Difference between revisions of "1978 USAMO Problems"

Revision as of 14:26, 17 September 2012

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

There are 9 delegates at a conference, each speaking at most three languages. Given any three delegates, at least 2 speak a common language. Show that there are three delegates with a common language.

Solution

@@ Line 41: / Line 41: @@
 ==Problem 4==
-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?
+(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?
 [[1978 USAMO Problems/Problem 4 | Solution]]

1978 USAMO (Problems • Resources)
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Difference between revisions of "1978 USAMO Problems"

Revision as of 14:26, 17 September 2012

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also