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Difference between revisions of "1975 Canadian MO Problems/Problem 4"

(Created page with "== Problem 4 == For a positive number such as <math>3.27</math>, <math>3</math> is referred to as the integral part of the number and <math>.27</math> as the decimal part. Fin...")
 
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== Solution ==
 
== Solution ==
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None yet!
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Let <math>a</math> be the integer part and <math>b</math> be the decimal part, thus, we have the G.P.
  
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{{Old CanadaMO box|before=First question|num-a=2|year=1975}}
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<math>(b,a,a+b)</math>
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So,
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<math>b(a+b)=a^2</math>
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<math>ab+b^2=a^2</math>
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There we have a quadratic equation. We must isolate <math>a</math> or <math>b</math>.
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<math>a^2-ab-b^2=0</math>
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<math>b=\frac{-a\pm a\sqrt{5}}{2}</math>
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If the number must be positive, thus we'll consider only the solution
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<math>b=\frac{-a+a\sqrt{5}}{2}</math>
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As <math>b</math> is the decimal part, then it must be lower than 1
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<math>b<1</math>
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<math>\frac{-a+a\sqrt{5}}{2}<1</math>
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<math>a<\frac{2}{\sqrt{5}-1}</math>
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<math>a<\frac{1+\sqrt{5}}{2}</math>
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This is the golden ratio and it's approximately <math>1.618</math>.
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As <math>a</math> must be an integer, thus <math>a=1</math>. Therefore
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<math>b=\frac{-a+a\sqrt{5}}{2}</math>
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<math>b=\frac{-1+\sqrt{5}}{2}</math>
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To find our number we must sum <math>a+b</math>, so
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<math>a+b</math>
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<math>1+\frac{-1+\sqrt5}{2}</math>
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<math>\boxed{\frac{1+\sqrt{5}}{2}}</math>
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The number we're searching for is the golden ratio.
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{{Old CanadaMO box|num-b = 3|num-a=5|year=1975}}

Latest revision as of 11:15, 24 January 2018

Problem 4

For a positive number such as $3.27$, $3$ is referred to as the integral part of the number and $.27$ as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

Solution

Let $a$ be the integer part and $b$ be the decimal part, thus, we have the G.P.


$(b,a,a+b)$


So,


$b(a+b)=a^2$

$ab+b^2=a^2$


There we have a quadratic equation. We must isolate $a$ or $b$.

$a^2-ab-b^2=0$

$b=\frac{-a\pm a\sqrt{5}}{2}$


If the number must be positive, thus we'll consider only the solution

$b=\frac{-a+a\sqrt{5}}{2}$


As $b$ is the decimal part, then it must be lower than 1


$b<1$

$\frac{-a+a\sqrt{5}}{2}<1$

$a<\frac{2}{\sqrt{5}-1}$

$a<\frac{1+\sqrt{5}}{2}$


This is the golden ratio and it's approximately $1.618$.


As $a$ must be an integer, thus $a=1$. Therefore


$b=\frac{-a+a\sqrt{5}}{2}$

$b=\frac{-1+\sqrt{5}}{2}$


To find our number we must sum $a+b$, so


$a+b$

$1+\frac{-1+\sqrt5}{2}$

$\boxed{\frac{1+\sqrt{5}}{2}}$


The number we're searching for is the golden ratio.


1975 Canadian MO (Problems)
Preceded by
Problem 3 		1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • Followed by
Problem 5


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