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AMC 10 USA Math Olympiad

Average Problem from AMC 10A – 2020 -Problem No. 6

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

What is Average ?


In mathematics and statistics, average refers to the sum of a group of values divided by n, where n is the number of values in the group. An average is also known as a mean.

Try this sum from AMC 10 – 2020


Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the same forward and backward. Then 2 hours later , the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this 2 – hour period ?

A) 50 B) 55 C)60 D) 65 E) 70

American Mathematics Competition 10 (AMC 10B), {2020}, {Problem Number 6}

Average

4 out of 10

Mathematics can be fun

Knowledge Graph


Average- knowledge graph

Use some hints


Do you really need any hint ???

Try this out:

In order to get the smallest palindrome greater than 15951 , we need to raise the middle digit. If we were to raise any of the digits after the middle, we would be forced to also raise a digit before the middle to keep it a palindrome, making it unnecessarily larger.

So what can we do here ?

We can raise 9 to the next largest value, 10 , but obviously, that’s not how place value works, so we’re in the 16000 s now . To keep this a palindrome, our number is now 16061.

If you really need the final hint this can be the life saver for this sum :

So Megan drove 16061 – 15951 = 110 miles . Since this happened over 2 hours , she drove at \(\frac {110}{2}\) = 55 mph .

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