Difference between revisions of "1977 USAMO Problems"
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| + | Problems from the '''1977 [[United States of America Mathematical Olympiad | USAMO]]'''. | ||
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==Problem 1== | ==Problem 1== | ||
Determine all pairs of positive integers <math> (m,n)</math> such that | Determine all pairs of positive integers <math> (m,n)</math> such that | ||
<math> (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})</math> is divisible by <math> (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})</math>. | <math> (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})</math> is divisible by <math> (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})</math>. | ||
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| + | [[1977 USAMO Problems/Problem 1 | Solution]] | ||
==Problem 2== | ==Problem 2== | ||
<math> ABC</math> and <math> A'B'C'</math> are two triangles in the same plane such that the lines <math> AA',BB',CC'</math> are mutually parallel. Let <math> [ABC]</math> denotes the area of triangle <math> ABC</math> with an appropriate <math> \pm</math> sign, etc.; prove that | <math> ABC</math> and <math> A'B'C'</math> are two triangles in the same plane such that the lines <math> AA',BB',CC'</math> are mutually parallel. Let <math> [ABC]</math> denotes the area of triangle <math> ABC</math> with an appropriate <math> \pm</math> sign, etc.; prove that | ||
<cmath> 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].</cmath> | <cmath> 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].</cmath> | ||
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| + | [[1977 USAMO Problems/Problem 2 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
If <math> a</math> and <math> b</math> are two of the roots of <math> x^4\plus{}x^3\minus{}1\equal{}0</math>, prove that <math> ab</math> is a root of <math> x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0</math>. | If <math> a</math> and <math> b</math> are two of the roots of <math> x^4\plus{}x^3\minus{}1\equal{}0</math>, prove that <math> ab</math> is a root of <math> x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0</math>. | ||
| + | |||
| + | [[1977 USAMO Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent. | Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent. | ||
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| + | [[1977 USAMO Problems/Problem 4 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
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<cmath> (a + b + c + d + e)(\frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} + \frac {1}{e}) \le 25 + 6\left(\sqrt {\frac {p}{q}} - \sqrt {\frac {q}{p}}\right)^2</cmath> | <cmath> (a + b + c + d + e)(\frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} + \frac {1}{e}) \le 25 + 6\left(\sqrt {\frac {p}{q}} - \sqrt {\frac {q}{p}}\right)^2</cmath> | ||
and determine when there is equality. | and determine when there is equality. | ||
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| + | [[1977 USAMO Problems/Problem 5 | Solution]] | ||
| + | |||
| + | = See also = | ||
| + | *[[USAMO Problems and Solutions]] | ||
| + | |||
| + | {{USAMO box|year=1977|before=[[1976 USAMO]]|after=[[1978 USAMO]]}} | ||
Revision as of 12:43, 30 September 2011
Problems from the 1977 USAMO.
Problem 1
Determine all pairs of positive integers 
such that
$(1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ (Error compiling LaTeX. Unknown error_msg) is divisible by $(1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$ (Error compiling LaTeX. Unknown error_msg).
Problem 2
and 
are two triangles in the same plane such that the lines 
are mutually parallel. Let ![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
denotes the area of triangle with an appropriate 
sign, etc.; prove that
\[3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].\] (Error compiling LaTeX. Unknown error_msg)
Problem 3
If 
and 
are two of the roots of $x^4\plus{}x^3\minus{}1\equal{}0$ (Error compiling LaTeX. Unknown error_msg), prove that 
is a root of $x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$ (Error compiling LaTeX. Unknown error_msg).
Problem 4
Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.
Problem 5
If 
are positive numbers bounded by 
and 
, i.e, if they lie in ![$[p,q], 0 < p$](http://latex.artofproblemsolving.com/b/7/d/b7db0d1c44b17ab521242db4599465126d5ad617.png)
, prove that
![\[(a + b + c + d + e)(\frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} + \frac {1}{e}) \le 25 + 6\left(\sqrt {\frac {p}{q}} - \sqrt {\frac {q}{p}}\right)^2\]](http://latex.artofproblemsolving.com/6/3/3/6335c58fcf19184291f5c4449812dc094c8b65bf.png)
and determine when there is equality.
See also
| 1977 USAMO (Problems • Resources) | ||
| Preceded by 1976 USAMO |
Followed by 1978 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
