Prove that \[\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\frac{2}{n}},\] and let $n \geq 1$ be an integer. Prove that this inequality is only possible in the case $n = 1.$

Prove that a function $f:\mathbb{R}_+\to\mathbb{R}$ satisfies \[f(x+y)-f(x)-f(y)=f\left(\frac{1}{x}+\frac{1}{y}\right)\]if and only if it satisfies $f(xy)=f(x)+f(y)$.

Let $P$ be a point inside a tetrahedron $ABCD$ and let $S_A,S_B,S_C,S_D$ be the centroids (i.e. centers of gravity) of the tetrahedra $PBCD,PCDA,PDAB,PABC$. Show that the volume of the tetrahedron $S_AS_BS_CS_D$ equals $1/64$ the volume of $ABCD$.

Adam has RM2010 in his bank account. He donates RM10 to charity every day. His first donation is on Monday. On what day will he donate his last RM10?

The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root? $\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $

Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property: For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$

Which one satisfies the equation $\sqrt[3]{6+\sqrt{x}} + \sqrt[3]{6-\sqrt{x}} = \sqrt[3]{3}$ ? $ \textbf{(A)}\ 27 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 63$

The squares of a chessboard are numbered in the following way. The upper left corner is numbered 1. The two squares on the next diagonal from top-right to bottom-left are numbered 2 and 3. The three squares on the next diagonal are numbered 4, 5 and 6, and so on. The two squares on the second-to-last diagonal are numbered 62 and 63, and the lower right corner is numbered 64. Peter puts eight pebbles on the squares of the chessboard in such a way that there is exactly one pebble in each column and each row. Then he moves each pebble to a square with a number greater than that of the original square. Can it happen that there is still exactly one pebble in each column and each row? [i](8 points)[/i]

Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$ (variation but same answer) [hide=Answer]3[/hide]

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

Points $M$, $N$, $P$ and $Q$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ of the inscribed quadrilateral $ABCD$ with intersection point $O$ of its diagonals. Let $K$ be the second intersection point of the circumscribed circles of $MOQ$ and $NOP$. Prove that $OK\perp AC$.

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

On the basis of VSEPR theory, what geometry is predicted for the central sulfur atom in $\ce{SOCl2}$? $ \textbf{(A) }\text{bent}\qquad\textbf{(B) }\text{T-shaped}\qquad\textbf{(C) }\text{trigonal planar} \qquad\textbf{(D) }\text{trigonal pyramidal} \qquad$

Find all real solutions of the equation $(x -4) (x^2 - 8x + 14)^2 = (x - 4)^3$.

The angle $\angle XOY =\alpha $ and the points $A$ and $B$ on OY are given such that $OA = a$ and $OB = b$ with $a > b$. A circle passes through the points $A$ and $B$ and is tangent to $OX$. a) Calculate the radius of that circle in terms of $a, b$ and $\alpha $. b) If $a$ and $b$ are constants and $\alpha $ varies, show that the minimum value of the radius of the circle is $\frac{a-b}{2}$.

Let there be given a set of $1996$ equal circles in the plane, no two of them having common interior points. Prove that there exists a circle touching at most three other circles.

Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.

$ 15 $ teams participate in a soccer league. Each team plays each of the remaining teams exactly once. If a team beats another team in a match they receive $ 3 $ points, while the loser receives $ 1 $ point. In the event of a tie, both teams receive $ 2 $ points. When all possible league matches are held, the following can be observed: $\bullet$ No two teams have finished with the same amount of points. $\bullet$ Each team finished the league with at least $ 21 $ points. Let $W$ be the team that finished the league with the highest score. Determine how many points $W$ scored and show that there were at least four ties in the league.

Let $ABCD$ be an isosceles trapezoid inscribed in circle $\omega$, such that $AD \| BC$. Point $E$ is chosen on the arc $BC$ of $\omega$ not containing $A$. Let $BC$ and $DE$ intersect at $F$. Show that if $E$ is chosen such that $EB = EC$, the area of $AEF$ is maximized.

Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.

Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1,a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$? ${ \textbf{(A)}\ \dfrac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \dfrac52\qquad\textbf{(C)}\ \dfrac{3+\sqrt{5}}{2}\qquad\textbf{(D)}}\ \dfrac{3+\sqrt{6}}{2}\qquad\textbf{(E)}\ 3 $

Prove that if equations $$x^2 + mx + n = 0 \,\,\,\, and\,\, \,\, x^2 + px + q = 0$$ have a common root, there is a relationship between the coefficients of these equations $$ (n - q)^2 - (m - p) (np - mq) = 0.$$

Evaluate $\sum^{15}_{k=0}\left(2^{560}(-1)^k \cos^{560}\left( \frac{k\pi}{16}\right)\right) \pmod{17}.$