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Difference between revisions of "2021 JMPSC Sprint Problems/Problem 19"

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Note that if the <math>M</math> changes to <math>A</math> in just 14 minutes, <math>M:AT</math> is at the end of an hour, meaning that <math>A</math> is either <math>4</math> or <math>5</math>. If <math>A=4</math>, T must be <math>0</math> if <math>14</math> minutes later is a new hour. However, at time <math>14</math> minutes after <math>M:40</math> will be <math>M:54</math> instead of a new hour number. Thus, <math>A=5</math>. If in <math>14</math> minutes it is due to be <math>5:TM</math>, then it is between 4-o'clock and 5-o
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'clock currently. Thus, from <math>M:AT</math>, <math>M=4</math>. From <math>5:T4</math> (<math>A:TM</math>), we can guess that <math>T=0</math>. Then we have <math>M:AT=4:50</math> and <math>A:TM=5:04</math>. Checking, the times are indeed 14 minutes apart! We want <math>A \times M</math>, so the answer is <math>5 \times 4 = \boxed{20}</math>.
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~MathHayden
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(wow this is my *one* actual contribution to the AoPS wiki!)

Revision as of 01:27, 11 July 2021

Problem

As an April Fool’s prank, Sean hacks his teacher’s digital clock and switches each digit to a certain letter. Right now, the hacked clock displays $\textbf{M:AT}$. $14$ minutes later, it displays $\textbf{A:TM}$. If no two digits represent the same letter, find the value of $\textbf{A} \times \textbf{M}.$

Solution

Note that if the $M$ changes to $A$ in just 14 minutes, $M:AT$ is at the end of an hour, meaning that $A$ is either $4$ or $5$. If $A=4$, T must be $0$ if $14$ minutes later is a new hour. However, at time $14$ minutes after $M:40$ will be $M:54$ instead of a new hour number. Thus, $A=5$. If in $14$ minutes it is due to be $5:TM$, then it is between 4-o'clock and 5-o 'clock currently. Thus, from $M:AT$, $M=4$. From $5:T4$ ($A:TM$), we can guess that $T=0$. Then we have $M:AT=4:50$ and $A:TM=5:04$. Checking, the times are indeed 14 minutes apart! We want $A \times M$, so the answer is $5 \times 4 = \boxed{20}$.

~MathHayden

(wow this is my *one* actual contribution to the AoPS wiki!)

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