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Try this beautiful problem from the PRMO, 2019 based on Rational Number and Integer.
Rational Number and Integer – PRMO 2019
let the rational number \(\frac{p}{q}\) be closest to but not equal to \(\frac{22}{7}\) among all rational numbers with denominator < 100, find p-3q.
- is 107
- is 14
- is 840
- cannot be determined from the given information
Key Concepts
Rational number
Algebra
Integer
Check the Answer
But try the problem first…
Answer: is 14.
Source
Suggested Reading
PRMO, 2019, Question 9
Higher Algebra by Hall and Knight
Try with Hints
First hint
|\(\frac{22}{7}-\frac{p}{q}\)|=|\(\frac{22q-7p}{7q}\)| then |22q-7p|=1 for smallest value
Second Hint
and q=99 then p=311
Final Step
p-3q=311-(3)(99)=311-297=14.
Other useful links
- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s
Related Program
Math Olympiad Program… taught by Olympians and researchers