Question 93
In a tall hat there are one hundred notes, numbered from $1$ to $100$. You want to have three notes with the property that each of the three numbers is smaller than the sum of the other two. For example, the three notes numbered $10$, $15$, and $20$ would be suitable (as $10 \lt 15 + 20$, $15 \lt 10 + 20$, and $20 \lt 10 + 15$), but the notes numbered $3$, $4$, and $7$ would not (as $7$ is not smaller than $3 + 4$). You may (without looking at the numbers on the notes) take some notes from the hat.
What is the smallest number of notes you have to take to be sure to have three notes that meet your wish?
What is the smallest number of notes you have to take to be sure to have three notes that meet your wish?