Problems from the 1972 USAMO.

Problem 1

The symbols $(a,b,\ldots,g)$ and $[a,b,\ldots, g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $a,b,\ldots, g$ . For example, $(3,6,18)=3$ and $[6,15]=30$ . Prove that

$\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}$ .

Solution

Problem 2

A given tetrahedron $ABCD$ is isosceles, that is, $AB=CD, AC=BD, AD=BC$ . Show that the faces of the tetrahedron are acute-angled triangles.

Solution

Problem 3

A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ( $n>1$ ), the product of the $n$ numbers selected will be divisible by 10.

Solution

Problem 4

Let $R$ denote a non-negative rational number. Determine a fixed set of integers $a,b,c,d,e,f$ , such that for every choice of $R$ ,

$\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|$

Solution

Problem 5

A given convex pentagon $ABCDE$ has the property that the area of each of the five triangles $ABC$ , $BCD$ , $CDE$ , $DEA$ , and $EAB$ is unity. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.

$[asy] size(80); defaultpen(fontsize(7)); pair A=(0,7), B=(5,4), C=(3,0), D=(-3,0), E=(-5,4), P; P=extension(B,D,C,E); draw(D--E--A--B--C--D--B--E--C); label("A",A,(0,1));label("B",B,(1,0));label("C",C,(1,-1));label("D",D,(-1,-1));label("E",E,(-1,0));label("P",P,(0,1)); [/asy]$

Solution

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

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See also