2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 11 Easy

The sequence 1, x, y, z is arithmetic. The sequence 1, p,q,z is geometric. Both sequences are strictly increasing and contain only integers, and z is as small as possible. What is the value of x + y + z + p + q ?

  • A.

    66

  • B.

    91

  • C.

    103

  • D.

    132

  • E.

    149

Answer:E

Let d be the common difference and r be the common ratio, where d, r \in \mathbb{N}_+ with d, r > 1.

The equation 1 + 3d = r^3 holds. Since r^3 \equiv 1 \pmod{3}, we have r \equiv 1 \pmod{3}, implying r \geq 4.

For r = 4: d = 21, yielding x = 22, y = 43, z = 64, p = 4, q = 16. x + y + z + p + q = 22 + 43 + 64 + 4 + 16 = 149

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