1988 USAMO Problems

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 divisble 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

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1988 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5