1967 IMO Problems/Problem 1

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.

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]

1967 IMO (Problems) • Resources
Preceded by First question	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 2
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1967 IMO Problems/Problem 1

Solution

Remarks (added by pf02, September 2024)

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