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1989 USAMO Problems: Difference between revisions

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Problems from the '''1989 [[USAMO]].'''
==Problem 1==
==Problem 1==
For each positive integer <math>n</math>, let
For each positive integer <math>n</math>, let
Line 8: Line 10:
<math>U_n = \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}</math>.
<math>U_n = \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}</math>.
</div>
</div>
Find, with proof, integers <math>0 < a,\ b,\ c,\ d < 1000000</math> such that <math>\displaystyle T_{1988} = a S_{1989} - b</math> and <math>\displaystyle U_{1988} = c S_{1989} - d</math>.
Find, with proof, integers <math>0 < a,\ b,\ c,\ d < 1000000</math> such that <math>T_{1988} = a S_{1989} - b</math> and <math>U_{1988} = c S_{1989} - d</math>.


[[1989 USAMO Problems/Problem 1 | Solution]]
[[1989 USAMO Problems/Problem 1 | Solution]]
Line 28: Line 30:
==Problem 5==
==Problem 5==
Let <math>u</math> and <math>v</math> be real numbers such that
Let <math>u</math> and <math>v</math> be real numbers such that
<math>
<math>
(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.
(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.
</math>
</math>
Determine, with proof, which of the two numbers, <math>u</math> or <math>v</math>, is larger.
Determine, with proof, which of the two numbers, <math>u</math> or <math>v</math>, is larger.


[[1989 USAMO Problems/Problem 5 | Solution]]
[[1989 USAMO Problems/Problem 5 | Solution]]


== See also ==
== See Also ==
* [[1989 USAMO]]
{{USAMO box|year=1989|before=[[1988 USAMO]]|after=[[1990 USAMO]]}}
{{MAA Notice}}

Latest revision as of 17:52, 18 July 2016

Problems from the 1989 USAMO.

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 $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.

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

1989 USAMO (Problems • Resources)
Preceded by
1988 USAMO 		Followed by
1990 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America.

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