TH-TMO 2006

Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$ . Find $x$ .

Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$ , and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$ . Compute $s^4 -18s^2 - 8s$ .

Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$

Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$ .

Let $p_n$ be the $n$ -th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$ . Compute gcd $(a_1, a_2, ... , a_{25})$

Compute the remainder when $\underbrace{\hbox{11...1}}_{\hbox{1862}}$ is divided by $2006$

Find the smallest positive integer $n$ such that $2549 | n^{2545} - 2$ .

How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer rooot?

Find the number of triples of sets $(A, B, C)$ such that $A \cup B \cup C = \{1, 2, 3, ... , 2549\}$

Six people, with distinct weights, want to form a triangular position where there are three people in the bottom row, two in the middle row, and one in the top row, and each person in the top two rows must weigh less than both of their supports. How many distinct formations are there?

In May, the traffic police wants to select 10 days to patrol, but no two consecutive days can be selected. How many ways are there for the traffic police to select patrol days?