Estonian Mathematical Olympiad

We show that the probability of listening to the same song multiple times is greater than $0.5$, which is equivalent to the probability of listening to $14$ distinct songs being less than $0.5$. The latter probability can be expressed as $$1\cdot 0.99\cdot 0.98\cdot 0.97\cdot 0.96\cdot 0.95\cdot 0.94\cdot 0.93\cdot 0.92\cdot 0.91\cdot 0.9\cdot 0.89\cdot 0.88\cdot \mathrm{0.87.}$$

To prove the bound, notice that $$0.93\cdot 0.87=(0.9+0.03)(0.9-0.03)=0.9^2-0.03^2<0.9^2,$$ $$0.92\cdot 0.88=(0.9+0.02)(0.9-0.02)=0.9^2-0.02^2<0.9^2,$$ $$0.91\cdot 0.89=(0.9+0.01)(0.9-0.01)=0.9^2-0.01^2<0.9^2.$$ Thus the product above is less than $0.9^7$. We will now show that $0.9^7<0.5$. Using the property $0.9\cdot \overline{0.ab}=\overline{0.ab}-\overline{0.0ab}\le \overline{0.ab}-\overline{0.0a}$ repeatedly, we get $$\begin{array}{rl}0.9^7& =0.9\cdot 0.9\cdot 0.9\cdot 0.9\cdot 0.9\cdot 0.9\cdot 0.9\\ & =0.81\cdot 0.9\cdot 0.9\cdot 0.9\cdot 0.9\cdot 0.9\\ & \le 0.73\cdot 0.9\cdot 0.9\cdot 0.9\cdot 0.9\\ & \le 0.66\cdot 0.9\cdot 0.9\cdot 0.9\\ & \le 0.6\cdot 0.9\cdot 0.9\\ & =0.54\cdot 0.9\\ & \le 0.49<\mathrm{0.5.}\end{array}$$

This finishes the proof.