Estonian Mathematical Olympiad

Question

Today, on September 23, 2023, twins Mari and Jüri received a total of 5000 candies for their 10th birthdays. Starting from this day, their mother allows them both to take candies once per day, such that the amount of candies taken by any child on any day is less than their age in full years (on their birthday, they already use their new age). Neither child can resist taking at least one candy every day. Jüri allows Mari to take candies first on every day. The children agreed that whoever takes the last candy, has to buy new candies. Which child can ensure that they do not have to buy new candies, no matter how the other child takes their candies?

Accepted Answer

We will show that Mari can avoid taking the last candy. To achieve this, she will take 8 candies today, and on every following day, she will take candies so that along with Jüri's candies from the previous day, the total is the age of the children on the previous day. Since 2024 is a leap year, they will be 10 years old for 366 days; thus on the 11th birthday of the children, there will be 5000 - 8 - 366 10 = 1332 candies remaining. After 121 more days, there will be 1332 - 121 11 = 1 candy remaining after Mari has taken her candies for the day. Thus Jüri has no choice but to take the last candy.