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Difference between revisions of "1994 USAMO Problems"

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[[1994 USAMO Problems/Problem 4|Solution]]
 
[[1994 USAMO Problems/Problem 4|Solution]]
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==Problem 5==
 
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Let <math>\, |U|, \, \sigma(U) \,</math> and <math>\, \pi(U) \,</math> denote the number of elements, the sum, and the product, respectively, of a finite set <math>\, U \,</math> of positive integers.  (If <math>\, U \,</math> is the empty set, <math>\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1</math>.) Let <math>\, S \,</math> be a finite set of positive integers. As usual, let <math>\, \binom{n}{k} \,</math> denote <math>\, n! \over k! \, (n - k)!</math>. Prove that
 
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<cmath>\sum_{U \subseteq S} ( - 1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S)</cmath>
 
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for all integers <math>\, m \geq \sigma(S)</math>.
 
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[[1994 USAMO Problems/Problem 5|Solution]]
 
  
 
== Resources ==
 
== Resources ==

Revision as of 22:39, 13 April 2011

Problems of the 1994 USAMO.

Problem 1

Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n + 1}) \,$ contains at least one perfect square.

Solution

Problem 2

The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, ... red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, ... red, yellow, blue?

Solution

Problem 3

A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB = CD = EF$ and diagonals $AD$, $BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $CP/PE = (AC/CE)^2$.

Solution

Problem 4

Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j = 1}^n a_j \geq \sqrt {n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$

\[\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n} \right).\]

Solution

Resources

1994 USAMO (Problems • Resources)
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1993 USAMO 		Followed by
1995 USAMO
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All USAMO Problems and Solutions
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