The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.) [b](a)[/b] Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$. [b](b)[/b] Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point. Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.

The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$. Đỗ Thanh Sơn

After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$

One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 \plus{} 110^5 \plus{} 84^5 \plus{} 27^5 \equal{} n^5.\] Find the value of $ n$.

(Mathematicians A to Z) Below are the names of 26 mathematicians, one for each letter of the alphabet. Your answer to this question should be a subset of $\{A,B,\cdots,Z\}$, where each letter represents the corresponding mathematician. If two mathematicians in your subset have birthdates that are within $20$ years of each other, then your score is $0$. Otherwise, your score is $\max(3(k-3),0)$ where $k$ is the number of elements in your set. \[\begin{tabular}{cc}Niels Abel & Isaac Newton\\Etienne Bezout & Nicole Oresme \\ Augustin-Louis Cauchy & Blaise Pascal \\ Rene Descartes & Daniel Quillen \\ Leonhard Euler & Bernhard Riemann\\ Pierre Fatou & Jean-Pierre Serre \\ Alexander Grothendieck & Alan Turing \\ David Hilbert & Stanislaw Ulam \\ Kenkichi Iwasawa & John Venn \\ Carl Jacobi & Andrew Wiles \\ Andrey Kolmogorov & Leonardo Ximenes \\ Joseph-Louis Lagrange & Shing-Tung Yau \\ John Milnor & Ernst Zermelo\end{tabular}\]