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Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 2"
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==Solution==
 
==Solution==
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We find that it is possible to construct the maximal <math>\boxed{16}</math> points, where each side of one quadrilteral intersects all four sides of the other quadrilateral.
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We find that it is possible to construct the maximal <math>\boxed{16}</math> points, where each side of one quadrilateral intersects all four sides of the other quadrilateral.
 
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[[File:Invites2.png|500px]]
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[[File:Invites2D.png|500px]]
 
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~samrocksnature
 
~samrocksnature
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==Solution 2==
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Take two concave quadrilaterals. Call two lines "somewhat parallel" if the different in their slopes is less than <math>\frac{1}{2}</math>. An arrow has approximately <math>4</math> lines which are "somewhat" parallel, which means <math>2</math> arrows that are <math>90^o</math> to each other form <math>4 \cdot 4 = \boxed{16}</math> intersections.
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<math>\linebreak</math>
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~Geometry285
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== Solution 3 ==
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A line can intersect <math>4</math> other non-parallel lines <math>4</math> times. If we draw <math>2</math> quadrilaterals with non-parallel sides, it would then be possible to get <math>4 \cdot 4 = \boxed{16}</math> intersections.
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~Mathdreams
  
 
==See also==
 
==See also==
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#[[2021 JMPSC Invitational Problems|Other 2021 JMPSC Invitational Problems]]
+
#[[2021 JMPSC Invitationals Problems|Other 2021 JMPSC Invitationals Problems]]
−
#[[2021 JMPSC Invitational Answer Key|2021 JMPSC Invitational Answer Key]]
+
#[[2021 JMPSC Invitationals Answer Key|2021 JMPSC Invitationals Answer Key]]
 
#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
 
#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
 
{{JMPSC Notice}}
 
{{JMPSC Notice}}

Latest revision as of 11:45, 12 July 2021

Problem

Two quadrilaterals are drawn on the plane such that they share no sides. What is the maximum possible number of intersections of the boundaries of the two quadrilaterals?

Solution

We find that it is possible to construct the maximal $\boxed{16}$ points, where each side of one quadrilateral intersects all four sides of the other quadrilateral.

Invites2D.png

~samrocksnature

Solution 2

Take two concave quadrilaterals. Call two lines "somewhat parallel" if the different in their slopes is less than $\frac{1}{2}$. An arrow has approximately $4$ lines which are "somewhat" parallel, which means $2$ arrows that are $90^o$ to each other form $4 \cdot 4 = \boxed{16}$ intersections. $\linebreak$ ~Geometry285

Solution 3

A line can intersect $4$ other non-parallel lines $4$ times. If we draw $2$ quadrilaterals with non-parallel sides, it would then be possible to get $4 \cdot 4 = \boxed{16}$ intersections.

~Mathdreams

See also

  1. Other 2021 JMPSC Invitationals Problems
  2. 2021 JMPSC Invitationals Answer Key
  3. All JMPSC Problems and Solutions

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. JMPSC.png

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