Let X be the set consisting of twenty positive integers n, n+2, ,...

$X=\{n, n+2, \ldots, n+38\}$ $a, b, c \in X$ For any $a, b, c$ (i) Triangle should be formed (ii) Triangle should be acute $\rightarrow$ only one angle can be obtuse at max (i) let $a \leq b \leq c$ $\Rightarrow$ for triangle $a+b>c$ for all possible combination $\Rightarrow$ even if $a, b$ are smallest $a=b=n$ $\Rightarrow n+n>n+38$ $\Rightarrow n>38 \Rightarrow$ triangle will form (ii) now using cosine formula largest side longest angle $\Rightarrow \cos c=\frac{a^2+b^2-c^2}{2 a b}>0$ for acute $\Delta$ $\Rightarrow \quad a^2+b^2-c^2>0$ for acute $\Delta \forall a, b, c \in X$ $n^2+(n)^2-(n+38)^2>0$ $$\begin{aligned} & \Rightarrow \quad n^2-76 n-38^2>0 \\ & \Rightarrow \quad n>91.74 \\ & \Rightarrow \quad n=92 \end{aligned}$$