Difference between revisions of "2024 SSMO Accuracy Round Problems/Problem 1"
(Created page with "==Problem== Let a time of day be three-full if exactly three of its digits are <math>3</math>s when displayed on a <math>12</math>-hour clock in the <math>hh:mm:ss</math> for...") |
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| + | Note that <math>hh</math> is a number from <math>01</math> to <math>12,</math> <math>mm</math> and <math>ss</math> are both integers from <math>00</math> to <math>59.</math> There are <math>45,14,1</math> two-digit integers from <math>00</math> to <math>59</math> that have <math>0,1,2</math> <math>3's,</math> respectively. If <math>hh</math> contains a <math>3,</math> then there are <math>(45)(1)+(14)(14)+(1)(45) = 286</math> possibilities. Otherwise, there are <math>(11)((1)(14)+(14)(1)) = 308</math> possibilities. Since the all times on the <math>12</math>-hour clock appear twice each day, our answer is <math>2(286+308) = \boxed{1188}.</math> | ||
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| + | ~SMO_Team | ||
Latest revision as of 14:30, 10 September 2025
Problem
Let a time of day be three-full if exactly three of its digits are 
s when displayed on a 
-hour clock in the 
format. How many seconds of the day are three-full?
Solution
Note that 
is a number from 
to 
and 
are both integers from 
to 
There are 
two-digit integers from to 
that have 
respectively. If contains a 
then there are 
possibilities. Otherwise, there are 
possibilities. Since the all times on the -hour clock appear twice each day, our answer is 
~SMO_Team
