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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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<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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<asy>
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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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An integer <math>n</math> will be called ''good'' if we can write
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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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(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 19:06, 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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