OIMU 2009

A line through a vertex of a non-degenerate triangle cuts it in two similar triangles with $\sqrt{3}$ as the ratio between correspondent sides. Find the angles of the given triangle.

Let $x_1,\cdots, x_n$ be nonzero vectors of a vector space $V$ and $\varphi:V\to V$ be a linear transformation such that $\varphi x_1 = x_1$ , $\varphi x_k = x_k - x_{k-1}$ for $k = 2, 3,\ldots,n$ . Prove that the vectors $x_1,\ldots,x_n$ are linearly independent.

Let $a, b, c, d, e \in \mathbb{R}^+$ and $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$ be given by $f(x, y) = (ax)(by)(c- dx- ey)$ . Find the maximum value of $f$ .

Given two positive integers $m,n$ , we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$ -slippery if it has the following properties:

i) $f$ is continuous; ii) $f(0) = 0$ , $f(m) = n$ ; iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$ , then $t_2-t_1 \in \{0,m\}$ .

Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$ -slippery.

Let $\mathbb{N}$ and $\mathbb{N}^*$ be the sets containing the natural numbers/positive integers respectively.

We define a binary relation on $\mathbb{N}$ by $a\acute{\in}b$ iff the $a$ -th bit in the binary representation of $b$ is $1$ .

We define a binary relation on $\mathbb{N}^*$ by $a\tilde{\in}b$ iff $b$ is a multiple of the $a$ -th prime number $p_a$ .

i) Prove that there is no bijection $f:\mathbb{N}\to \mathbb{N}^*$ such that $a\acute{\in}b\Leftrightarrow f(a)\tilde{\in}f(b)$ . ii) Prove that there is a bijection $g:\mathbb{N}\to \mathbb{N}^*$ such that $(a\acute{\in}b \vee b\acute{\in}a)\Leftrightarrow (g(a)\tilde{\in}g(b) \vee g(b)\tilde{\in}g(a))$ .

Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials

$f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$

have integer coefficients.

Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$.

Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j)\right|=1$

Let $G$ be a group such that every subgroup of $G$ is subnormal. Suppose that there exists $N$ normal subgroup of $G$ such that $Z(N)$ is nontrivial and $G/N$ is cyclic. Prove that $Z(G)$ is nontrivial. ( $Z(G)$ denotes the center of $G$ ).Note: A subgroup $H$ of $G$ is subnormal if there exist subgroups $H_1,H_2,\ldots,H_m=G$ of $G$ such that $H\lhd H_1\lhd H_2 \lhd \ldots \lhd H_m= G$ ( $\lhd$ denotes normal subgroup).