum total score is $4n - 1$. Find $P/Q$.

Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$. Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.

Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.

An equilateral triangle $ABC$ has sides of length $n$, a positive integer. Divide the triangle into equilateral triangles of length $ 1$, drawing parallel lines (at distance $ 1$) to all sides of the triangle. A path is a continuous path, starting at the triangle with vertex $A$ and always crossing from one small triangle to another on the side that both triangles share, in such a way that it never passes through a small triangle twice. Find the maximum number of triangles that can be visited.

The natural numbers $k \geqslant n$ are given. Peter has $n{}$ objects and $N{}$ special ways in which he likes to lay them out in a row from left to right. He noticed that for any non-empty subset $A{}$ of these objects containing $|A| \leqslant k$ objects, and any element $a\in A$, there are exactly $N/|A|$ special ways for which element $a{}$ is the leftmost in the set $A{}$. Prove that, under the same conditions on $A{}$ and $a{}$, for any integer $m =1,2,\ldots,|A|$ there are exactly $N/|A|$ special ways for which the element $a{}$ is the $m^{\text{th}}$ from the left in the set $A{}$.

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.

Let $f:[0,\infty)\to\mathbb R$ be a differentiable function with $|f(x)|\le M$ and $f(x)f'(x)\ge\cos x$ for $x\in[0,\infty)$, where $M>0$. Prove that $f(x)$ does not have a limit as $x\to\infty$.

Let $a>0$. Find the minimum value of $\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx$

Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose \[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\] holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.

The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.

[u]Set 2[/u] [b]2.1[/b] A school has $50$ students and four teachers. Each student has exactly one teacher, such that two teachers have $10$ students each and the other two teachers have $15$ students each. You survey each student in the school, asking the number of classmates they have (not including themself or the teacher). What is the average of all $50$ responses? [b]2.2[/b] Let $T$ be the answer from the previous problem. A ball is thrown straight up from the ground, reaching (maximum) height $T+1$. Then the ball bounces on the ground and rebounds to height $T-1$. The ball continues bouncing indefinitely, and the height of each bounce is $r$ times the height of the previous bounce for some constant $r$. What is the total vertical distance that the ball travels? [b]2.3[/b] Let $T$ be the answer from the previous problem. The polynomial equation $$x^3 + x^2 - (T + 1)x + (T- 1) = 0$$ has one (integer) solution for x which does not depend on $T$ and two solutions for $x$ which do depend on $T$. Find the greatest solution for $x$ in this equation. (Hint: Find the independent solution for $x$ while you wait for $T$.) PS. You should use hide for answers.

Find the number of quadratic polynomials $ax^2 + bx +c$ which satisfy the following: (a) $a,b,c$ are distinct; (b) $a,b,c \in \{ 1,2,3,\cdots 1999 \}$; (c) $x+1$ divides $ax^2 + bx+c$.

Consider $6$ points located at $P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$. Let $R$ be the region consisting of [i]all[/i] points in the plane whose distance from $P_0$ is smaller than that from any other $P_i$, $i=1,2,3,4,5$. Find the perimeter of the region $R$.