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Difference between revisions of "1972 USAMO Problems"

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Problems from the '''1972 [[United States of America Mathematical Olympiad | USAMO]]'''.
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Problems from the '''1972 [[United States of America Mathematical Olympiad|USAMO]]'''.
  
 
==Problem 1==
 
==Problem 1==
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<center><math>\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}</math>.</center>
 
<center><math>\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}</math>.</center>
  
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[[1972 USAMO Problems/Problem 1 | Solution]]
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[[1972 USAMO Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
 
A given tetrahedron <math>ABCD</math> is isosceles, that is, <math>AB=CD, AC=BD, AD=BC</math>. Show that the faces of the tetrahedron are acute-angled triangles.
 
A given tetrahedron <math>ABCD</math> is isosceles, that is, <math>AB=CD, AC=BD, AD=BC</math>. Show that the faces of the tetrahedron are acute-angled triangles.
  
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[[1972 USAMO Problems/Problem 2 | Solution]]
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[[1972 USAMO Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==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 <math>n</math> selections (<math>n>1</math>), the product of the <math>n</math> numbers selected will be divisible by 10.
 
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 <math>n</math> selections (<math>n>1</math>), the product of the <math>n</math> numbers selected will be divisible by 10.
  
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[[1972 USAMO Problems/Problem 3 | Solution]]
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[[1972 USAMO Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
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<center><math>\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|</math></center>
 
<center><math>\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|</math></center>
  
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[[1972 USAMO Problems/Problem 4 | Solution]]
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[[1972 USAMO Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
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<asy>
 
<asy>
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size(80);
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size(120);
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defaultpen(fontsize(7));
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defaultpen(fontsize(10));
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pair A=(0,7), B=(5,4), C=(3,0), D=(-3,0), E=(-5,4), P;
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pair A=dir(90), B=dir(90-72), C=dir(90-2*72), D=dir(90-3*72), E=dir(90-4*72);
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P=extension(B,D,C,E);
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draw(A--B--C--D--E--cycle);
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draw(D--E--A--B--C--D--B--E--C);
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draw(A--C--E--B--D--cycle);
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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));
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label("A",A,A);label("B",B,B);label("C",C,C);label("D",D,D);label("E",E,E);
 
</asy>
 
</asy>
  
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[[1972 USAMO Problems/Problem 5 | Solution]]
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[[1972 USAMO Problems/Problem 5|Solution]]
  
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= See also =
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== See Also ==
 
{{USAMO box|year=1972|before=First USAMO|after=[[1973 USAMO]]}}
 
{{USAMO box|year=1972|before=First USAMO|after=[[1973 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 14:28, 2 June 2018

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(120); defaultpen(fontsize(10)); pair A=dir(90), B=dir(90-72), C=dir(90-2*72), D=dir(90-3*72), E=dir(90-4*72); draw(A--B--C--D--E--cycle); draw(A--C--E--B--D--cycle); label("A",A,A);label("B",B,B);label("C",C,C);label("D",D,D);label("E",E,E); [/asy]

Solution

See Also

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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