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

Difference between revisions of "1967 IMO Problems/Problem 1"

m
Line 81: Line 81:
 
If we take <math>a > 1</math> then the problem is a genuine problem, and
 
If we take <math>a > 1</math> then the problem is a genuine problem, and
 
there is something to prove.
 
there is something to prove.
 +
 +
3. As we will see in the proofs I give below, we don't need to
 +
know that <math>\triangle ABD</math> is acute.  All we need is that <math>\alpha</math>
 +
is acute.
 +
 +
In fact, it is possible to modify <math>S2</math> to a statement <math>S3</math>
 +
similar to <math>S2</math> so that <math>S1</math> and <math>S3</math> are equivalent without
 +
any assumption on <math>alpha</math>.  I will not go into this, I will
 +
just give a hint:  Denote <math>\beta = \angle ABC</math>.  If <math>\alpha</math>
 +
is acute, <math>\beta</math> is obtuse, and we can easily reformulate
 +
<math>S2</math> in terms of <math>\beta</math>.
 +
 +
4. Below, I will give two solutions.  Solution 2 is one I
 +
carried out myself.  Solution 3 is one inspired by an idea
 +
by feliz shown on the web page
 +
https://artofproblemsolving.com/community/c6h21154p137323
 +
The author calls it a solution, but it is so confused that
 +
it can hardly be called a solution.  The idea though is
 +
good and and nice, and it yields a nice solution.
 +
 +
 +
==Solution 2==
 +
  
  

Revision as of 13:01, 17 September 2024

Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\Delta ABD$ is acute, prove that the four circles of radius $1$ with centers $A$, $B$, $C$, $D$ cover the parallelogram if and only if

$a\leq \cos \alpha+\sqrt{3}\sin \alpha$ $\ \ \ \ \ \ \ \ \ \ (1)$

Solution

To start our proof we draw a parallelogram with the requested sides. We notice that by drawing the circles with centers A, B, C, D that the length of $a$ must not exceed 2 (the radius for each circle) or the circles will not meet and thus not cover the parallelogram.

To prove our conjecture we draw a parallelogram with $a=2$ and draw a segment $DB$ so that $\angle ADB=90^{\circ}$

This is the parallelogram which we claim has the maximum length on $a$ and the highest value on any one angle.

We now have two triangles inside a parallelogram with lengths $1, 2$ and $x$, $x$ being segment $DB$. Using the Pythagorean theorem we conclude:

$1^2+x^2=2^2\\x=\sqrt{3}$

Using trigonometric functions we can compute:

$cos\alpha=\frac{1}{2}\\sin\alpha=\frac{\sqrt{3}}{2}$

Notice that by applying the $arcsine$ and $arccos$ functions, we can conclude that our angle $\alpha=60^{\circ}$

To conclude our proof we make sure that our values match the required values for maximum length of $a$

$a\leq\cos\alpha+\sqrt{3}\sin\alpha\\\\a\leq\frac{1}{2}+\sqrt{3}\cdot \frac{\sqrt{3}}{2}\\\\a\leq 2$

Notice that as $\angle\alpha$ decreases, the value of (1) increases beyond 2. We can prove this using the law of sines. Similarly as $\angle\alpha$ increases, the value of (1) decreases below 2, confirming that (1) is only implied when $\Delta ABD$ is acute.

--Bjarnidk 02:16, 17 May 2013 (EDT)


Remarks (added by pf02, September 2024)

1. I am sorry to be so harshly critical, but the solution above is deeply flawed. Not only it has errors, but the logic is flawed.

It shows that when $a = 2, \alpha = \frac{\pi}{3}$ the parallelogram is covered by the circles of radius $1$ centered at $A, B, C, D$, and the inequality in the problem is true. (Even this is incomplete, while giving too many, unnecessary details.) (Note that this is not a case which satisfies the conditions of the problem since $\triangle ABD$ is right, not acute.)

In the last two lines it gives some reasoning about other values of $\alpha$ which is incomprehensible to this reader.

In one short sentence: this is not a solution.

2. The problem itself is mildly flawed. To see this, denote $S1, S2$ the following two statements:

S1: The parallelogram $ABCD$ is covered by the four circles of radius $1$ centered at $A, B, C, D$.

S2: We have $a \le \cos \alpha + \sqrt{3} \sin \alpha$.

The problem says that if $\triangle ABD$ is acute, $S1$ and $S2$ are equivalent, i.e. they imply each other.

Notice that $S2$ can be rewritten as $a \le 2 \cos \left( \alpha - \frac{\sqrt{\pi}}{3} \right)$.

Now notice that if $a \le 1$ then S1 is obviously true.

Also, notice that if $a \le 1$ and $\alpha \in \left( 0, \frac{\sqrt{\pi}}{2} \right)$ then $S2$ as true. Indeed $\left( \alpha - \frac{\sqrt{\pi}}{3} \right) \in \left( -\frac{\sqrt{\pi}}{3}, \frac{\sqrt{\pi}}{6} \right)$, so $\cos$ is $> \frac{1}{2}$ on this interval, so the right hand side of $S2$ is $> 1 \ge a$.

We see that if $a \le 1$ and $\triangle ABD$ is acute, both $S1$ and $S2$ are true. We can not say that one implies the other in the usual meaning of the word "imply": the two statements just happen to be both true.

If we take $a > 1$ then the problem is a genuine problem, and there is something to prove.

3. As we will see in the proofs I give below, we don't need to know that $\triangle ABD$ is acute. All we need is that $\alpha$ is acute.

In fact, it is possible to modify $S2$ to a statement $S3$ similar to $S2$ so that $S1$ and $S3$ are equivalent without any assumption on $alpha$. I will not go into this, I will just give a hint: Denote $\beta = \angle ABC$. If $\alpha$ is acute, $\beta$ is obtuse, and we can easily reformulate $S2$ in terms of $\beta$.

4. Below, I will give two solutions. Solution 2 is one I carried out myself. Solution 3 is one inspired by an idea by feliz shown on the web page https://artofproblemsolving.com/community/c6h21154p137323 The author calls it a solution, but it is so confused that it can hardly be called a solution. The idea though is good and and nice, and it yields a nice solution.


Solution 2

TO BE CONTINUED. I AM SAVING MID WAY SO AS NOT TO LOSE WORK DONE SO FAR.

A solution can also be found here [1]

See Also

1967 IMO (Problems) • Resources
Preceded by
First question 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 2
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1967_IMO_Problems/Problem_1&oldid=227407"