Difference between revisions of "2022 AIME II Problems/Problem 9"
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| + | ==Problem== | ||
| + | Let <math>\ell_A</math> and <math>\ell_B</math> be two distinct parallel lines. For positive integers <math>m</math> and <math>n</math>, distinct points <math>A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m</math> lie on <math>\ell_A</math>, and distinct points <math>B_1, B_2, B_3, \ldots, B_n</math> lie on <math>\ell_B</math>. Additionally, when segments <math>\overline{A_iB_j}</math> are drawn for all <math>i=1,2,3,\ldots, m</math> and <math>j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n</math>, no point strictly between <math>\ell_A</math> and <math>\ell_B</math> lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when <math>m=7</math> and <math>n=5</math>. The figure shows that there are 8 regions when <math>m=3</math> and <math>n=2</math>. | ||
| + | <asy> | ||
| + | import geometry; | ||
| + | size(10cm); | ||
| + | draw((-2,0)--(13,0)); | ||
| + | draw((0,4)--(10,4)); | ||
| + | label("$\ell_A$",(-2,0),W); | ||
| + | label("$\ell_B$",(0,4),W); | ||
| + | point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); | ||
| + | draw(B1--A1--B2); | ||
| + | draw(B1--A2--B2); | ||
| + | draw(B1--A3--B2); | ||
| + | label("$A_1$",A1,S); | ||
| + | label("$A_2$",A2,S); | ||
| + | label("$A_3$",A3,S); | ||
| + | label("$B_1$",B1,N); | ||
| + | label("$B_2$",B2,N); | ||
| + | label("1",centroid(A1,B1,I1)); | ||
| + | label("2",centroid(B1,I1,I3)); | ||
| + | label("3",centroid(B1,B2,I3)); | ||
| + | label("4",centroid(A1,A2,I1)); | ||
| + | label("5",(A2+I1+I2+I3)/4); | ||
| + | label("6",centroid(B2,I2,I3)); | ||
| + | label("7",centroid(A2,A3,I2)); | ||
| + | label("8",centroid(A3,B2,I2)); | ||
| + | dot(A1); | ||
| + | dot(A2); | ||
| + | dot(A3); | ||
| + | dot(B1); | ||
| + | dot(B2); | ||
| + | </asy> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | ==See Also== | ||
| + | {{AIME box|year=2022|n=II|num-b=8|num-a=10}} | ||
| + | {{MAA Notice}} | ||
Revision as of 08:26, 18 February 2022
Problem
Let 
and 
be two distinct parallel lines. For positive integers 
and 
, distinct points 
lie on , and distinct points 
lie on . Additionally, when segments 
are drawn for all 
and 
, no point strictly between and lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when 
and 
. The figure shows that there are 8 regions when 
and 
.
![[asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("$\ell_A$",(-2,0),W); label("$\ell_B$",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("$A_1$",A1,S); label("$A_2$",A2,S); label("$A_3$",A3,S); label("$B_1$",B1,N); label("$B_2$",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]](http://latex.artofproblemsolving.com/5/6/c/56c4dfb57fc6e7e474f648f52cc8ce4669b2a526.png)
Solution
See Also
| 2022 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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. 
