MathDB

NT game, nxn board (Ukraine 1998 Grade 9 P7)

Source:

6/5/2021

Two players alternately write numbers in the cells of an

n\times n

square board. Hereby, in the intersection of the

i

-th row and the

j

-th column the first player may write the greatest common divisor of numbers

i

and

j

, whereas the second player may write their least common multiple. When the board is filled up, the numbers of the first column are divided by

1

, those of the second column by

2

, etc., those of the last column by

n

. Then the product of the obtained numbers in the board is computed. If the result is smaller than

1

, the first player wins, otherwise the second player wins. Which player has a winning strategy?

number theorycombinatoricsgame

finite number of infinite rooms

Source: Ukraine 1998 Grade 10 P7

6/5/2021

Baron Munchausen claims that he can accomodate an arbitrary set of guests in the rooms of his castle in such a way that the guests in each room either all know each other or all do not know each other. Is his claim true? (Assume that the rooms are large enough to accomodate any number of people, but that there are finitely many rooms.)

infinityset theoryBaron Munchausen

externally tangent spheres

Source: Ukraine 1998 Grade 11 P7

6/6/2021

Two spheres are externally tangent at point

P

. The segments

AB

and

CD

touch the spheres with

A

and

C

lying on the first sphere and

B

and

D

on the second. Let

M

and

N

be the projections of the midpoints of segments

AC

and

BD

on the line connecting the centers of the spheres. Prove that

PM=PN

.

spheregeometry3D geometry

Problem 7

Problems(3)