CEMC Fermat, 2014

Number theory Difficulty 3.5 AMC 10/12 Find the answer Canada

At the start of a 5 hour trip, the odometer in Jill’s car indicates that her car had already been driven 13 831 km. The integer 13 831 is a palindrome, because it is the same when read forwards or backwards. At the end of the 5 hour trip, the odometer reading was another palindrome. If Jill never drove faster than 80 km/h, her greatest possible average speed was closest to

Pick one

Solution

Since Jill never drove faster than 80 km/h over her 5 hour drive, then she could not have driven more than $5 \times 80 = 400$ km.

Since the initial odometer reading was 13 831 km, then the final odometer reading is no more than $13\,831 + 400 = 14\,231$ km.

Determining her greatest possible average speed can be done by first determining the greatest possible distance that she could have travelled, which can be done by determining the greatest possible odometer reading.

Knowing that the final odometer reading was also a palindrome, we want to determine the greatest palindrome less than $14\,231$ . This is $14\,141$ . (To find this, we begin by trying to find palindromes that are at least 14 000. Such palindromes end with 41, so are of the form $14\,x41$ . The greatest such integer less than 14 231 is $14\,141$ .)

Since Jill’s greatest possible final odometer reading was $14\,141$ , then she would have travelled $14\,141-13\,831 = 310$ km, and so her greatest possible average speed was $\frac{310}{5}=62$ km/h.

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