Difference between revisions of "2001 IMO Problems/Problem 5"
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==Solution 2== | ==Solution 2== | ||
− | Refer to the image in Solution 1 | + | Refer to the image in Solution 1 without any construction |
− | |||
\begin{align*} | \begin{align*} | ||
− | \text{Set: } & \angle | + | \text{Set: } & \angle ABY = \angle YBC = x, \quad \angle YCB = 120^\circ - 2x. \ |
− | \text{Observe: } & \angle | + | \text{Observe: } & \angle AYB = 120^\circ - x, \quad \angle AXB = 150^\circ - 2x. \ |
\text{Using the Law of Sines, we get: } & \ | \text{Using the Law of Sines, we get: } & \ | ||
− | & | + | & AY = AB \cdot \frac{\sin x^\circ}{\sin(120^\circ - x)}, \ |
− | & | + | & BX = AB \cdot \frac{\sin 30^\circ}{\sin(150^\circ - 2x)}, \ |
− | & | + | & YB = AB \cdot \frac{\sin 60^\circ}{\sin(120^\circ - x)}. \ |
− | \text{So, the relation } AB + | + | \text{So, the relation } AB + BX &= AY + AB \text{ is the same as saying} \ |
& 1 + \frac{\sin 30^\circ}{\sin(150^\circ - 2x)} = \frac{\sin x + \sin 60^\circ}{\sin(120^\circ - x)}. \ | & 1 + \frac{\sin 30^\circ}{\sin(150^\circ - 2x)} = \frac{\sin x + \sin 60^\circ}{\sin(120^\circ - x)}. \ | ||
\text{We have } & \sin x + \sin 60^\circ = 2 \sin\left(\frac{1}{2}(x + 60^\circ)\right) \cos\left(\frac{1}{2}(x - 60^\circ)\right). \ | \text{We have } & \sin x + \sin 60^\circ = 2 \sin\left(\frac{1}{2}(x + 60^\circ)\right) \cos\left(\frac{1}{2}(x - 60^\circ)\right). \ |
Revision as of 11:28, 11 May 2024
Contents
[hide]Problem
is a triangle. lies on and bisects angle . lies on and bisects angle . Angle is . . Find all possible values for angle .
Solution1
Let be on extension of and . Let be on and , then Since , is equilateral. Let , then, We claim that must be on , i.e., . If is not on , then , which leads to , and is equilateral, which is not possible. With that, we have, in , , , and .
Solution by .
Solution 2
Refer to the image in Solution 1 without any construction
~Lakshya Pamecha
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See also
2001 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |