TH-TMO 2007

Let $a, b, c$ be complex numbers such that $a+b+c = 1$ , $a^2+b^2+c^2 = 2$ and $a^3+b^3+c^3 = 3$ . Find the value of $a^4 + b^4 + c^4$ .

Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$ . Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$ .

Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$ . What is the value of $f(85)$ ?

Find the smallest positive integer $n$ such that the equation $\sqrt3 z^{n+1} - z^n - 1 = 0$ has a root on the unit circle.

Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$ .

An alien with four feet wants to wear four identical socks and four identical shoes, where on each foot a sock must be put on before a shoe. How many ways are there for the alien to wear socks and shoes?

Let $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of $S$ ?

The sum $$\sum_{k=84}^{8000}{k \choose 84}{{8084 - k} \choose 84}$$ can be written as a binomial coefficient $a \choose b$ for integers $a, b$ . Find a possible pair $(a, b)$

Compute the remainder when $222!^{111} + 111^{222!} + 111!^{222} + 222^{111!}$ is divided by $2007$ .

What is the smallest positive integer with $24$ positive divisors?

Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.

Let $p_k$ be the $k$ -th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$ .