3. The numbers 1, 2, 3, ..., 10 are written in a circle in some order. Petya calculated 10 sums of all triples of adjacent numbers and wrote the smallest of the calculated numbers on the board. What is the largest number that could have been written on the board?
Answer: 15. Solution: First, we prove that the written number is not greater than 15. We single out the number 10, and divide the remaining 9 numbers into three groups of three consecutive numbers. The sum of the numbers in these three groups is $1+2+3+\ldots+9=45$, so in at least one of the considered groups, the sum of the numbers is not greater than 45: $3=15$. An example of the arrangement of numbers, in which the number written by Petya is 15, is as follows: 1-5-9-2-7-6-8-3-4-10-(1). The given example is not the only one!