Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

1969 IMO Problems/Problem 5

Revision as of 19:54, 2 August 2024 by Pf02 (talk | contribs)

Problem

Given $n > 4$ points in the plane such that no three are collinear, prove that there are at least $(n-3)(n-4)/2$ convex quadrilaterals whose vertices are four of the given points.

Solution

Orient the points so that one of them corresponds to the origin (A), another lies on the y-axis (B), and all the others are in either quandrant I or IV. (In other words, you are creating a boundary line.) Select one of the points (C) which minimizes the angle BAC. You cannot have two of them because then they would be collinear. Also no points lie inside the area of ABC because if such a point D did exist, then the angle DAC would be less than BAC - contradiction. So there are none of the other n-3 points inside the area of ABC.

Now we will show that for any two of the remaining n-3 points, you can create a convex quadrilateral containing two of them and two of A,B,C.

Now select two arbitrary points from the remaining n-3. Call them E and F, and draw a line though them; call this line l. If l does not intersect ABC, then ABCEF is convex and you choose the quadrilateral ABEF. The other case is that it separates ABC such that one of A,B,C is one one side of l and the other two are on the other side. Without loss of generality, A is on one side of l while B and C are on the other side. Consider the quadrilateral BCEF (or CBEF, depending on orientation.) We know that BC are on the same side of EF. Also, since no points are in the third quadrant, we know that EF are on the same side of BC. Now we can choose either BCEF or CBEF to be our convex quadrilateral.

So we now have that for the given ABC, all pairs of the other n points give a distinct convex quadrilateral. So we have n-3 points to choose from, and we must choose two of them. So this gives us at least C(n-3,2) = $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals.

The above solution was posted and copyrighted by Philip_Leszczynski. The original thread for this problem can be found here: [1]

Remarks (added by pf02, August 2024)

The solution given above is incorrect. This is pointed out very clearly and explicitly by DHu on https://aops.com/community/p364185. I will not repeat the argument here. DHu also gives the idea for a correct proof.

On the same page (https://aops.com/community/p364185), manuel gives an idea for a proof of a stronger result. His proof is somewhere between vague, incorrect and incomplete.

Below, I will give two solutions. The first is just an expansion of DHu's idea. The second one is carrying out manuel's idea.

Solution

We start by proving that given $5$ points, there is at least one convex quadrilateral formed by $4$ of them. Denote the $5$ points by $A, B, C, D, E$. Without loss of generality, we can chose to label the points so that $C, D, E$ are to the right of the line $line(AB)$. Further, we chose to label the point $C$ so that $D, E$ are above the line $line(AC)$.

Notice that from the choice of $A, B, C$ it follows that $\angle(BAC) < 2\pi$.

(We could have chosen $A, B, C$ also by considering the convex hull of the set of 5 points. It will be either a triangle, or a rectangle, or a pentagon, so it will definitely have angles which are $< 2\pi$, and we would have chosen $A, B, C$ that way.)

The picture below shows this choice of labels.

No of quads.png

The line $line(DE)$ must intersect at least one of $line(AB)$ and $line(AC)$. Without loss of generality, we can assume it intersects $line(AB)$. In general, it will also intersect $line(AC)$. The argument we are going to give for the case when $line(DE)$ intersects both $line(AB)$ and $line(AC)$ is identical to the argument needed in the limiting case (when $line(DE) \parallel line(AC)$) and is left as an exercise to the reader.

There are three cases:

Case 1, when $line(DE)$ intersects $segment(AB)$, but it does not intersect $segment(AC)$. This is shown on the first 3 pictures (one picture would have been enough, but I made all of them, for emphasis). In this case the quadrilateral formed by the points $A, D, E, C$ is convex. (Note that this also applies to the case when $line(DE) \parallel line(AC)$.)

Case 2, when $line(DE)$ intersects both $segment(AB)$ and $segment(AC)$. This is shown in the 4th picture. In this case the quadrilateral formed by the points $B, D, E, C$ is convex.

Case 3, when $line(DE)$ does not intersect either of $segment(AB)$ or $segment(AC)$. This is shown in the 5th picture. In this case the quadrilateral formed by the points $B, D, E, C$ is convex. (Note that this also applies to the case when $line(DE) \parallel line(AC)$.)

We thus proved that in any group of 5 points, there is a subgroup of 4 points which forms a convex quadrilateral.

Important remark: In fact, we proved more: if $D, E$ are inside the angle formed by $line(AB)$ and $line(AC)$ (in the sense that $D, E$ are on the same side of $line(AB)$ and on the same side of $line(AC)$, then $D, E$ are both part of the convex quadrilateral.


[TO BE CONTINUED. I AM JUST SAVING THIS BEFORE IT IS FINISHED, SO I DON'T LOSE WORK DONE SO FAR.]

The original thread for this problem can be found here: https://aops.com/community/p364185

See Also

1969 IMO (Problems) • Resources
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 6
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1969_IMO_Problems/Problem_5&oldid=224305"