Track / Stage 5 / 130 of 400 #730 of 2000

Problem 730

AIME late

Combinatorics Difficulty 5.3 Find the answer

3. At a humanitarian action, chocolate balls are being sold packed in boxes. Ana and Bela each bought 2 packages, and Cico bought the last three packages. Soon, Dado joined, disappointed that all the balls were sold. Ana, Bela, and Cico then fairly divided the balls into four parts. Dado paid 245 kn for his share. How will Ana, Bela, and Cico fairly divide that amount?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

First solution.

There were a total of $2+2+3=7$ boxes, so we conclude that each person should pay for $\frac{7}{4}$ boxes. $\quad 2$ POINTS Since Dado left 245 kn for his share, we conclude that a quarter of the balls in a box is worth $245 \mathrm{kn} \div 7 = 35 \mathrm{kn}$ .

Ana paid for two boxes of chocolate balls (8 quarters), so we conclude that she spent $8 \cdot 35 \mathrm{kn} = 280 \mathrm{kn}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.