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# If all the internal angles of the convex polygon are distinct integers

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If all the internal angles of the convex polygon are distinct integers  [#permalink]

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14 Jan 2019, 11:25
1
00:00

Difficulty:

95% (hard)

Question Stats:

19% (02:46) correct 81% (01:27) wrong based on 16 sessions

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If all the internal angles of the convex polygon are distinct integers, what is the maximum number of sides the polygon can have?

I) The smallest exterior angle of the polygon is $$70^{\circ}$$.
II) The average of all internal angles = $$108^{\circ}$$.
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Re: If all the internal angles of the convex polygon are distinct integers  [#permalink]

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14 Jan 2019, 16:37
zubair123 wrote:
If all the internal angles of the convex polygon are distinct integers, what is the maximum number of sides the polygon can have?

I) The smallest exterior angle of the polygon is $$70^{\circ}$$.
II) The average of all internal angles = $$108^{\circ}$$.

Statement 1: The smallest exterior angle of the polygon is $$70^{\circ}$$.

it means that the largest interior angle is 110.

as all the interior angles are distinct integers, then the sum of angles would be at most (110+109+108+ ...)
so the sum of angles of a polygon with 'n' sides <110n,

As the formula of sum of interior angles of a polygon with n sides = (n – 2)*180,
then (n – 2)*180 < 110n
180n - 360 < 110n
n<5.14
so the maximum number of sides is 5
(sufficient)

Statement 2: The average of all internal angles = $$108^{\circ}$$.
then the sum of all interior angles = 108n
and 108n = (n-2)*180
108n = 180n - 360
72n = 360
n = 5
(sufficient)

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Re: If all the internal angles of the convex polygon are distinct integers   [#permalink] 14 Jan 2019, 16:37
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