Difference between revisions of "1968 AHSME Problems/Problem 30"

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== Solution 1 ==
 
== Solution 1 ==
Notice how <math>P_2</math> can pass through each line segment of <math>P_1</math> at most twice? To have more than two intersections, the line passing through <math>P_1</math> would have a zigzag shape which is impossible for convex polyons. Therefore, the intersections does not depend on <math>P_2</math> and the answer is <math>\fbox{A}</math>
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Notice how <math>P_2</math> can pass through each line segment of <math>P_1</math> at most twice. To have more than two intersections, the line passing through <math>P_1</math> would have a zigzag shape which is impossible for convex polygons. Therefore, the intersections does not depend on <math>P_2</math> and the answer is <math>\fbox{A}</math>
  
 
== Solution 2 ==
 
== Solution 2 ==

Revision as of 20:09, 1 November 2018

Problem

Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1\le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is:

$\text{(A) } 2n_1\quad \text{(B) } 2n_2\quad \text{(C) } n_1n_2\quad \text{(D) } n_1+n_2\quad \text{(E) } \text{none of these}$

Solution 1

Notice how $P_2$ can pass through each line segment of $P_1$ at most twice. To have more than two intersections, the line passing through $P_1$ would have a zigzag shape which is impossible for convex polygons. Therefore, the intersections does not depend on $P_2$ and the answer is $\fbox{A}$

Solution 2

Try to get the answer experimentally. Draw two of the simplest shapes: a square and a triangle and maximize the number of intersections. You will discover it is 6, and the only expression provided that will give 6 is $\fbox{A}$

See also

1968 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 29
Followed by
Problem 31
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 26 27 28 29 30
All AHSME Problems and Solutions

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