Difference between revisions of "1977 USAMO Problems"

(Created page with "==Problem 1== 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\...")
 
(Problem 5: fixed LaTeXed equation)
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==Problem 5==
 
==Problem 5==
 
If <math> a,b,c,d,e</math> are positive numbers bounded by <math> p</math> and <math> q</math>, i.e, if they lie in <math> [p,q], 0 < p</math>, prove that
 
If <math> a,b,c,d,e</math> are positive numbers bounded by <math> p</math> and <math> q</math>, i.e, if they lie in <math> [p,q], 0 < p</math>, prove that
<cmath> (a \plus{} b \plus{} c \plus{} d \plus{} e)(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \plus{} \frac {1}{d} \plus{} \frac {1}{e}) \le 25 \plus{} 6\left(\sqrt {\frac {p}{q}} \minus{} \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.

Revision as of 22:26, 19 August 2011

Problem 1

Determine all pairs of positive integers $(m,n)$ such that $(1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ (Error compiling LaTeX. ! Undefined control sequence.) is divisible by $(1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$ (Error compiling LaTeX. ! Undefined control sequence.).

Problem 2

$ABC$ and $A'B'C'$ are two triangles in the same plane such that the lines $AA',BB',CC'$ are mutually parallel. Let $[ABC]$ denotes the area of triangle $ABC$ with an appropriate $\pm$ 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. ! Undefined control sequence.)

Problem 3

If $a$ and $b$ are two of the roots of $x^4\plus{}x^3\minus{}1\equal{}0$ (Error compiling LaTeX. ! Undefined control sequence.), prove that $ab$ is a root of $x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$ (Error compiling LaTeX. ! Undefined control sequence.).

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 $a,b,c,d,e$ are positive numbers bounded by $p$ and $q$, i.e, if they lie in $[p,q], 0 < p$, prove that \[(a + b + c + d + e)(\frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} + \frac {1}{e}) \le 25 + 6\left(\sqrt {\frac {p}{q}} - \sqrt {\frac {q}{p}}\right)^2\] and determine when there is equality.

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