4

Part of 1997 Irish Math Olympiad

Problems(2)

easy inequality

Source: Ireland 1997

a,b,c

be nonnegative real numbers. Suppose that a\plus{}b\plus{}c\ge abc. Prove that: a^2\plus{}b^2\plus{}c^2 \ge abc.

inequalitiesinequalities proposed

determine the number of elements

Source: Ireland 1997

S

be the set of natural numbers

n

satisfying the following conditions:

(i)

n

1000

(ii)

all the digits of

n

(iii)

any two adjacent digits of

n

2

. Determine the number of elements of

S

combinatorics unsolvedcombinatorics