Problems from the 1973 USAMO.

Problem 1

Two points $P$ and $Q$ lie in the interior of a regular tetrahedron $ABCD$ . Prove that angle $PAQ<60^o$ .

Problem 2

Let $\{X_n\}$ and $\{Y_n\}$ denote two sequences of integers defined as follows:

$X_0=1$ , $X_1=1$ , $X_{n+1}=X_n+2X_{n-1}$ $(n=1,2,3,\dots),$ $Y_0=1$ , $Y_1=7$ , $Y_{n+1}=2Y_n+3Y_{n-1}$ $(n=1,2,3,\dots)$ .

Thus, the first few terms of the sequences are:

$X:1, 1, 3, 5, 11, 21, \dots$ , $Y:1, 7, 17, 55, 161, 487, \dots$ .

Prove that, except for the "1", there is no term which occurs in both sequences.

Solution

Problem 3

Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?

Solution

Problem 4

Determine all the roots, real or complex, of the system of simultaneous equations

$x+y+z=3$ ,

$x^2+y^2+z^2=3$ ,

$x^3+y^3+z^3=3$ .

Solution

Problem 5

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

Solution

1973 USAMO (Problems • Resources)
Preceded by 1972 USAMO	Followed by 1974 USAMO
1 • 2 • 3 • 4 • 5 • 6
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1973 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See also