KVPY SX SB Exam 19-Nov-2017 Question Paper

Question : 7 of 120

Marks: +1, -0

The number of 5-tuples (a, b, c, d, e) of positive integers such that
I. a,b,c,d,e are the measures of angles of a convex pentagon in degrees ;
II. a ≤ b ≤ c ≤ d ≤ e ;
III a, b, c, d, e are in arithmetic progression is

Validate

Solution:

a + b + c + d + e = 540
Let say first term = a
Common difference d

n/2[2a + (n -1)d] = 540

⇒

a + 2d = 108

Case-1
d = 0
a = l08
(l08, 108, 108, l08, 108)
Case-2
d = 1
(106, 107, 108, 109, 110)
Similarly it goes up to d = 35
for d > 35, interior angle > 180°
which is not possible
So form d = 0 to d = 35
total 36 tuples are possible

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