Maths Olympiad Prep

Track / Stage 5 / 176 of 400 #776 of 1964

Problem 776

AIME late
Combinatorics Difficulty 5.4 Find the answer

If you had asked me such a question, I would have answered you only that my mother dreamed of having no fewer than 19 children, but she did not manage to fulfill her dream; however, I had three times as many sisters as cousins, and brothers - half as many as sisters. How many children did my mother have?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Next problem →

Official solution

16. From the conditions of the problem, it follows that the number of my sisters is divisible by both 3 and 2. Therefore, it is divisible by 6. Hence, the total number of children is

[( number divisible by 6)(1+1/2)]+1 [(\text { number divisible by } 6) \cdot(1+1 / 2)]+1

(the last 1 corresponds to myself). This number must be strictly less than 19, so it inevitably equals 6.

Therefore, I had 6 sisters and 3 brothers, and my mother had a total of 10 children.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.