Difference between revisions of "2016 AMC 12A Problems/Problem 25"
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==Solution== | ==Solution== | ||
− | Consider <math>f(2)</math>. The numbers left on the blackboard will show the hundreds place at the end. In order for the hundreds place to differ by 2, the difference between two perfect squares needs to be at least <math>100</math>. Calculus <math>\left(\frac{\text{d}}{\text{d}x} x^2=2x\right)</math> and a bit of thinking says this first happens at <math>x\ge 100/2 = 50</math>. The perfect squares from here go: <math>2500, 2601, 2704, 2809\dots</math>. Note that the ones and tens also make the perfect squares, <math>1^2,2^2,3^2\dots</math>. After the ones and tens make <math>100</math>, the hundreds place will go up by <math>2</math>, thus reaching our goal. Since <math>10^2=100</math>, the last perfect square to be written will be <math>\left(50+10\right)^2=60^2=3600</math>. The missing number is one less than the number of hundreds <math>(k=2)</math> of <math>3600</math>, or <math>35</math>. | + | Consider <math>f(2)</math>. The numbers left on the blackboard will show the hundreds place at the end. In order for the hundreds place to differ by 2, the difference between two perfect squares needs to be at least <math>100</math>. Calculus <math>\left(\frac{\text{d}}{\text{d}x} x^2=2x\right)</math> and a bit of thinking says this first happens at <math>x\ge 100/2 = 50</math>*. The perfect squares from here go: <math>2500, 2601, 2704, 2809\dots</math>. Note that the ones and tens also make the perfect squares, <math>1^2,2^2,3^2\dots</math>. After the ones and tens make <math>100</math>, the hundreds place will go up by <math>2</math>, thus reaching our goal. Since <math>10^2=100</math>, the last perfect square to be written will be <math>\left(50+10\right)^2=60^2=3600</math>. The missing number is one less than the number of hundreds <math>(k=2)</math> of <math>3600</math>, or <math>35</math>. |
Now consider f(4). Instead of the difference between two squares needing to be <math>100</math>, the difference must now be <math>10000</math>. This first happens at <math>x\ge 5000</math>. After this point, similarly, <math>\sqrt{10000}=100</math> more numbers are needed to make the <math>10^4</math> th's place go up by <math>2</math>. This will take place at <math>\left(5000+100\right)^2=5100^2= 26010000</math>. Removing the last four digits (the zeros) and subtracting one yields <math>2600</math> for the skipped value. | Now consider f(4). Instead of the difference between two squares needing to be <math>100</math>, the difference must now be <math>10000</math>. This first happens at <math>x\ge 5000</math>. After this point, similarly, <math>\sqrt{10000}=100</math> more numbers are needed to make the <math>10^4</math> th's place go up by <math>2</math>. This will take place at <math>\left(5000+100\right)^2=5100^2= 26010000</math>. Removing the last four digits (the zeros) and subtracting one yields <math>2600</math> for the skipped value. | ||
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There is no carrying in this addition. Therefore each <math>f(k)</math> adds <math>2 + 5 + 1 = 8</math> to the sum of the digits. | There is no carrying in this addition. Therefore each <math>f(k)</math> adds <math>2 + 5 + 1 = 8</math> to the sum of the digits. | ||
Since <math>2n = 2016</math>, <math>n = 1008</math>, and <math>8n = 8064</math>, or <math>\boxed{\textbf{(E)}\text{ 8064}}</math>. | Since <math>2n = 2016</math>, <math>n = 1008</math>, and <math>8n = 8064</math>, or <math>\boxed{\textbf{(E)}\text{ 8064}}</math>. | ||
+ | |||
+ | Addendum: *You could also use the fact that <cmath>(x+1)^2 = x^2 +2x+1</cmath> | ||
+ | In other words, the difference between <math>x^2</math> and <math>(x+1)^2</math> is equal to <math>2x+1</math>. We can set the inequality <math>2x+1 \geq 100</math>. Obviously, the first integer <math>x</math> that satisfies this is 50. | ||
+ | This way, while being longer, is IMO more motivated and doesn't use calculus. | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2016|ab=A|num-b=24|after=Last Problem}} | {{AMC12 box|year=2016|ab=A|num-b=24|after=Last Problem}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 22:10, 16 December 2017
Problem
Let be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with k+1 digits. Every time Bernardo writes a number, Silvia erases the last k digits of it. Bernardo then writes the next perfect square, Silvia erases the last k digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let f(k) be the smallest positive integer not written on the board. For example, if k = 1, then the numbers that Bernardo writes are , and the numbers showing on the board after Silvia erases are 1, 2, 3, 4, and 6, and thus f(1) = 5. What is the sum of the digits of ?
Solution
Consider . The numbers left on the blackboard will show the hundreds place at the end. In order for the hundreds place to differ by 2, the difference between two perfect squares needs to be at least . Calculus and a bit of thinking says this first happens at *. The perfect squares from here go: . Note that the ones and tens also make the perfect squares, . After the ones and tens make , the hundreds place will go up by , thus reaching our goal. Since , the last perfect square to be written will be . The missing number is one less than the number of hundreds of , or .
Now consider f(4). Instead of the difference between two squares needing to be , the difference must now be . This first happens at . After this point, similarly, more numbers are needed to make the th's place go up by . This will take place at . Removing the last four digits (the zeros) and subtracting one yields for the skipped value.
In general, each new value of will add two digits to the "" and one digit to the "". This means that the last number Bernardo writes for is , the last for will be , and so on until . Removing the last digits as Silvia does will be the same as removing trailing zeroes on the number to be squared. This means that the last number on the board for is , is , and so on. So the first missing number is The squaring will make a "" with two more digits than the last number, a "" with one more digit, and a "". The missing number is one less than that, so the "1" will be subtracted from . In other words, .
Therefore:
And so on. The sum is:
+ , with repetitions each of "" and "". There is no carrying in this addition. Therefore each adds to the sum of the digits. Since , , and , or .
Addendum: *You could also use the fact that In other words, the difference between and is equal to . We can set the inequality . Obviously, the first integer that satisfies this is 50. This way, while being longer, is IMO more motivated and doesn't use calculus.
See Also
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
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All AMC 12 Problems and Solutions |
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