Difference between revisions of "1968 IMO Problems/Problem 1"
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+ | ==Solution 2== | ||
− | + | (Note: this proof is an expansion by pf02 of an outline of a solution | |
+ | posted here before.) | ||
− | |||
In a given triangle <math>ABC</math>, let <math>A=2B</math>, <math>\implies C=180-3B</math>, and <math>\sin C=\sin 3B</math>. | In a given triangle <math>ABC</math>, let <math>A=2B</math>, <math>\implies C=180-3B</math>, and <math>\sin C=\sin 3B</math>. | ||
Then | Then | ||
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<math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}</math>. | <math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}</math>. | ||
− | + | Denote this ratio by <math>r</math>; we have <math>\sin A = ra, \sin B = rb, \sin C = rc</math>. | |
− | + | Substitute in <math>\sin ^2 A = \sin B(\sin B + \sin C)</math> and simplify by <math>r^2</math>. | |
− | + | We get <math>(*)</math>. | |
At this point, notice that <math>(*)</math> is equivalent to the equality | At this point, notice that <math>(*)</math> is equivalent to the equality | ||
<math>a^2c = b(a^2 + c^2 - b^2)</math> from Solution 1. Indeed, the latter | <math>a^2c = b(a^2 + c^2 - b^2)</math> from Solution 1. Indeed, the latter | ||
can be rewritten as <math>a^2(c - b) = b(c + b)(c - b)</math>, and we know | can be rewritten as <math>a^2(c - b) = b(c + b)(c - b)</math>, and we know | ||
− | that <math>c \ne b</math>. So we could simply quote the fact proven in | + | that <math>c \ne b</math>. So we could simply quote the fact (proven in |
− | Solution 1 that if <math>a, b, c</math> are consecutive integers and | + | Solution 1) that if <math>a, b, c</math> are consecutive integers and |
− | <math>a^2 = b(c + b)</math>, then <math>b = 4, c = 5, a = 6</math> is the only solution. | + | <math>a^2 = b(c + b)</math>, then <math>b = 4, c = 5, a = 6</math> is the only solution |
+ | which could be the sides of a triangle. | ||
− | For the sake of completeness, and for fun, | + | For the sake of completeness, and for fun, I give a slightly |
different proof here. | different proof here. | ||
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numbers are ordered. The six possibilities are: | numbers are ordered. The six possibilities are: | ||
− | 1: <math>a = b - 2, c = b - 1, b</math> | + | 1: <math>\ \ a = b - 2, c = b - 1, b</math> |
− | 2: <math>c = b - 2, a = b - 1, b</math> | + | 2: <math>\ \ c = b - 2, a = b - 1, b</math> |
− | 3: <math>a = b - 1, b, c = b + 1</math> | + | 3: <math>\ \ a = b - 1, b, c = b + 1</math> |
− | 4: <math>c = b - 1, b, a = b + 1</math> | + | 4: <math>\ \ c = b - 1, b, a = b + 1</math> |
− | 5: <math>b, a = b + 1, c = b + 2</math> | + | 5: <math>\ \ b, a = b + 1, c = b + 2</math> |
− | 6: <math>b, c = b + 1, a = b + 2</math> | + | 6: <math>\ \ b, c = b + 1, a = b + 2</math> |
− | For each case, we substitute <math>a, c</math> in <math>(*)</math>, get an equation | + | For each case, we could substitute <math>a, c</math> in <math>(*)</math>, get an |
− | in <math>b</math>, solve it, and get all the possible solutions. | + | equation in <math>b</math>, solve it, and get all the possible solutions. |
− | shortcut, notice that (*) implies that <math>b|a^2</math>. If <math>a, b</math> | + | As a shortcut, notice that <math>(*)</math> implies that <math>b|a^2</math>. If |
− | are consecutive integers, then they are relatively prime, so | + | <math>a, b</math> are consecutive integers, then they are relatively |
− | <math>b|a^2</math> can not be true unless <math>b = 1</math>. In this case the | + | prime, so <math>b|a^2</math> can not be true unless <math>b = 1</math>. In this |
− | triangle would have sides <math>1, 2, 3</math>, which is impossible. | + | case the triangle would have sides <math>1, 2, 3</math>, which is |
− | This eliminates cases 2, 3, 4 and 5. | + | impossible. This eliminates cases 2, 3, 4 and 5. |
In case 1, <math>(*)</math> becomes | In case 1, <math>(*)</math> becomes | ||
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<math>(b + 2)^2 = b(b + b + 1)</math>, or <math>b^2 - 3b - 4 = 0</math>. | <math>(b + 2)^2 = b(b + b + 1)</math>, or <math>b^2 - 3b - 4 = 0</math>. | ||
+ | The solutions are <math>-1, 4</math>. The value <math>b = -1</math> is impossible. | ||
+ | Thus, we get the unique triangle <math>a = 6, b = 4, c = 5</math>. | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
==Solution 3== | ==Solution 3== | ||
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Extend <math>AC</math> to <math>D</math> such that <math>CD = BC = a.</math> Then <math>\angle CDB = \frac{\angle ACB}{2} = \angle ABC</math>, so <math>ABC</math> and <math>ADB</math> are similar by AA Similarity. Hence, <math>c^2 = b(a+b)</math>. Then proceed as in Solution 2, as only algebraic manipulations are left. | Extend <math>AC</math> to <math>D</math> such that <math>CD = BC = a.</math> Then <math>\angle CDB = \frac{\angle ACB}{2} = \angle ABC</math>, so <math>ABC</math> and <math>ADB</math> are similar by AA Similarity. Hence, <math>c^2 = b(a+b)</math>. Then proceed as in Solution 2, as only algebraic manipulations are left. | ||
+ | |||
+ | |||
+ | ==Solution 4== | ||
+ | |||
+ | Note: Adding this 4th solution is justified by the fact | ||
+ | that it is extremely straightforward, and by the fact | ||
+ | that it shows that there are exactly two triangles for | ||
+ | which the sides differ by <math>1</math> (i.e. they are | ||
+ | <math>x, x + 1, x + 2</math> for some <math>x</math>), and the condition | ||
+ | on the angles is satisfied (that one is twice the other). | ||
+ | But only one of the solutions has <math>x</math> integer. | ||
+ | |||
+ | So, let us start by assuming that two angles are | ||
+ | <math>\alpha, 2\alpha</math> and the sides are <math>x, x + 1, x + 2</math>. | ||
+ | We will want to apply the [[Law of Sines]]: | ||
+ | |||
+ | <math>\frac{x}{\sin A} = \frac{x + 1}{\sin B} = \frac{x + 2}{\sin C}</math> | ||
+ | |||
+ | The angles <math>A, B, C</math> should be so that <math>\sin A \le \sin B \le \sin C</math>, | ||
+ | but we don't know how to map <math>\{A, B, C\}</math> to | ||
+ | <math>\{\alpha, 2\alpha, \pi - 3\alpha\}</math>. One thing we know, is that | ||
+ | <math>\sin \alpha < \sin 2\alpha</math>. Indeed, if <math>\alpha \le \pi/4</math> the | ||
+ | inequality is true because <math>\sin</math> is increasing on <math>[0, \pi/2]</math>. | ||
+ | Now note that <math>\alpha < \pi/3</math> since otherwise <math>\pi - 3\alpha</math> | ||
+ | could not be the angle of a triangle. So, if | ||
+ | <math>\pi/4 < \alpha < \pi/3</math> then <math>\pi/2 < 2\alpha < 2\pi/3</math> | ||
+ | and <math>\sin \alpha < \sqrt{3}/2 < \sin 2\alpha</math>. | ||
+ | |||
+ | That means we will have to consider three possibilities: | ||
+ | |||
+ | 1: <math>\ \ \frac{x}{\sin \alpha} = \frac{x + 1}{\sin 2\alpha} = \frac{x + 2}{\sin (\pi - 3\alpha)}</math> | ||
+ | |||
+ | 2: <math>\ \ \frac{x}{\sin \alpha} = \frac{x + 1}{\sin (\pi - 3\alpha)} = \frac{x + 2}{\sin 2\alpha}</math> | ||
+ | |||
+ | 3: <math>\ \ \frac{x}{\sin (\pi - 3\alpha)} = \frac{x + 1}{\sin \alpha} = \frac{x + 2}{\sin 2\alpha}</math> | ||
+ | |||
+ | Using the identities | ||
+ | <math>\sin (\pi - \theta) = \sin \theta, \sin 2\theta = 2\sin \theta \cos \theta</math> | ||
+ | and | ||
+ | <math>\sin 3\theta = 3\sin \theta - 4\sin^3 \theta = \sin \theta\ (4\cos^2 \theta - 1)</math> | ||
+ | and simplifying by <math>\sin \alpha</math> the three cases become | ||
+ | |||
+ | 1: <math>\ \ x = \frac{x + 1}{2\cos \alpha} = \frac{x + 2}{4\cos^2 \alpha - 1}</math> | ||
+ | |||
+ | 2: <math>\ \ x = \frac{x + 1}{4\cos^2 \alpha - 1} = \frac{x + 2}{2\cos \alpha}</math> | ||
+ | |||
+ | 3: <math>\ \ \frac{x}{4\cos^2 \alpha - 1} = x + 1 = \frac{x + 2}{2\cos \alpha}</math> | ||
+ | |||
+ | Each case is a system of two equations in two unknowns, <math>x, \cos \alpha</math>. | ||
+ | We will solve each system, obtain all possible solutions, and chose those | ||
+ | values for which <math>x, x + 1, x + 2</math> and <math>\alpha, 2\alpha, \pi - 3\alpha</math> | ||
+ | can be the sides and angles of a triangle. | ||
+ | |||
+ | Case 1: Compute <math>x</math> from <math>x = \frac{x + 1}{2\cos \alpha}</math>. We get | ||
+ | <math>x = \frac{1}{2\cos \alpha - 1}</math>. Substitute <math>x</math> in | ||
+ | <math>x = \frac{x + 2}{4\cos^2 \alpha - 1}</math>. After doing all the computations | ||
+ | we get <math>4\cos^2 \alpha - 4\cos \alpha = 0</math>. The roots are <math>\cos \alpha = 0</math> | ||
+ | and <math>\cos \alpha = 1</math>. None are acceptable if <math>\alpha</math> is an angle of a | ||
+ | triangle. So case 1 yields no solutions. | ||
+ | |||
+ | Case 2: Compute <math>x</math> from <math>x = \frac{x + 2}{2\cos \alpha}</math>. We get | ||
+ | <math>x = \frac{2}{2\cos \alpha - 1}</math>. Substitute <math>x</math> in | ||
+ | <math>x = \frac{x + 1}{4\cos^2 \alpha - 1}</math>. After doing all the computations | ||
+ | we get <math>8\cos^2 \alpha - 2\cos \alpha - 3 = 0</math>. The solutions are | ||
+ | <math>\cos \alpha = \frac{3}{4}</math> and <math>\cos \alpha = -\frac{1}{2}</math>. Only | ||
+ | <math>\cos \alpha = \frac{3}{4}</math> is acceptable, and it yields | ||
+ | <math>x = \frac{2}{\frac{6}{4} - 1} = 4</math>. Thus <math>4, 5, 6</math> is a possible | ||
+ | solution to the problem. | ||
+ | |||
+ | Case 3: Compute <math>x</math> from <math>x + 1 = \frac{x + 2}{2\cos \alpha}</math>. We get | ||
+ | <math>x = \frac{2 - 2\cos \alpha}{2\cos \alpha - 1}</math>. Substitute <math>x</math> in | ||
+ | <math>x + 1 = \frac{x}{4\cos^2 \alpha - 1}</math>. After doing all the computations | ||
+ | we get <math>4\cos^2 \alpha + 2\cos \alpha - 3 = 0</math>. The roots are | ||
+ | <math>\cos \alpha = \frac{-1 \pm \sqrt{13}}{4} \approx 0.65, -1.15</math>. | ||
+ | The positive value is for an <math>\alpha</math> acceptable as an angle in | ||
+ | a triangle (it is a little over <math>\pi/4</math>), and yields | ||
+ | <math>x = \frac{\sqrt{13} + 1}{2}</math>. We can easily verify that | ||
+ | <math>x, x + 1, x + 2</math> can be the sides of a triangle. (Indeed, | ||
+ | they are | ||
+ | <math>\frac{\sqrt{13} + 1}{2}, \frac{\sqrt{13} + 3}{2}, \frac{\sqrt{13} + 5}{2}</math> | ||
+ | and | ||
+ | <math>\frac{\sqrt{13} + 1}{2} + \frac{\sqrt{13} + 3}{2} > \frac{\sqrt{13} + 5}{2}</math>). | ||
+ | However, they are not integer, so are not solutions to the problem. | ||
+ | |||
+ | The only solution to the problem is the triangle with sides <math>4, 5, 6</math> | ||
+ | from case 2. | ||
+ | |||
+ | (Solution by pf02, August 2024) | ||
+ | |||
==See Also== | ==See Also== | ||
{{IMO box|year=1968|before=First Problem|num-a=2}} | {{IMO box|year=1968|before=First Problem|num-a=2}} | ||
− | + | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] |
Latest revision as of 11:07, 24 August 2024
Problem
Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.
Solution 1
In triangle , let , , , , and . Using the Law of Sines gives that
Therefore . Using the Law of Cosines gives that
This can be simplified to . Since , , and are positive integers, . Note that if is between and , then is relatively prime to and , and cannot possibly divide . Therefore is either the least of the three consecutive integers or the greatest.
Assume that is the least of the three consecutive integers. Then either or , depending on if or . If , then is 1 or 2. couldn't be 1, for if it was then the triangle would be degenerate. If is 2, then , but and must be 3 and 4 in some order, which means that this triangle doesn't exist. therefore cannot divide , and so must divide . If then , so is 1, 2, or 4. Clearly cannot be 1 or 2, so must be 4. Therefore . This shows that and , and the triangle has sides that measure 4, 5, and 6.
Now assume that is the greatest of the three consecutive integers. Then either or , depending on if or . is absurd, so , and . Therefore is 1, 2, or 4. However, all of these cases are either degenerate or have been previously ruled out, so cannot be the greatest of the three consecutive integers. This shows that there is exactly one triangle with this property - and it has side lengths of 4, 5, and 6.
Solution 2
(Note: this proof is an expansion by pf02 of an outline of a solution posted here before.)
In a given triangle , let , , and . Then
Hence,
Indeed, we know from the Law of Sines that
.
Denote this ratio by ; we have . Substitute in and simplify by . We get .
At this point, notice that is equivalent to the equality from Solution 1. Indeed, the latter can be rewritten as , and we know that . So we could simply quote the fact (proven in Solution 1) that if are consecutive integers and , then is the only solution which could be the sides of a triangle.
For the sake of completeness, and for fun, I give a slightly different proof here.
We have six possibilities, depending on how the three consecutive numbers are ordered. The six possibilities are:
1:
2:
3:
4:
5:
6:
For each case, we could substitute in , get an equation in , solve it, and get all the possible solutions. As a shortcut, notice that implies that . If are consecutive integers, then they are relatively prime, so can not be true unless . In this case the triangle would have sides , which is impossible. This eliminates cases 2, 3, 4 and 5.
In case 1, becomes
, or .
This has solutions . The value is impossible. The value yields , which is impossible.
In case 6, becomes
, or .
The solutions are . The value is impossible. Thus, we get the unique triangle .
Solution 3
NO TRIGONOMETRY!!!
Let be the side lengths of a triangle in which
Extend to such that Then , so and are similar by AA Similarity. Hence, . Then proceed as in Solution 2, as only algebraic manipulations are left.
Solution 4
Note: Adding this 4th solution is justified by the fact that it is extremely straightforward, and by the fact that it shows that there are exactly two triangles for which the sides differ by (i.e. they are for some ), and the condition on the angles is satisfied (that one is twice the other). But only one of the solutions has integer.
So, let us start by assuming that two angles are and the sides are . We will want to apply the Law of Sines:
The angles should be so that , but we don't know how to map to . One thing we know, is that . Indeed, if the inequality is true because is increasing on . Now note that since otherwise could not be the angle of a triangle. So, if then and .
That means we will have to consider three possibilities:
1:
2:
3:
Using the identities and and simplifying by the three cases become
1:
2:
3:
Each case is a system of two equations in two unknowns, . We will solve each system, obtain all possible solutions, and chose those values for which and can be the sides and angles of a triangle.
Case 1: Compute from . We get . Substitute in . After doing all the computations we get . The roots are and . None are acceptable if is an angle of a triangle. So case 1 yields no solutions.
Case 2: Compute from . We get . Substitute in . After doing all the computations we get . The solutions are and . Only is acceptable, and it yields . Thus is a possible solution to the problem.
Case 3: Compute from . We get . Substitute in . After doing all the computations we get . The roots are . The positive value is for an acceptable as an angle in a triangle (it is a little over ), and yields . We can easily verify that can be the sides of a triangle. (Indeed, they are and ). However, they are not integer, so are not solutions to the problem.
The only solution to the problem is the triangle with sides from case 2.
(Solution by pf02, August 2024)
See Also
1968 IMO (Problems) • Resources | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |