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Difference between revisions of "1979 USAMO Problems"

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==Problem 3==
 
==Problem 3==
 
<math>a_1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at  
 
<math>a_1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at  
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random. Its value is <math>a</math>. Another member is picked at random, independently of the first. Its value is <math>b</math>. Then a third value, <math>c</math>. Show that the probability that <math>a \plus{ } b \plus{ } c</math> is divisible by <math>3</math> is at least <math>\frac14</math>.
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random. Its value is <math>a</math>. Another member is picked at random, independently of the first. Its value is <math>b</math>. Then a third value, <math>c</math>. Show that the probability that <math>a + b + c</math> is divisible by <math>3</math> is at least <math>\frac14</math>.
  
 
[[1979 USAMO Problems/Problem 3 | Solution]]
 
[[1979 USAMO Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
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<math>P</math> lies between the rays <math>OA</math> and <math>OB</math>. Find <math>Q</math> on <math>OA</math> and <math>R</math> on <math>OB</math> collinear with <math>P</math> so that <math>\frac{1}{PQ}\plus{} \frac{1}{PR}</math> is as large as possible.
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<math>P</math> lies between the rays <math>OA</math> and <math>OB</math>. Find <math>Q</math> on <math>OA</math> and <math>R</math> on <math>OB</math> collinear with <math>P</math> so that <math>\frac{1}{PQ} + \frac{1}{PR}</math> is as large as possible.
  
 
[[1979 USAMO Problems/Problem 4 | Solution]]
 
[[1979 USAMO Problems/Problem 4 | Solution]]
  
 
==Problem 5==
 
==Problem 5==
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Let <math>A_1,A_2,...,A_{n+1}</math> be distinct subsets of <math>[n]</math> with <math>|A_1|=|A_2|=\cdots =|A_n|=3</math>.  Prove that <math>|A_i\cap A_j|=1</math> for some pair <math>\{i,j\}</math>.
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Let <math>A_1,A_2,...,A_{n+1}</math> be distinct subsets of <math>[n]</math> with <math>|A_1|=|A_2|=\cdots =|A_{n+1}|=3</math>.  Prove that <math>|A_i\cap A_j|=1</math> for some pair <math>\{i,j\}</math>. Note that <math>[n] = \{1, 2, 3, ..., n\}</math>, or, alternatively, <math>\{x: 1 \le x \le n\}</math>.
  
 
[[1979 USAMO Problems/Problem 5 | Solution]]
 
[[1979 USAMO Problems/Problem 5 | Solution]]
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== See Also ==
 
== See Also ==
 
{{USAMO box|year=1979|before=[[1978 USAMO]]|after=[[1980 USAMO]]}}
 
{{USAMO box|year=1979|before=[[1978 USAMO]]|after=[[1980 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 12:04, 24 December 2015

Problems from the 1979 USAMO.

Problem 1

Determine all non-negative integral solutions $(n_1,n_2,\dots , n_{14})$ if any, apart from permutations, of the Diophantine Equation $n_1^4+n_2^4+\cdots +n_{14}^4=1599$.

Solution

Problem 2

$N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$. $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides).

Solution

Problem 3

$a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b + c$ is divisible by $3$ is at least $\frac14$.

Solution

Problem 4

$P$ lies between the rays $OA$ and $OB$. Find $Q$ on $OA$ and $R$ on $OB$ collinear with $P$ so that $\frac{1}{PQ} + \frac{1}{PR}$ is as large as possible.

Solution

Problem 5

Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $[n]$ with $|A_1|=|A_2|=\cdots =|A_{n+1}|=3$. Prove that $|A_i\cap A_j|=1$ for some pair $\{i,j\}$. Note that $[n] = \{1, 2, 3, ..., n\}$, or, alternatively, $\{x: 1 \le x \le n\}$.

Solution

See Also

1979 USAMO (Problems • Resources)
Preceded by
1978 USAMO 		Followed by
1980 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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