Difference between revisions of "1967 IMO Problems/Problem 4"
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(More about this later.) | (More about this later.) | ||
+ | The quantities <math>a_0, b_0, c_0, \angle B, \angle C</math> are given. From | ||
+ | this data, <math>\angle A_0, \angle B_0, \angle C_0, \angle A</math> are known and | ||
+ | constructible. We will compute <math>BC</math> in terms of <math>\angle \alpha</math> and | ||
+ | these quantities. This will be a function in the variable <math>\alpha</math>, | ||
+ | and we will find the value of <math>\alpha</math> for which this function attains | ||
+ | its maximum. | ||
+ | |||
+ | We will start by computing <math>A_0B</math>. We will use the law of sines in | ||
+ | <math>\triangle A_0C_0B</math>. We get | ||
+ | <math>\frac{A_0B}{\sin (\pi - B - \alpha)} = \frac{b_0}{\sin B}</math>, and a | ||
+ | similar equality from <math>\triangle A_0B_0C</math> (for <math>A_0C</math>). We obtain | ||
+ | |||
+ | <math>BC = A_0B + A_0C = \frac{b_0}{\sin B} \sin (B + \alpha) + | ||
+ | \frac{c_0}{\sin C} \sin (A_0 - C + \alpha) = f(\alpha)</math> | ||
+ | |||
+ | We can now proceed in two ways. We could use the formula for linear | ||
+ | combination of sine functions with same period but different phase | ||
+ | shifts (see | ||
+ | https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinations | ||
+ | or https://mathworld.wolfram.com/HarmonicAdditionTheorem.html , (13)-(23)) | ||
+ | or use calculus to find <math>\alpha</math> for which <math>f(\alpha)</math> has its maximum value. | ||
+ | |||
+ | With the first method, we would obtain that | ||
+ | <math>f(\alpha) = D \sin (\alpha + \theta)</math> for certain <math>D</math> and <math>\theta</math>, | ||
+ | and we would choose <math>\alpha</math> such that <math>\alpha + \theta = \pi/2</math>. | ||
+ | But we will use calculus, as a more mainstream approach. Compute the | ||
+ | derivative <math>f'(\alpha)</math> and consider the equation <math>f'(\alpha) = 0</math>. | ||
+ | Use the formula for <math>\cos</math> of sum of angles, and rearrange terms. | ||
+ | |||
+ | We have | ||
+ | |||
+ | <math>\cos \alpha \cdot \left [ \frac{b_0}{\sin B} \cdot \cos B + | ||
+ | \frac{c_0}{\sin C} \cdot \cos (A_0 - C) \right ]= | ||
+ | \sin \alpha \cdot \left [ \frac{b_0}{\sin B} \cdot \sin B + | ||
+ | \frac{c_0}{\sin C} \cdot \sin (A_0 - C) \right ]</math> | ||
+ | |||
+ | Finally, | ||
+ | |||
+ | <math>\alpha = \arctan \frac{b_0 \cos B \sin C + c_0 \sin B \cos (A_0 - C)} | ||
+ | {b_0 \sin B \sin C + c_0 \sin B \sin (A_0 - C)}</math> | ||
Revision as of 18:16, 3 September 2024
Let and be any two acute-angled triangles. Consider all triangles that are similar to (so that vertices , , correspond to vertices , , , respectively) and circumscribed about triangle (where lies on , on , and on ). Of all such possible triangles, determine the one with maximum area, and construct it.
Solution
We construct a point inside s.t. , where are a permutation of . Now construct the three circles . We obtain any of the triangles circumscribed to and similar to by selecting on , then taking , and then (a quick angle chase shows that are also colinear).
We now want to maximize . Clearly, always has the same shape (i.e. all triangles are similar), so we actually want to maximize . This happens when is the diameter of . Then , so will also be the diameter of . In the same way we show that is the diameter of , so everything is maximized, as we wanted.
This solution was posted and copyrighted by grobber. The thread can be found here: [1]
Solution 2
Since all the triangles circumscribed to are similar, the one with maximum area will be the one with maximum sides, or equivalently, the one with maximum side . So we will try to maximize .
The plan is to find the value of which maximizes .
Note that for any we can construct the line through which forms the angle with . We can construct points on this line, and lines through these points which form angles with the line, and which pass through respectively. Since are acute, is between and these lines will meet at a point such that is between and is between .
(More about this later.)
The quantities are given. From this data, are known and constructible. We will compute in terms of and these quantities. This will be a function in the variable , and we will find the value of for which this function attains its maximum.
We will start by computing . We will use the law of sines in . We get , and a similar equality from (for ). We obtain
We can now proceed in two ways. We could use the formula for linear combination of sine functions with same period but different phase shifts (see https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinations or https://mathworld.wolfram.com/HarmonicAdditionTheorem.html , (13)-(23)) or use calculus to find for which has its maximum value.
With the first method, we would obtain that for certain and , and we would choose such that . But we will use calculus, as a more mainstream approach. Compute the derivative and consider the equation . Use the formula for of sum of angles, and rearrange terms.
We have
Finally,
(Solution by pf02, September 2024)
TO BE CONTINUED. I AM SAVING MID WAY SO AS NOT TO LOSE WORK DONE SO FAR.
See Also
1967 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |