At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? $\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

Find the product of the non-real roots of the equation \[(x^2-3)^2+5(x^2-3)+6=0.\] $\begin{array}{@{\hspace{0em}}l@{\hspace{13.7em}}l@{\hspace{13.7em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }-1\\\\ \textbf{(D) }2&\textbf{(E) }-2&\textbf{(F) }3\\\\ \textbf{(G) }-3&\textbf{(H) }4&\textbf{(I) }-4\\\\ \textbf{(J) }5&\textbf{(K) }-5&\textbf{(L) }6\\\\ \textbf{(M) }-6&\textbf{(N) }3+2i&\textbf{(O) }3-2i\\\\ \textbf{(P) }\dfrac{-3+i\sqrt3}2&\textbf{(Q) }8&\textbf{(R) }-8\\\\ \textbf{(S) }12&\textbf{(T) }-12&\textbf{(U) }42\\\\ \textbf{(V) }\text{Ying} & \textbf{(W) }2007 &\end{array}$

There is a list of $n$ football matches. Determine how many possible columns, with an even number of draws, there are.

Prove there exist positive integers a,b,c for which $a+b+c=1998$, the gcd is maximized, and $0<a<b\leq c<2a$. Find those numbers. Are they unique?

Let $p$ be an odd prime number. Let $g$ be a primitive root of unity modulo $p$. Find all the values of $p$ such that the sets $A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\}$ and $B=\left\{g^m:1\le m\le\frac{p-1}2\right\}$ are equal modulo $p$.

Evaluate the following definite integrals: (a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$ (b) $\int_0^{\frac 13} xe^{3x}\ dx$ (c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$ (d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

Let $a$ and $b$ be positive integers. Show that if $a \cdot b$ is even, then there are positive integers $c$ and $d$ with $a^2 + b^2 + c^2 = d^2$; if, on the other hand, $a\cdot b$ is odd, there are no such positive integers $c$ and $d$.

Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that $$f\left(xy+f(y)\right)f(x)=x^2f(y)+f(xy)$$ for all $x,y \in \mathbb{R}$ [i]Proposed by chengbilly[/i]

Determine if there exist positive integers $a_1,a_2,...,a_{10}$, $b_1,b_2,...,b_{10}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,10\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 12+\sum\limits_{i\in S}b_i \right)$.

For each real number $a$ and integer $n \ge 1$ prove the inequality $a^n +\frac{1}{a^n} -2 \ge n^2 \left(a +\frac{1}{a} -2\right)$ and find the cases of equality.

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Let $a, b, n$ be positive integers with $\gcd(a, b)=1$. Prove that \[\sum_{k}\left\{ \frac{ak+b}{n}\right\}=\frac{n-1}{2},\] where $k$ runs through a complete system of residues modulo $m$.

a) Given $555$ weights: of $1$ g, $2$ g, $3$ g, . . . , $555$ g, divide them into three piles of equal mass. b) Arrange $81$ weights of $1^2, 2^2, . . . , 81^2$ (all in grams) into three piles of equal mass.

What is the sum of the possible values for the complex number $a$ such that the coefficient of the $x^5$ term in the power series expansion of $\tfrac{x^3+ax^2+3x-4}{2x^2+ax+2}$ is $1?$

Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.

If $\log_{a}\sqrt2<1$, then the range value of $a$ is________.

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled. [i]Proposed by Alexander A. Polyansky, Russia[/i]

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.

Let $A, B, C, D$ be four distinct points in the plane such that the length of the six line segments $AB, AC, AD, BC, BD, CD$ form a $2$-element set ${a, b}$. If $a > b$, determine all the possible values of $\frac ab$.

The plane is painted in red or blue color. Prove that you have a rectangle with the corners of the same color.

Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $x,y\in{{\mathbb{R}}}$ holds $f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$ [i]Proposed by Matija Bucić[/i]

Given points $P_1, P_2,\cdots,P_7$ on a straight line, in the order stated (not necessarily evenly spaced). Let $P$ be an arbitrarily selected point on the line and let $s$ be the sum of the undirected lengths \[PP_1, PP_2, \cdots , PP_7.\] Then $s$ is smallest if and only if the point $P$ is: $\textbf{(A)}\ \text{midway between }P_1\text{ and }P_7\qquad \textbf{(B)}\ \text{midway between }P_2\text{ and }P_6\qquad \textbf{(C)}\ \text{midway between }P_3\text{ and }P_5 \qquad$ $ \textbf{(D)}\ \text{at }P_4 \qquad \textbf{(E)}\ \text{at }P_1$

We call a set “sum free” if no two elements of the set add up to a third element of the set. What is the maximum size of a sum free subset of $\{ 1, 2, \ldots , 2n - 1 \}$.

Find all pairs of positive integers $(x,y)$ with the following property: If $a,b$ are relative prime and positive divisors of $ x^3 + y^3$, then $a+b - 1$ is divisor of $x^3+y^3$. (Cyprus)

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.