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Difference between revisions of "1978 USAMO Problems"

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Problems from the '''1978 [[United States of America Mathematical Olympiad | USAMO]]'''.
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==Problem 1==
 
==Problem 1==
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Given that <math>a,b,c,d,e</math> are real numbers such that
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<math>a+b+c+d+e=8</math>,
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<math>a^2+b^2+c^2+d^2+e^2=16</math>.
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Determine the maximum value of <math>e</math>.
  
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The sum of 5 real numbers is 8 and the sum of their squares is 16. What is the largest possible
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[[1978 USAMO Problems/Problem 1 | Solution]]
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value for one of the numbers?
 
  
 
==Problem 2==
 
==Problem 2==
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Two square maps cover exactly the same area of terrain on different scales. The smaller map is
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<math>ABCD</math> and <math>A'B'C'D'</math> are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point <math>O</math> on the small map that lies directly over point <math>O'</math> of the large map such that <math>O</math> and <math>O'</math> each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for <math>O</math>.
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placed on top of the larger map and inside its borders. Show that there is a unique point on the top
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map which lies exactly above the corresponding point on the lower map. How can this point be
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<asy>
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constructed?
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defaultpen(linewidth(0.7)+fontsize(10));
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real theta = -100, r = 0.3; pair D2 = (0.3,0.76);
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string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare);
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for(int i = 0; i < lbl.length; ++i) {
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pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5;
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label("$"+lbl[i]+"'$", P, Q);
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label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q);
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}</asy>
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[[1978 USAMO Problems/Problem 2 | Solution]]
  
 
==Problem 3==
 
==Problem 3==
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You are told that all integers from <math>33</math> to <math>73</math> inclusive can be expressed as a sum of positive
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An integer <math>n</math> will be called ''good'' if we can write
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integers whose reciprocals sum to 1. Show that the same is true for all integers greater than <math>73</math>.
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<math>n=a_1+a_2+\cdots+a_k</math>,
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where <math>a_1,a_2, \ldots, a_k</math> are positive integers (not necessarily distinct) satisfying
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<math>\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_k}=1</math>.
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Given the information that the integers 33 through 73 are good, prove that every integer <math>\ge 33</math> is good.
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[[1978 USAMO Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
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Show that if the angle between each pair of faces of a tetrahedron is equal, then the tetrahedron is regular. Does a tetrahedron have to be regular if five of the angles are equal?
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(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
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(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
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[[1978 USAMO Problems/Problem 4 | Solution]]
  
 
==Problem 5==
 
==Problem 5==
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Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.
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[[1978 USAMO Problems/Problem 5 | Solution]]
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== See Also ==
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{{USAMO box|year=1978|before=[[1977 USAMO]]|after=[[1979 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 18:06, 3 July 2013

Problems from the 1978 USAMO.

Problem 1

Given that $a,b,c,d,e$ are real numbers such that

$a+b+c+d+e=8$,

$a^2+b^2+c^2+d^2+e^2=16$.

Determine the maximum value of $e$.

Solution

Problem 2

$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.

[asy] defaultpen(linewidth(0.7)+fontsize(10)); real theta = -100, r = 0.3; pair D2 = (0.3,0.76); string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare); for(int i = 0; i < lbl.length; ++i) { pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5; label("$"+lbl[i]+"'$", P, Q); label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]

Solution

Problem 3

An integer $n$ will be called good if we can write

$n=a_1+a_2+\cdots+a_k$,

where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying

$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_k}=1$.

Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.

Solution

Problem 4

(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.

(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?

Solution

Problem 5

Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.

Solution

See Also

1978 USAMO (Problems • Resources)
Preceded by
1977 USAMO 		Followed by
1979 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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