Circle $\omega$ has radius 10 with center $O$. Let $P$ be a point such that $PO=6$. Let the midpoints of all chords of $\omega$ through $P$ bound a region of area $R$. Find the value of $\lfloor 10R \rfloor$. [i]Proposed by Andrew Zhao[/i]

The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.

Let \[\begin{array}{ccccccccccc}A&=&5\cdot 6&-&6\cdot 7&+&7\cdot 8&-&\cdots&+&2003\cdot 2004,\\B&=&1\cdot 10&-&2\cdot 11&+&3\cdot 12&-&\cdots&+&1999\cdot 2008.\end{array}\] Find the value of $A-B$.

A toy factory produces several kinds of clay toys. The toys are painted in $k$ colours. [i]Diversity[/i] of a colour is the number of [i]different[/i] toys of that colour. (Thus, if there are $5$ blue cats, $7$ blue mice and nothing else is blue, the diversity of colour blue is $2$.) The painting protocol requires that [i]each colour is used and the diversities of each two colours are different[/i]. The toys in the store could be painted according to the protocol. However, a batch of clay Cheburashkas arrived at the store before painting (there were no Cheburashkas before). The number of Cheburashkas is not less that the number of the toys of any other kind. The total number of all toys, including Cheburashkas, is at least $\frac{(k+1)(k+2)}{2}$. Prove that now the toys can be painted in $k + 1$ colours according to the protocol. [i]Proposed by F. Petrov[/i]

The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )

Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers n. Prove that f is a polynomial.

Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80-320i$, $yz = 60$, and $zx = -96+24i$, where $i = \sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x+y+z = a+bi$. Find $a^2 + b^2$.

A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^3$ . Find the sides of the initial trapezoid.

Prove that for arbitary positive integer $ n\geq 4$, there exists a permutation of the subsets that contain at least two elements of the set $ G_{n} \equal{} \{1,2,3,\cdots,n\}$: $ P_{1},P_{2},\cdots,P_{2^n \minus{} n \minus{} 1}$ such that $ |P_{i}\cap P_{i \plus{} 1}| \equal{} 2,i \equal{} 1,2,\cdots,2^n \minus{} n \minus{} 2.$

Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$, $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$, where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$. Prove that the set $A$ has [i]exactly[/i] two elements. [i]Marius Gherghu, Slatina[/i]