Difference between revisions of "1969 Canadian MO Problems/Problem 1"

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(The denominator is irrelevant since it never equals zero)
 
(The denominator is irrelevant since it never equals zero)
  
From <math>a_1/b_1=a_2b_2,</math> <math>a_1^nb_2^n=a_2^nb_1^n.</math> Similarly, <math>a_1^nb_3^n=a_3^nb_1^n</math> from <math>a_1/b_1=a_3/b_3.</math>  
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From <math>a_1/b_1=a_2/b_2,</math> <math>a_1^nb_2^n=a_2^nb_1^n.</math> Similarly, <math>a_1^nb_3^n=a_3^nb_1^n</math> from <math>a_1/b_1=a_3/b_3.</math>  
  
 
Hence, <math>a_2^nb_1^n-a_1^nb_2^n=a_3^nb_1^n-a_1^nb_3^n=0</math> and our proof is complete.
 
Hence, <math>a_2^nb_1^n-a_1^nb_2^n=a_3^nb_1^n-a_1^nb_3^n=0</math> and our proof is complete.
 
{{Old CanadaMO box|before=First question|num-a=2|year=1969}}
 
{{Old CanadaMO box|before=First question|num-a=2|year=1969}}

Latest revision as of 15:06, 18 October 2015

Problem

Show that if $a_1/b_1=a_2/b_2=a_3/b_3$ and $p_1,p_2,p_3$ are not all zero, then $\left(\frac{a_1}{b_1} \right)^n=\frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}$ for every positive integer $n.$

Solution

Instead of proving the two expressions equal, we prove that their difference equals zero.

Subtracting the LHS from the RHS, $0=\frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}-\frac{a_1^n}{b_1^n}.$

Finding a common denominator, the numerator becomes $b_1^n(p_1a_1^n+p_2a_2^n+p_3a_3^n)-a_1^n(p_1b_1^n+p_2b_2^n+p_3b_3^n)=p_2(a_2^nb_1^n-a_1^nb_2^n)+p_3(a_3^nb_1^n-a_1^nb_3^n)=0.$ (The denominator is irrelevant since it never equals zero)

From $a_1/b_1=a_2/b_2,$ $a_1^nb_2^n=a_2^nb_1^n.$ Similarly, $a_1^nb_3^n=a_3^nb_1^n$ from $a_1/b_1=a_3/b_3.$

Hence, $a_2^nb_1^n-a_1^nb_2^n=a_3^nb_1^n-a_1^nb_3^n=0$ and our proof is complete.

1969 Canadian MO (Problems)
Preceded by
First question
1 2 3 4 5 6 7 8 Followed by
Problem 2


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