2007 AIME I Problems/Problem 5

Revision as of 23:55, 14 March 2007 by Me@home (talk | contribs) (Solution)

Problem

The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.

For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature?

Solution

Examine $F - 32$ modulo 9.

Solution 2

Hint. Consider the identity $Round(ax)=Round(xRound(a/Round(ax))$ its something like that...

See also

2007 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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All AIME Problems and Solutions