2011 District Round (Round II)

Part of District Round (Round II)

Subcontests

(4)

1

Changing numbers on a cube so that they're all equal

M

be a set of six distinct positive integers whose sum is

60

. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding

1

to the numbers on those faces. Determine the number of sets

M

for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.

1

Find m & n such that (m+n)^2 divides 4(mn+1)

(m, n)

of positive integers for which

4 (mn +1)

is divisible by

(m + n)^2

1

Eight-digit multiples of 4 with & without the digit 1

Among all eight-digit multiples of four, are there more numbers with the digit

1

or without the digit

1

in their decimal representation?

1

Hexagon with twice the area of a triangle

ABC

denote a triangle with area

S

U

be any point inside the triangle whose vertices are the midpoints of the sides of triangle

ABC

A'

B'

C'

denote the reflections of

A

B

C

, respectively, about the point

U

. Prove that hexagon

AC'BA'CB'

2S