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Difference between revisions of "1988 USAMO Problems"

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==Problem 1==
 
==Problem 1==
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The repeating decimal <math>0.ab\cdots k\overline{pq\cdots u}=\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime integers, and there is at least one decimal before the repeating part. Show that <math>n</math> is divisble by 2 or 5 (or both). (For example, <math>0.011\overline{36}=0.01136363636\cdots=\frac 1{88}</math>, and 88 is divisible by 2.)
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The repeating decimal <math>0.ab\cdots k\overline{pq\cdots u}=\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime integers, and there is at least one decimal before the repeating part. Show that <math>n</math> is divisible by 2 or 5 (or both). (For example, <math>0.011\overline{36}=0.01136363636\cdots=\frac 1{88}</math>, and 88 is divisible by 2.)
  
 
[[1988 USAMO Problems/Problem 1|Solution]]
 
[[1988 USAMO Problems/Problem 1|Solution]]

Latest revision as of 15:33, 5 February 2018

Problems from the 1988 USAMO.

Problem 1

The repeating decimal $0.ab\cdots k\overline{pq\cdots u}=\frac mn$, where $m$ and $n$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that $n$ is divisible by 2 or 5 (or both). (For example, $0.011\overline{36}=0.01136363636\cdots=\frac 1{88}$, and 88 is divisible by 2.)

Solution

Problem 2

The cubic polynomial $x^3+ax^2+bx+c$ has real coefficients and three real roots $r\ge s\ge t$. Show that $k=a^2-3b\ge 0$ and that $\sqrt k\le r-t$.

Solution

Problem 3

Let $X$ be the set $\{ 1, 2, \cdots , 20\}$ and let $P$ be the set of all 9-element subsets of $X$. Show that for any map $f: P\mapsto X$ we can find a 10-element subset $Y$ of $X$, such that $f(Y-\{k\})\neq k$ for any $k$ in $Y$.

Solution

Problem 4

$\Delta ABC$ is a triangle with incenter $I$. Show that the circumcenters of $\Delta IAB$, $\Delta IBC$, and $\Delta ICA$ lie on a circle whose center is the circumcenter of $\Delta ABC$.

Solution

Problem 5

Let $p(x)$ be the polynomial $(1-x)^a(1-x^2)^b(1-x^3)^c\cdots(1-x^{32})^k$, where $a, b, \cdots, k$ are integers. When expanded in powers of $x$, the coefficient of $x^1$ is $-2$ and the coefficients of $x^2$, $x^3$, ..., $x^{32}$ are all zero. Find $k$.

Solution

See Also

1988 USAMO (Problems • Resources)
Preceded by
1987 USAMO 		Followed by
1989 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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