MK-MO 1995

Let $a_0$ be a real number. The sequence $\{a_n \}$ is given by $a_ {n + 1} = 3 ^ n-5a_n$ , $n = 0,1,2, \ldots$ . a) Express the general member $a_n$ through $a_0$ and $n.$ b) Find such $a_0,$ that $a_ {n + 1}> a_n,$ for every $n.$

Let $a,$ $b$ , and $c$ be sides in a triangle, a $h_a,$ $h_b$ , and $h_c$ are the corresponding altitudes. Prove that $h ^ 2_a + h ^ 2_b + h ^ 2_c \leq \frac{3}{4} (a ^ 2 + b ^ 2 + c ^ 2).$ When is the equation valid?

Prove that the product of $8$ consecutive natural numbers can never be a fourth power of natural number.

On a $30 \times30$ square board or placed figures of shape 1 (of 5 squares) (in all four possible positions) and shaped figures of shape 2 (of 4 squares) . The figures do not overlap, they do not pass through the edges of the board and the squares of which they are drawn lie exactly through the squares of the board. a) Prove that the board can be fully covered using $100$ figures of both shapes. b) Prove that if there are already $50$ shaped figures on the board of shape 1, then at least one more figure can be placed on the board. c) Prove that if there are already $28$ figures of both shapes on the board then at least one more figure of both shapes can be placed on the board.