1967 IMO Problems/Problem 1
Let be a parallelogram with side lengths , , and with . If is acute, prove that the four circles of radius with centers , , , cover the parallelogram if and only if
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 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 and draw a segment so that
This is the parallelogram which we claim has the maximum length on and the highest value on any one angle.
We now have two triangles inside a parallelogram with lengths and , being segment . Using the Pythagorean theorem we conclude:
Using trigonometric functions we can compute:
Notice that by applying the and functions, we can conclude that our angle
To conclude our proof we make sure that our values match the required values for maximum length of
Notice that as decreases, the value of (1) increases beyond 2. We can prove this using the law of sines. Similarly as increases, the value of (1) decreases below 2, confirming that (1) is only implied when is acute.
--Bjarnidk 02:16, 17 May 2013 (EDT)
Remarks (added by pf02, September 2024)
. 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 the parallelogram is covered by the circles of radius centered at , 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 is right, not acute.)
In the last two lines it gives some reasoning about other values of which is incomprehensible to this reader.
In one short sentence: this is not a solution.
. The problem itself is mildly flawed. To see this, denote the following two statements:
S1: The parallelogram is covered by the four circles of radius centered at .
S2: We have .
The problem says that if is acute, and are equivalent, i.e. they imply each other.
Notice that can be rewritten as .
Now notice that if then S1 is obviously true. See the picture below:
Also, notice that if and then is true as well. Indeed , so is on this interval, so the right hand side of is .
We see that if and is acute, both and 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 then the problem is a genuine problem, and there is something to prove.
. In the proofs I give below, we will see where we need that is acute. We will see that is needed for the technicalities of the proof. The fact that is acute will be needed at one crucial point in the proof.
In fact, it is possible to modify to a statement similar to so that and are equivalent without any assumption on . I will not go into this, I will just give a hint: Denote . If is acute, is obtuse, and we can easily reformulate in terms of .
. Below, I will give two solutions. Solution 2 is one I carried out myself and relies on a straightforward computation. Solution 3 (it happens to be similar to Solution 2) is inspired by an idea by feliz shown on the web page https://artofproblemsolving.com/community/c6h21154p137323 The author calls their text a solution, but it is quite confused, so I would not call it a good solution. The idea though is good and nice, and it yields a nice solution.
Solution 2
We can assume . Indeed, refer to Remark 2 above to see that if there is nothing to prove.
Note that instead of the statement we can consider the following statement :
: the circles of radius centered at cover .
This is equivalent to because of the symmetry between and .
Let be the intersection above of the circles of radius centered at . The three circles cover if an only if is inside the circle of radius 1 centered at .
This needs an explanation: Let be the midpoint of , and consider . All the vertices of this triangle are in the circle centered at , so the whole triangle is inside this circle. It is obvious that is inside the circle centered at , and that are inside the circles centered at .
We will now use that is inside the circle of radius 1 centered at if an only if .
The plan is to calculate in terms of and impose this condition. Let , and . From the right triangle we have . From the right triangle we have
(Note that here we used the fact that is acute. These equalities would look slightly differently otherwise.)
Now look at the condition , or equivalently . Making all the computations and simplifications, we have .
Now I would like to square both sides. In order to get an equivalent inequality, we need to know that . This follows from the fact that is acute. Indeed, denote . From the law of sines in we have . Successively this becomes or or . From here we see that implies the right hand side is positive, so .
Going back to our inequality, we can square both sides, and after rearranging terms we get that if and only if
.
View this as an equation of degree in . The value of the polynomial in is when is between its solutions, that is
.
Note that . If then , and it follows that .
On the other hand, remember that we are in the case , so the left inequality is always true. It follows that
(i.e. the three circles of radius centered at cover ) if an only if .
(Solution by pf02, September 2024)
Solution 3
This solution is very similar to Solution 2, except that we choose another point instead of . This will in fact simplify the proof. Start like in Solution 2.
We can assume . Indeed, refer to Remark 2 above to see that if there is nothing to prove.
Note that instead of the statement we can consider the following statement :
: the circles of radius centered at cover .
This is equivalent to because of the symmetry between and .
Let be the center of the circle circumscribed to . Let , be the midpoints of . The three circles cover if an only if is inside the circle of radius 1 centered at .
This needs an explanation. In fact, since , the point is inside or on the circle centered at if and only if , if and only if is in or inside the circles centered at . Since , , so the triangles are inside the circles centered at respectively. By drawing we can easily verify that the whole triangle is inside the circles centered at .
Note that in the above argument we used that is inside , which is true because the triangle is acute.
Denote the radius of the circle circumscribed to . From the law of sines, we have , and from the law of cosines we have . So translates to
Since , we can simplify this inequality, and get
.
But this is exactly the inequality we encountered in Solution 2, so proving that this is equivalent to
is identical to what has been done above.
(Solution based on an idea by feliz; see link in Remark 4, or below.)
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 |