1989 USAMO Problems

Problem 1

For each positive integer $n$ , let

$S_n = 1 + \frac 12 + \frac 13 + \cdots + \frac 1n$

$T_n = S_1 + S_2 + S_3 + \cdots + S_n$

$U_n = \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}$ .

Find, with proof, integers $0 < a,\ b,\ c,\ d < 1000000$ such that $\displaystyle T_{1988} = a S_{1989} - b$ and $\displaystyle U_{1988} = c S_{1989} - d$ .

Solution

Problem 2

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.

Solution

Problem 3

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$ , with real coefficients $c_k$ . Suppose that $|P(i)| < 1$ . Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$ .

Solution

Problem 4

Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$ . If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$ .

Solution

Problem 5

Let $u$ and $v$ be real numbers such that $(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.$ Determine, with proof, which of the two numbers, $u$ or $v$ , is larger.

Solution

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1989 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See also