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 X , not necessarily distinct, form the sides of an acute-angled triangle is:

Step-by-step solution

X= n, n+2, , n+38 a, b, c X For any a, b, c (i) Triangle should be formed (ii) Triangle should be acute only one angle can be obtuse at max (i) let a b c for triangle a+b>c for all possible combination even if a, b are smallest a=b=n n+n>n+38 n>38 triangle will form (ii) now using cosine formula largest side longest angle c= a^2+b^2-c^2 2 a b >0 for acute a^2+b^2-c^2>0 for acute a, b, c Xn^2+(n)^2-(n+38)^2>0 aligned & n^2-76 n-38^2>0 & n>91.74 & n=92 aligned