Difference between revisions of "2016 AIME I Problems/Problem 14"
(→Solution) |
(+diagram) |
||
Line 12: | Line 12: | ||
(Solution by gundraja) | (Solution by gundraja) | ||
− | + | <asy>size(12cm);draw((0,0)--(7,3));draw(box((0,0),(7,3)),dotted); | |
+ | for(int i=0;i<8;++i)for(int j=0;j<4;++j){dot((i,j),linewidth(1));draw(box((i-.1,j-.1),(i+.1,j+.1)),linewidth(.5));draw(circle((i,j),.1),linewidth(.5));} | ||
+ | </asy> | ||
== See also == | == See also == | ||
{{AIME box|year=2016|n=I|num-b=13|num-a=15}} | {{AIME box|year=2016|n=I|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 09:43, 5 March 2016
Problem
Centered at each lattice point in the coordinate plane are a circle radius and a square with sides of length whose sides are parallel to the coordinate axes. The line segment from to intersects of the squares and of the circles. Find .
Solution
First note that and so every point of the form is on the line. Then consider the line from to . Translate the line so that is now the origin. There is one square and one circle that intersect the line around . Then the points on with an integral -coordinate are, since has the equation :
We claim that the lower right vertex of the square centered at lies on . Since the square has side length , the lower right vertex of this square has coordinates . Because , lies on . Since the circle centered at is contained inside the square, this circle does not intersect . Similarly the upper left vertex of the square centered at is on . Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between and that intersect . Since there are segments from to , the above count is yields circles. Since every lattice point on is of the form where , there are lattice points on . Centered at each lattice point, there is one square and one circle, hence this counts squares and circles. Thus .
(Solution by gundraja)
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.