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1989 USAMO Problems

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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

See also

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