MathDB

1

rotate a cube around diagonal 60^o, volume of intersection wanted

C

is rotated around one of its diagonals for the angle

\pi /3

to form a cube

C'

. Find the volume of the intersection of

C

and

C'

.

4

1

triangle of projections of P on perp. bisectors of ABC similar to ABC

Let

P

be a point in the plane of a triangle

ABC

, different from its circumcenter. Prove that the triangle whose vertices are the projections of

P

on the perpendicular bisectors of the sides of

ABC

, is similar to

ABC

.

3

1

two sequences of coins in infinite chessboard , a_n in n row, b_n in n column

Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences (

a_n

) and (

b_n

) of positive integers (

n \in N

). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are

a_n

coins in the

n

-th row and

b_n

coins in the

n

-th column for all

n

.

2

1

3x^2 - px + q = 0 , where p.q are primes and has two rational solutions

Find all pairs

(p,q)

of positive primes such that the equation

3x^2 - px + q = 0

has two distinct rational roots.

1

area of region bounded by 3 semicircles

Let be given points

A,B,C

on a line, with

C

between

A

and

B

. Three semicircles with diameters

AC,BC,AB

are drawn on the same side of line

ABC

. The perpendicular to

AB

at

C

meets the circle with diameter

AB

at

H

. Given that

CH =\sqrt2

, compute the area of the region bounded by the three semicircles.

5

1

Muirhead theorem??? (Italy 1993)

Prove the following inequality for any positive real numbers a,b,c not exceeding 1

a^2b+b^2c+c^2a+1\ge a^2+b^2+c^2

1993 ITAMO

Subcontests

rotate a cube around diagonal 60^o, volume of intersection wanted

triangle of projections of P on perp. bisectors of ABC similar to ABC

two sequences of coins in infinite chessboard , a_n in n row, b_n in n column

3x^2 - px + q = 0 , where p.q are primes and has two rational solutions

area of region bounded by 3 semicircles

Muirhead theorem??? (Italy 1993)