2020 AMC 10B Problem 6

P1-10~2 minPrint

Bases & Digits Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

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Key insight

A palindrome is fixed by its first three digits. After 159 comes 160, so the next palindrome is 16061, which is 110 miles later.

Solution

1. A five-digit palindrome is fixed once you know its first three digits: the last two digits just copy the first two in reverse. For example, the first three digits $159$ give $15951$ .
2. So the next palindrome comes from the next set of first three digits. After $159$ comes $160$ , which gives $16061$ .
3. Check that nothing was skipped. Any palindrome between would have to start with $159$ , and the only one starting with $159$ is $15951$ itself. (For example, $15961$ is not a palindrome, because reversed it reads $16951$ .)
4. The distance driven is $16061 - 15951 = 110$ miles.
5. It took $2$ hours, so the average speed is
$\frac{110}{2} = 55 \text{ miles per hour}.$

The answer is $\boxed{\textbf{(B)}\ 55}$ .

Why this works

A palindrome is completely decided by its front half. To find the next one, increase the front half by one and mirror it. After that, average speed is just distance divided by time.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

A five-digit palindrome has the form $\overline{abcba}$ , determined by its first three digits. The current reading is $159\,51$ with first three digits $159$ . The next palindrome comes from the next three-digit prefix, $160$ , giving $16061$ . (Nothing lies between: changing only the last two digits would break the mirror, and the middle digit $9$ cannot increase.)

Distance traveled: $16061 - 15951 = 110$ miles, in $2$ hours. Average speed $= \dfrac{110}{2} = 55$ miles per hour.

The answer is $\boxed{\textbf{(B)}\ 55}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 8: Base Numbers
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Number Theory, Chapter 9: Base Number Arithmetic
Introduction to Algebra, Chapter 7: Proportion (§7.4)