Difference between revisions of "1984 USAMO Problems"

Latest revision as of 11:33, 18 July 2016

Problems from the 1984 USAMO.

Problem 1

The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$ . Determine the value of $k$ .

Solution

Problem 2

The geometric mean of any set of $m$ non-negative numbers is the $m$ -th root of their product.

$\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?

$\quad (\text{ii})\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?

Solution

Problem 3

$P$ , $A$ , $B$ , $C$ , and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$ , where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$ .

Solution

Problem 4

A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

Solution

Problem 5

$P(x)$ is a polynomial of degree $3n$ such that

$\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}$

Determine $n$ .

Solution

@@ Line 1: / Line 1: @@
+Problems from the '''1984 [[USAMO]].'''
 ==Problem 1==
 The product of two of the four roots of the quartic equation <math>x^4 - 18x^3 + kx^2+200x-1984=0</math> is <math>-32</math>. Determine the value of <math>k</math>.
@@ Line 14: / Line 16: @@
 ==Problem 3==
-<math>P, A, B, C,</math> and <math>D</math> are five distinct points in space such that <math>\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta</math>, where <math>\theta</math> is a given acute angle. Determine the greatest and least values of <math>\angle APC + \angle BPD</math>.
+<math>P</math>, <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> are five distinct points in space such that <math>\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta</math>, where <math>\theta</math> is a given acute angle. Determine the greatest and least values of <math>\angle APC + \angle BPD</math>.
 [[1984 USAMO Problems/Problem 3 | Solution]]
 ==Problem 4==
-A dfficult mathematical competition consisted of a Part I and a Part II with a combined total of <math>28</math> problems. Each contestant solved <math>7</math> problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
+A difficult mathematical competition consisted of a Part I and a Part II with a combined total of <math>28</math> problems. Each contestant solved <math>7</math> problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
 [[1984 USAMO Problems/Problem 4 | Solution]]

1984 USAMO (Problems • Resources)
Preceded by 1983 USAMO	Followed by 1985 USAMO
1 • 2 • 3 • 4 • 5
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Difference between revisions of "1984 USAMO Problems"

Latest revision as of 11:33, 18 July 2016

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also