AMC 10 Daily Practice Round 2

Complete problem set with solutions and individual problem pages

Problem 10 Easy

Cards labeled from 1 to 9 are distributed to students A, B, and C, three cards each person without repetition. The following are the conversations between the students.

A: The three numbers on my cards form an arithmetic sequence.

B: Me too.

C: Only my cards do not form an arithmetic sequence.

Suppose that everyone is telling the truth, find the minimum of the sum of the numbers in three cards of C.

  • A.

    6

  • B.

    7

  • C.

    8

  • D.

    9

  • E.

    10

Answer:D

As the numbers of A and B form arithmetic sequences, their sums must be multiples of 3. The sum of 9 cards is 45, which is also a multiple of 3. Therefore, the sum of C is also a multiple of 3. The sum of C can not be 6=(1+2+3), so the sum is at least 9. Check and find 9 is the correct answer.

A: (9,8,7)

B: (5,4,3)

C: (1,2,6)

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