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Difference between revisions of "1996 USAMO Problems"

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Problems of the [[1996 USAMO | 1996]] [[USAMO]].
 
Problems of the [[1996 USAMO | 1996]] [[USAMO]].
  
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=Day 1=
 
==Problem 1==
 
==Problem 1==
 
Prove that the average of the numbers <math>n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)</math> is <math>\cot 1^{\circ}</math>.
 
Prove that the average of the numbers <math>n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)</math> is <math>\cot 1^{\circ}</math>.
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[[1996 USAMO Problems/Problem 3|Solution]]
 
[[1996 USAMO Problems/Problem 3|Solution]]
  
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=Day 2=
 
==Problem 4==
 
==Problem 4==
 
An <math>n</math>-term sequence <math>(x_1, x_2, \ldots, x_n)</math> in which each term is either 0 or 1 is called a ''binary sequence of length'' <math>n</math>.  Let <math>a_n</math> be the number of binary sequences of length <math>n</math> containing no three consecutive terms equal to 0, 1, 0 in that order.  Let <math>b_n</math> be the number of binary sequences of length <math>n</math> that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that <math>b_{n+1} = 2a_n</math> for all positive integers <math>n</math>.
 
An <math>n</math>-term sequence <math>(x_1, x_2, \ldots, x_n)</math> in which each term is either 0 or 1 is called a ''binary sequence of length'' <math>n</math>.  Let <math>a_n</math> be the number of binary sequences of length <math>n</math> containing no three consecutive terms equal to 0, 1, 0 in that order.  Let <math>b_n</math> be the number of binary sequences of length <math>n</math> that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that <math>b_{n+1} = 2a_n</math> for all positive integers <math>n</math>.
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== See Also ==
 
== See Also ==
 
{{USAMO newbox|year=1996|before=[[1995 USAMO]]|after=[[1997 USAMO]]}}
 
{{USAMO newbox|year=1996|before=[[1995 USAMO]]|after=[[1997 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 07:10, 19 July 2016

Problems of the 1996 USAMO.

Day 1

Problem 1

Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$.

Solution

Problem 2

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

Solution

Problem 3

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

Solution

Day 2

Problem 4

An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.

Solution

Problem 5

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

Solution

Problem 6

Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$.

Solution

See Also

1996 USAMO (Problems • Resources)
Preceded by
1995 USAMO 		Followed by
1997 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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