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Difference between revisions of "1995 USAMO Problems"

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==Problem 3==
 
==Problem 3==
  
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Given a nonisosceles, nonright triangle <math>\, ABC, \,</math> let <math>\, O \,</math> denote the center of its circumscribed circle, and let <math>\, A_1, \, B_1, \,</math> and <math>\, C_1 \,</math> be the midpoints of sides <math>\, BC, \, CA, \,</math> and <math>\, AB, \,</math> respectively.  Point <math>\, A_2 \,</math> is located on the ray <math>\, OA_1 \,</math> so that <math>\, \Delta OAA_1 \,</math> is similar to <math>\, \Delta OA_2A</math>.  Points <math>\, B_2 \,</math> and <math>\, C_2 \,</math> on rays <math>\, OB_1 \,</math> and <math>\, OC_1, \,</math> respectively, are defined similarly.  Prove that lines <math>\, AA_2, \, BB_2, \,</math> and <math>\, CC_2 \,</math> are concurrent, i.e. these three lines intersect at a point.
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Given a nonisosceles, nonright triangle <math>\, ABC, \,</math> let <math>\, O \,</math> denote the center of its circumscribed circle, and let <math>\, A_1, \, B_1, \,</math> and <math>\, C_1 \,</math> be the midpoints of sides <math>\, BC, \, CA, \,</math> and <math>\, AB, \,</math> respectively.  Point <math>\, A_2 \,</math> is located on the ray <math>\, OA_1 \,</math> so that <math>\, \triangle OAA_1 \,</math> is similar to <math>\, \triangle OA_2A</math>.  Points <math>\, B_2 \,</math> and <math>\, C_2 \,</math> on rays <math>\, OB_1 \,</math> and <math>\, OC_1, \,</math> respectively, are defined similarly.  Prove that lines <math>\, AA_2, \, BB_2, \,</math> and <math>\, CC_2 \,</math> are concurrent, i.e. these three lines intersect at a point.
  
 
[[1995 USAMO Problems/Problem 3|Solution]]
 
[[1995 USAMO Problems/Problem 3|Solution]]
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[[1995 USAMO Problems/Problem 5|Solution]]
 
[[1995 USAMO Problems/Problem 5|Solution]]
  
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== Resources ==
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== See Also ==
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{{USAMO box|year=1995|before=[[1994 USAMO]]|after=[[1996 USAMO]]}}
 
{{USAMO box|year=1995|before=[[1994 USAMO]]|after=[[1996 USAMO]]}}
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* [http://www.artofproblemsolving.com/resources.php?c=182&cid=27&year=1995 1995 USAMO Problems on the resources page]
 
 
* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1995.tex 1995 USAMO Problems (TEX)]
 
* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1995.tex 1995 USAMO Problems (TEX)]
 
* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]
 
* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1995.pdf 1995 USAMO Problems (PDF)]
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{{MAA Notice}}

Latest revision as of 12:31, 4 July 2013

Problems of the 1995 USAMO.

Problem 1

Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.

Solution

Problem 2

A calculator is broken so that the only keys that still work are the $\, \sin, \; \cos,$ $\tan, \; \sin^{-1}, \; \cos^{-1}, \,$ and $\, \tan^{-1} \,$ buttons. The display initially shows 0. Given any positive rational number $\, q, \,$ show that pressing some finite sequence of buttons will yield $\, q$. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Solution

Problem 3

Given a nonisosceles, nonright triangle $\, ABC, \,$ let $\, O \,$ denote the center of its circumscribed circle, and let $\, A_1, \, B_1, \,$ and $\, C_1 \,$ be the midpoints of sides $\, BC, \, CA, \,$ and $\, AB, \,$ respectively. Point $\, A_2 \,$ is located on the ray $\, OA_1 \,$ so that $\, \triangle OAA_1 \,$ is similar to $\, \triangle OA_2A$. Points $\, B_2 \,$ and $\, C_2 \,$ on rays $\, OB_1 \,$ and $\, OC_1, \,$ respectively, are defined similarly. Prove that lines $\, AA_2, \, BB_2, \,$ and $\, CC_2 \,$ are concurrent, i.e. these three lines intersect at a point.

Solution

Problem 4

Suppose $\, q_0, \, q_1, \, q_2, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions:
(i) $\, m-n \,$ divides $\, q_m - q_n \,$ for $\, m > n \geq 0,$
(ii) there is a polynomial $\, P \,$ such that $\, |q_n| < P(n) \,$ for all $\, n$.
Prove that there is a polynomial $\, Q \,$ such that $\, q_n = Q(n) \,$ for all $\, n$.

Solution

Problem 5

Suppose that in a certain society, each pair of persons can be classified as either amicable or hostile. We shall say that each member of an amicable pair is a friend of the other, and each member of a hostile pair is a foe of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.

Solution

See Also

1995 USAMO (Problems • Resources)
Preceded by
1994 USAMO 		Followed by
1996 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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