1979 USAMO Problems

Problems from the 1979 USAMO.

Problem 1

Determine all non-negative integral solutions $(n_1,n_2,\dots , n_{14})$ if any, apart from permutations, of the Diophantine Equation $n_1^4+n_2^4+\cdots +n_{14}^4=1599$ .

Solution

Problem 2

$N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$ . $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides).

Solution

Problem 3

$a_1, a_2, \ldots, a_n$ 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 $\frac14$ .

Solution

Problem 4

$P$ lies between the rays $OA$ and $OB$ . Find $Q$ on $OA$ and $R$ on $OB$ collinear with $P$ so that $\frac{1}{PQ} + \frac{1}{PR}$ is as large as possible.

Solution

Problem 5

Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $[n]$ with $|A_1|=|A_2|=\cdots =|A_{n+1}|=3$ . Prove that $|A_i\cap A_j|=1$ for some pair $\{i,j\}$ . Note that $[n] = \{1, 2, 3, ..., n\}$ , or, alternatively, $\{x: 1 \le x \le n\}$ .

Solution

1979 USAMO (Problems • Resources)
Preceded by 1978 USAMO	Followed by 1980 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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1979 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also