1925 Eotvos Mathematical Competition

Part of Eotvos Mathematical Competition (Hungary)

Subcontests

(3)

1

12 divides (b - a)(c - a)(d - a)(d - c)(d - b)( c - b)

a,b, c,d

be four integers. Prove that the product of the six differences

b - a,c - a,d - a,d - c,d - b, c - b

is divisible by

12

1

maay zeros are at the end of the number 1000!

How maay zeros are there at the end of the number

1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?

1

iradius of right triangle <1/2 b and <1/4 a

r

be the radius of the inscribed circle of a right triangle

ABC

r

is less than half of either leg and less than one fourth of the hypotenuse.