1979 USAMO Problems/Problem 3

Problem

$a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is $a$ . Another member is picked at random, independently of the first. Its value is $b$ . Then a third value, $c$ . Show that the probability that $a + b +c$ is divisible by $3$ is at least $\frac14$ .

First Hint

The given problem is equivalent to proving that $4(x^3 + y^3 + z^3 + 6xyz) \ge (x + y + z)^3$ .

Second Hint

What do you do with homogenous inequalities? (A function $f$ is homogenous of degree $d$ if $f(kx) = k^df(x)$ for all x in the domain of $f$ .)

Solution

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1979 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5
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1979 USAMO Problems/Problem 3

Contents

Problem

First Hint

Second Hint

Solution

See Also