2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 10 Easy

Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the median is 2 greater than their arithmetic mean. What is the least possible value for the mode?

  • A.

    5

  • B.

    7

  • C.

    9

  • D.

    11

  • E.

    13

Answer:D

We have \text{mode}=\text{median}+2 and \text{median}=\text{mean}+2. If there are three modes, then \text{mode}=\text{median}, which is a contradiction. So there are two modes. Let the five numbers be a,b,c,d,d, respectively, with increasing order. Then d=c+2 and c=\frac{a+b+c+d+d}{5}+2.

4c=a+b+(c+2)\times 2+10

2c=a+b+14 \geqslant 8.5, so c_{min}=9 with d_{min}=9+2=11.

The five numbers will be 1,3,9,11,11.

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