Difference between revisions of "2018 AMC 10B Problems/Problem 25"
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+ | == Problem == | ||
How many <math>x</math> satisfy the equation <math>x^2 + 10,000\lfloor x \rfloor = 10,000x</math>? | How many <math>x</math> satisfy the equation <math>x^2 + 10,000\lfloor x \rfloor = 10,000x</math>? | ||
<math>\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201</math> | <math>\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201</math> | ||
+ | |||
+ | == Solution == | ||
+ | This rewrites itself to <math>x^2=10,000\{x\}</math>. | ||
+ | |||
+ | Graphing <math>y=10,000\{x\}</math> and <math>y=x^2</math> we see that the former is a set of line segments with slope <math>10,000</math> from <math>0</math> to <math>1</math> with a hole at <math>x=1</math>, then <math>1</math> to <math>2</math> with a whole at <math>x=2</math> etc. | ||
+ | |||
+ | Here is a graph of <math>y=x^2</math> and <math>y=16\{x\}</math> for visualization. | ||
+ | |||
+ | <asy> | ||
+ | import graph; | ||
+ | size(400); | ||
+ | xaxis("$x$",Ticks(Label(fontsize(8pt)),new real[]{-5,-4,-3, -2, -1,0,1 2,3, 4,5})); | ||
+ | yaxis("$y$",Ticks(Label(fontsize(8pt)),new real[]{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})); | ||
+ | real y(real x) {return x^2;} | ||
+ | draw(circle((-4,16), 0.1)); | ||
+ | draw(circle((-3,16), 0.1)); | ||
+ | draw(circle((-2,16), 0.1)); | ||
+ | draw(circle((-1,16), 0.1)); | ||
+ | draw(circle((0,16), 0.1)); | ||
+ | draw(circle((1,16), 0.1)); | ||
+ | draw(circle((2,16), 0.1)); | ||
+ | draw(circle((3,16), 0.1)); | ||
+ | draw(circle((4,16), 0.1)); | ||
+ | draw((-5,0)--(-4,16), black); | ||
+ | draw((-4,0)--(-3,16), black); | ||
+ | draw((-3,0)--(-2,16), black); | ||
+ | draw((-2,0)--(-1,16), black); | ||
+ | draw((-1,0)--(-0,16), black); | ||
+ | draw((0,0)--(1,16), black); | ||
+ | draw((1,0)--(2,16), black); | ||
+ | draw((2,0)--(3,16), black); | ||
+ | draw((3,0)--(4,16), black); | ||
+ | draw(graph(y,-4.2,4.2),green); | ||
+ | </asy> | ||
+ | |||
+ | Now notice that when <math>x=\pm 100</math> then graph has a hole at <math>(\pm 100,10,000)</math> which the equation <math>y=x^2</math> passes through and then continues upwards. Thus our set of possible solutions is bounded by <math>(-100,100)</math>. We can see that <math>y=x^2</math> intersects each of the lines once and there are <math>99-(-99)+1=199</math> lines for an answer of <math>\boxed{\text{(C)}~199}</math>. (Mudkipswims42) | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AMC10 box|year=2018|ab=B|num-b=24|after=Last Problem}} | ||
+ | {{AMC12 box|year=2018|ab=B|num-b=23|num-a=25}} | ||
+ | {{MAA Notice}} |
Revision as of 14:41, 16 February 2018
Problem
How many satisfy the equation ?
Solution
This rewrites itself to .
Graphing and we see that the former is a set of line segments with slope from to with a hole at , then to with a whole at etc.
Here is a graph of and for visualization.
Now notice that when then graph has a hole at which the equation passes through and then continues upwards. Thus our set of possible solutions is bounded by . We can see that intersects each of the lines once and there are lines for an answer of . (Mudkipswims42)
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.