Scoring penalties

At some point in Leonel Messi's career, the football player had less than 80% success when performing penalty kicks. Later in his career, he had more than 80% success when performing penalty kicks. Show that there was a moment in Leonel Messi's career, when he had exactly 80% success when performing penalty kicks.


Solution

Consider the last penalty kick with number divisible by 5, after which Leonel Messi had success rate of less than 80%. Therefore if he performed 5k times, he scored at most 4k-1 goals. After 5 more tries, his success rate was at most 4k+4/5k+5=80%, and since it could not be less than that, it had to be exactly 80%.