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Interior Angle Problem | AIME I, 1990 | Question 3

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Interior Angle.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Interior Angle.

Interior Angle Problem – AIME I, 1990


Let \(P_1\) be a regular r gon and \(P_2\) be a regular s gon \((r \geq s \geq 3)\) such that each interior angle of \(P_1\) is \(\frac{59}{58}\) as large as each interior angle of \(P_2\), find the largest possible value of s.

  • is 107
  • is 117
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Polygons

Algebra

Check the Answer


Answer: is 117.

AIME I, 1990, Question 3

Elementary Algebra by Hall and Knight

Try with Hints


Interior angle of a regular sided polygon=\(\frac{(n-2)180}{n}\)

or, \(\frac{\frac{(r-2)180}{r}}{\frac{(s-2)180}{s}}=\frac{59}{58}\)

or, \(\frac{58(r-2)}{r}=\frac{59(s-2)}{s}\)

or, 58rs-58(2s)=59rs-59(2r)

or, 118r-116s=rs

or, r=\(\frac{116s}{118-s}\)

for 118-s>0, s<118

or, s=117

or, r=(116)(117)

or, s=117.

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