Problem 1 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

For a real number $r$, let $[r]$ denote the maximal integer less or equal than $r$. Let $a$ and $b$ be two positive irrational numbers such that $\frac{1}{a} + \frac{1}{b} = 1$. Show that the two sequences of integers $[ax]$, $[bx]$ for $x = 1, 2, 3, \dots$ contain all natural numbers without repetition.

Problem 2 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $n \ge 2$ be an integer and consider the Fermat equation $$X^n + Y^n = Z^n,$$ $X, Y, Z \in \mathbb{C}[t]$. Find all nontrivial solution $(X,Y,Z)$ of the above equation in the sense that $X, Y, Z$ have no common zero and are not all constant.

Problem 3 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $p \ge 7$ be an odd prime number.
(a) Evaluate the rational number $\cos(\pi/7) \cdot \cos(2\pi/7) \cdot \cos(3\pi/7)$.
(b) Show that $\prod_{n=1}^{(p-1)/2} \cos(n\pi/p)$ is a rational number and determine its value.

Problem 4 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

For a positive integer $a$, consider the polynomial $$f_a = x^6 + 3 a x^4 + 3 x^3 + 3 a x^2 + 1.$$ Show that it is irreducible. Let $F$ be the splitting field of $f_a$. Show that its Galois group is solvable.

Problem 5 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Prove that a group of order $150$ is not simple.

Problem 6 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $V \cong \mathbb{C}^2$ be the standard representation of $SL_2(\mathbb{C})$.
(a) Show that the $n$-th symmetric power $V_n = \operatorname{Sym}^n V$ is irreducible.
(b) Which $V_n$ appear in the decomposition of the tensor product $V_2 \otimes V_3$ into irreducible representations?