parmenides51 wrote: A survey of participants was conducted at the Olympiad. $ 90\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $90\%$ of the participants liked the opening of the Olympiad. Each participant was known to enjoy at least two of these three events. Determine the percentage of participants who rated all three events positively. $a$ liked first and second $b$ liked first and opening $c$ liked second and opening $d$ liked all three Statement is : $\frac {a+b+d}{a+b+c+d}=0.9$ and $\frac {a+c+d}{a+b+c+d}=0.6$ and $\frac {b+c+d}{a+b+c+d}=0.9$ Adding, we get $\frac{2a+2b+2c+3d}{a+b+c+d}=2.4$ And so $\boxed{\frac d{a+b+c+d}=40\%}$