Let X be the set consisting of twenty positive integers n, $n$ + 2,…., $n$ + 38. The smallest value of $n$ for which any three numbers a, b, $c$ in X , not necessarily distinct, form the sides of an acute-angled triangle is:

Step 1: X = n, $n$ + 2, \dots, $n$ + 38

Step 2: a, b, $c$ in X

Step 3: For any a, b, c

Step 4: (i) Triangle should be formed

Step 5: (ii) Triangle should be acute

Step 6: → only one angle can be obtuse at max

Step 7: (i) let $a$ \le $b$ \le c

Step 8: => for triangle

Step 9: a + $b$ > $c$ for all possible combination

Step 10: => even if a, $b$ are smallest $a$ = $b$ = n

Step 11: => $n$ + $n$ > $n$ + 38

Step 12: => $n$ > 38 => triangle will from

Step 13: (ii) now using cosine formula largest side longest angle

Step 14: a2 + b2 − $c$ 2

Step 15: => cos $c$ =  0 for acute 

Step 16: => $a$ 2 + $b$ 2 − $c$ 2  0 for acute   a, b, $c$ in x

Step 17: n 2 + ( $n$ ) − ( $n$ + 38 )  0

Step 18: 2 2

Step 19: => n2 – 76n – 382 > 0

Step 20: => $n$ > 91.74

Step 21: => $n$ = 92