Maths Olympiad Prep

Track / Stage 5 / 122 of 400 #722 of 1964

Problem 722

AIME late
Combinatorics Difficulty 5.3 Find the answer

4. Masha has 2 kg of "Swallow" candies, 3 kg of "Truffle" candies, 4 kg of "Bird's Milk" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year's gifts she can make if each gift must contain 3 different types of candies, 100 grams of each?

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Answer: 45

Solution: Even if Masha puts "Citron" in all the gifts, she will still have 2+3+4 = 9 kg of candies left, and in each gift, she must put at least 200g (and if she doesn't put Citron in all of them, then more than 200g). This means there can be no more than 45 gifts. 45 gifts can be made if she creates 5 gifts of the type Swallow+Truffle+Citron, 15 gifts of the type Swallow+Bird's Milk + Citron, and

25 gifts of the type Truffle+Bird's Milk + Citron.

Comment for graders: Here, half a point can be given for the correct selection without proving that it is optimal.

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