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Difference between revisions of "2022 MMATHS Individual Round Problems/Problem 4"

(Created page with "==Problem== Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit perfect square!" Claire asks, "If I picked a di...")
 
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Claire says, "Now I know your favorite number!" What is Cat's favorite number?
 
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
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==Solution==
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It would be helpful to list some two-digit perfect squares. These are <math>16, 25, 36, 49, 64,</math> and <math>81</math>. We can eliminate <math>16</math> and <math>64</math>. Let's check each of the next ones.
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For <math>25</math>, we have <math>5-2 = 3</math>, <math>5+2 = 7</math>.
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For <math>36</math>, we have <math>6-3 = 3, 6+3 = 9</math>.
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For <math>49</math>, we have <math>9-4 = 5, 9+4 = 13</math>.
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For <math>64</math>, we have <math>6-4 = 2, 6+4 = 10</math>.
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We can clearly see that <math>25</math> doesn't work because <math>7</math> would identify it, nor does <math>49</math> because <math>13</math> would identify it, nor does <math>64</math> because <math>10</math> would identify it. We're left with only <math>36</math>. Therefore, our answer is <math>\boxed {36}</math>.
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-Arcticturn

Latest revision as of 00:44, 25 August 2023

Problem

Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit perfect square!"

Claire asks, "If I picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there a chance I'd know for certain what it is?"

Cat says, "Yes!" Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn't know my favorite number!

Claire says, "Now I know your favorite number!" What is Cat's favorite number?

Solution

It would be helpful to list some two-digit perfect squares. These are $16, 25, 36, 49, 64,$ and $81$. We can eliminate $16$ and $64$. Let's check each of the next ones.

For $25$, we have $5-2 = 3$, $5+2 = 7$.

For $36$, we have $6-3 = 3, 6+3 = 9$.

For $49$, we have $9-4 = 5, 9+4 = 13$.

For $64$, we have $6-4 = 2, 6+4 = 10$.

We can clearly see that $25$ doesn't work because $7$ would identify it, nor does $49$ because $13$ would identify it, nor does $64$ because $10$ would identify it. We're left with only $36$. Therefore, our answer is $\boxed {36}$.

-Arcticturn

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