Given a natural number $n> 3$. On the plane are considered convex $n$ - gons $F_1$ and $F_2$ such that on each side of $F_1$ lies one vertex of $F_2$ and no two vertices $F_1$ and $F_2$ coincide. For each $n$, determine the limits of the ratio of the areas of the polygons $F_1$ and $F_2$. For each $n$, determine the range of the areas of the polygons $F_1$ and $F_2$, if $F_1$ is a regular $n$-gon. Determine the set of values of this value for other partial cases of the polygon $F_1$.

Two (not necessarily different) numbers are chosen independently and at random from $\{1, 2, 3, \dots, 10\}$. On average, what is the product of the two integers? (Compute the expected product. That is, if you do this over and over again, what will the product of the integers be on average?)

Michael is at the centre of a circle of radius $100$ metres. Each minute, he will announce the direction in which he will be moving. Catherine can leave it as is, or change it to the opposite direction. Then Michael moves exactly $1$ metre in the direction determined by Catherine. Does Michael have a strategy which guarantees that he can get out of the circle, even though Catherine will try to stop him?

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

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]