2021/007) Three real numbers are given, Fractional part of product of every 2 of them is $\frac{1}{2}$. Prove that these numbers are irrational.

 Let the three numbers be a,b,c as product of every 2 numbers has fractional part $\frac{1}{2}$ so each of the product is an integer plus $\frac{1}{2}$ 

So there exists integers p,q r such that $ab = \frac{2p+1}{2}\cdots(1)$

 $bc = \frac{2q+1}{2}\cdots(2)$ 

 $ca = \frac{2r+1}{2}\cdots(3)$ 

 multiplying all 3 we get $(abc)^2 = \frac{(2p+1)(2q+1)(2r+1)}{8}$ 

 or $(abc) = \frac{\sqrt{(2p+1)(2q+1)(2r+1)}}{2\sqrt{2}}$ 

 Now numerator is square root of odd number and denominator is product of 2 and$\sqrt{2}$ so the number is irrational and as product of all 3 is irrational and product of 2 number is rational so 3rd number is irrational.

 this is true for each pair and hence all 3 numbers are irrational. Hence proved