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Difference between revisions of "1989 USAMO Problems"

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Problems from the '''1989 [[USAMO]].'''
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==Problem 1==
 
==Problem 1==
 
For each positive integer <math>n</math>, let
 
For each positive integer <math>n</math>, let
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<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>
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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>.
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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]]
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[[1989 USAMO Problems/Problem 3 | Solution]]
 
[[1989 USAMO Problems/Problem 3 | Solution]]
 
==Problem 4==
 
==Problem 4==
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Let <math>ABC</math> be an acute-angled triangle whose side lengths satisfy the inequalities <math>AB < AC < BC</math>. If point <math>I</math> is the center of the inscribed circle of triangle <math>ABC</math> and point <math>O</math> is the center of the circumscribed circle, prove that line <math>IO</math> intersects segments <math>AB</math> and <math>BC</math>.
  
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[[1989 USAMO Problems/Problem 4 | Solution]]
  
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[[1989 USAMO Problems/Problem 4 | Solution]]
 
 
==Problem 5==
 
==Problem 5==
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Let <math>u</math> and <math>v</math> be real numbers such that
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<math>
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(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.
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</math>
  
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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]]
  
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== See also ==
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== See Also ==
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* [[1989 USAMO]]
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{{USAMO box|year=1989|before=[[1988 USAMO]]|after=[[1990 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 18:52, 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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