MathDB

1

largest possible total length of the walls, on a mza of n^2 unit square cells

n

is divided into

n^2

unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell not crossing any wall. Find, in terms of

n

, the largest possible total length of the walls.

6

1

repeatedly draw out the balls one by one, 3 balls, coloring

There are three balls of distinct colors in a bag. We repeatedly draw out the balls one by one, the balls are put back into the bag after each drawing. What is the probability that, after

n

drawings, (a) exactly one color occured? (b) exactly two oclors occured? (c) all three colors occured?

5

1

exist different indices r,s such that a_r \ge a_s and b_r \ge b_s, over N

Let

a_1,a_2,...

and

b_1,b_2,..

. be two arbitrary infinite sequences of natural numbers. Prove that there exist different indices

r

and

s

such that

a_r \ge a_s

and

b_r \ge b_s

.

4

1

I_n is set of solutions x^2 −2xy+y^2−4^n = 0 , where y \in I_{n-1}

Given

I_0 = \{-1,1\}

, define

I_n

recurrently as the set of solutions

x

of the equations

x^2 -2xy+y^2- 4^n = 0

, where

y

ranges over all elements of

I_{n-1}

. Determine the union of the sets

I_n

over all nonnegative integers

n

.

3

1

construct right triangle given R and r

Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.

2

1

regular tetrahedron wanted

A tetrahedron has the property that the three segments connecting the pairs of midpoints of opposite edges are equal and mutually orthogonal. Prove that this tetrahedron is regular.

1

3x^5 +5x^3 -8x is divisible by 120

Show that

3x^5 +5x^3 -8x

is divisible by

120

for any integer

x

1987 ITAMO

Subcontests

largest possible total length of the walls, on a mza of n^2 unit square cells

repeatedly draw out the balls one by one, 3 balls, coloring

exist different indices r,s such that a_r \ge a_s and b_r \ge b_s, over N

I_n is set of solutions x^2 −2xy+y^2−4^n = 0 , where y \in I_{n-1}

construct right triangle given R and r

regular tetrahedron wanted

3x^5 +5x^3 -8x is divisible by 120

1987 ITAMO

Subcontests

largest possible total length of the walls, on a mza of n^2 unit square cells

repeatedly draw out the balls one by one, 3 balls, coloring

exist different indices r,s such that a_r \ge a_s and b_r \ge b_s, over N

I_n is set of solutions x^2 &minus;2xy+y^2&minus;4^n = 0 , where y \in I_{n-1}

construct right triangle given R and r

regular tetrahedron wanted

3x^5 +5x^3 -8x is divisible by 120

I_n is set of solutions x^2 −2xy+y^2−4^n = 0 , where y \in I_{n-1}