Difference between revisions of "1968 IMO Problems/Problem 1"

Latest revision as of 18:24, 10 November 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 $ABC$ , let $BC=a$ , $AC=b$ , $AB=c$ , $\angle ABC=\alpha$ , and $\angle BAC=2\alpha$ . Using the Law of Sines gives that

$\frac{b}{\sin{\alpha}}=\frac{a}{\sin{2\alpha}}\Rightarrow \frac{\sin{2\alpha}}{\sin{\alpha}}=2\cos{\alpha}=\frac{a}{b}$

Therefore $\cos{\alpha}=\frac{a}{2b}$ . Using the Law of Cosines gives that

$\cos{\alpha}=\frac{a^2+c^2-b^2}{2ac}=\frac{a}{2b}$

This can be simplified to $a^2c=b(a^2+c^2-b^2)$ . Since $a$ , $b$ , and $c$ are positive integers, $b|a^2c$ . Note that if $b$ is between $a$ and $c$ , then $b$ is relatively prime to $a$ and $c$ , and $b$ cannot possibly divide $a^2c$ . Therefore $b$ is either the least of the three consecutive integers or the greatest.

Assume that $b$ is the least of the three consecutive integers. Then either $b|b+2$ or $b|(b+2)^2$ , depending on if $a=b+2$ or $c=b+2$ . If $b|b+2$ , then $b$ is 1 or 2. $b$ couldn't be 1, for if it was then the triangle would be degenerate. If $b$ is 2, then $b(a^2+c^2-b^2)=42=a^2c$ , but $a$ and $c$ must be 3 and 4 in some order, which means that this triangle doesn't exist. therefore $b$ cannot divide $b+2$ , and so $b$ must divide $(b+2)^2$ . If $b|(b+2)^2$ then $b|(b+2)^2-b^2-4b=4$ , so $b$ is 1, 2, or 4. Clearly $b$ cannot be 1 or 2, so $b$ must be 4. Therefore $b(a^2+c^2-b^2)=180=a^2c$ . This shows that $a=6$ and $c=5$ , and the triangle has sides that measure 4, 5, and 6.

Now assume that $b$ is the greatest of the three consecutive integers. Then either $b|b-2$ or $b|(b-2)^2$ , depending on if $a=b-2$ or $c=b-2$ . $b|b-2$ is absurd, so $b|(b-2)^2$ , and $b|(b-2)^2-b^2+4b=4$ . Therefore $b$ is 1, 2, or 4. However, all of these cases are either degenerate or have been previously ruled out, so $b$ 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. $\blacksquare$

Solution 2

(Note: this proof is an expansion by pf02 of an outline of a solution posted here before.)

In a given triangle $ABC$ , let $A=2B$ , $\implies C=180-3B$ , and $\sin C=\sin 3B$ . Then

$\sin ^2 A = \sin ^2 2B = 2 \sin B \cos B \sin 2B = \sin B(\sin B + \sin 3B) = \sin B(\sin B + \sin C)$

Hence,

$a^2 = b(b + c)\ (*)$

Indeed, we know from the Law of Sines that

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$ .

Denote this ratio by $r$ ; we have $\sin A = ra, \sin B = rb, \sin C = rc$ . Substitute in $\sin ^2 A = \sin B(\sin B + \sin C)$ and simplify by $r^2$ . We get $(*)$ .

At this point, notice that $(*)$ is equivalent to the equality $a^2c = b(a^2 + c^2 - b^2)$ from Solution 1. Indeed, the latter can be rewritten as $a^2(c - b) = b(c + b)(c - b)$ , and we know that $c \ne b$ . So we could simply quote the fact (proven in Solution 1) that if $a, b, c$ are consecutive integers and $a^2 = b(c + b)$ , then $b = 4, c = 5, a = 6$ 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: $\ \ a = b - 2, c = b - 1, b$

2: $\ \ c = b - 2, a = b - 1, b$

3: $\ \ a = b - 1, b, c = b + 1$

4: $\ \ c = b - 1, b, a = b + 1$

5: $\ \ b, a = b + 1, c = b + 2$

6: $\ \ b, c = b + 1, a = b + 2$

For each case, we could substitute $a, c$ in $(*)$ , get an equation in $b$ , solve it, and get all the possible solutions. As a shortcut, notice that $(*)$ implies that $b|a^2$ . If $a, b$ are consecutive integers, then they are relatively prime, so $b|a^2$ can not be true unless $b = 1$ . In this case the triangle would have sides $1, 2, 3$ , which is impossible. This eliminates cases 2, 3, 4 and 5.

In case 1, $(*)$ becomes

$(b - 2)^2 = b(b + b - 1)$ , or $b^2 + 3b - 4 = 0$ .

This has solutions $1, -4$ . The value $b = -4$ is impossible. The value $b = 1$ yields $a = -1, c = 0$ , which is impossible.

In case 6, $(*)$ becomes

$(b + 2)^2 = b(b + b + 1)$ , or $b^2 - 3b - 4 = 0$ .

The solutions are $-1, 4$ . The value $b = -1$ is impossible. Thus, we get the unique triangle $a = 6, b = 4, c = 5$ .

Solution 3

NO TRIGONOMETRY!!!

Let $a, b, c$ be the side lengths of a triangle in which $\angle C = 2\angle B.$

Extend $AC$ to $D$ such that $CD = BC = a.$ Then $\angle CDB = \frac{\angle ACB}{2} = \angle ABC$ , so $ABC$ and $ADB$ are similar by AA Similarity. Hence, $c^2 = b(a+b)$ . 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 $1$ (i.e. they are $x, x + 1, x + 2$ for some $x$ ), and the condition on the angles is satisfied (that one is twice the other). But only one of the solutions has $x$ integer.

So, let us start by assuming that two angles are $\alpha, 2\alpha$ and the sides are $x, x + 1, x + 2$ . We will want to apply the Law of Sines:

$\frac{x}{\sin A} = \frac{x + 1}{\sin B} = \frac{x + 2}{\sin C}$

The angles $A, B, C$ should be so that $\sin A \le \sin B \le \sin C$ , but we don't know how to map $\{A, B, C\}$ to $\{\alpha, 2\alpha, \pi - 3\alpha\}$ . One thing we know, is that $\sin \alpha < \sin 2\alpha$ . Indeed, if $\alpha \le \pi/4$ the inequality is true because $\sin$ is increasing on $[0, \pi/2]$ . Now note that $\alpha < \pi/3$ since otherwise $\pi - 3\alpha$ could not be the angle of a triangle. So, if $\pi/4 < \alpha < \pi/3$ then $\pi/2 < 2\alpha < 2\pi/3$ and $\sin \alpha < \sqrt{3}/2 < \sin 2\alpha$ .

That means we will have to consider three possibilities:

1: $\ \ \frac{x}{\sin \alpha} = \frac{x + 1}{\sin 2\alpha} = \frac{x + 2}{\sin (\pi - 3\alpha)}$

2: $\ \ \frac{x}{\sin \alpha} = \frac{x + 1}{\sin (\pi - 3\alpha)} = \frac{x + 2}{\sin 2\alpha}$

3: $\ \ \frac{x}{\sin (\pi - 3\alpha)} = \frac{x + 1}{\sin \alpha} = \frac{x + 2}{\sin 2\alpha}$

Using the identities $\sin (\pi - \theta) = \sin \theta, \sin 2\theta = 2\sin \theta \cos \theta$ and $\sin 3\theta = 3\sin \theta - 4\sin^3 \theta = \sin \theta\ (4\cos^2 \theta - 1)$ and simplifying by $\sin \alpha$ the three cases become

1: $\ \ x = \frac{x + 1}{2\cos \alpha} = \frac{x + 2}{4\cos^2 \alpha - 1}$

2: $\ \ x = \frac{x + 1}{4\cos^2 \alpha - 1} = \frac{x + 2}{2\cos \alpha}$

3: $\ \ \frac{x}{4\cos^2 \alpha - 1} = x + 1 = \frac{x + 2}{2\cos \alpha}$

Each case is a system of two equations in two unknowns, $x, \cos \alpha$ . We will solve each system, obtain all possible solutions, and chose those values for which $x, x + 1, x + 2$ and $\alpha, 2\alpha, \pi - 3\alpha$ can be the sides and angles of a triangle.

Case 1: Compute $x$ from $x = \frac{x + 1}{2\cos \alpha}$ . We get $x = \frac{1}{2\cos \alpha - 1}$ . Substitute $x$ in $x = \frac{x + 2}{4\cos^2 \alpha - 1}$ . After doing all the computations we get $4\cos^2 \alpha - 4\cos \alpha = 0$ . The roots are $\cos \alpha = 0$ and $\cos \alpha = 1$ . None are acceptable if $\alpha$ is an angle of a triangle. So case 1 yields no solutions.

Case 2: Compute $x$ from $x = \frac{x + 2}{2\cos \alpha}$ . We get $x = \frac{2}{2\cos \alpha - 1}$ . Substitute $x$ in $x = \frac{x + 1}{4\cos^2 \alpha - 1}$ . After doing all the computations we get $8\cos^2 \alpha - 2\cos \alpha - 3 = 0$ . The solutions are $\cos \alpha = \frac{3}{4}$ and $\cos \alpha = -\frac{1}{2}$ . Only $\cos \alpha = \frac{3}{4}$ is acceptable, and it yields $x = \frac{2}{\frac{6}{4} - 1} = 4$ . Thus $4, 5, 6$ is a possible solution to the problem.

Case 3: Compute $x$ from $x + 1 = \frac{x + 2}{2\cos \alpha}$ . We get $x = \frac{2 - 2\cos \alpha}{2\cos \alpha - 1}$ . Substitute $x$ in $x + 1 = \frac{x}{4\cos^2 \alpha - 1}$ . After doing all the computations we get $4\cos^2 \alpha + 2\cos \alpha - 3 = 0$ . The roots are $\cos \alpha = \frac{-1 \pm \sqrt{13}}{4} \approx 0.65, -1.15$ . The positive value is for an $\alpha$ acceptable as an angle in a triangle (it is a little over $\pi/4$ ), and yields $x = \frac{\sqrt{13} + 1}{2}$ . We can easily verify that $x, x + 1, x + 2$ can be the sides of a triangle. (Indeed, they are $\frac{\sqrt{13} + 1}{2}, \frac{\sqrt{13} + 3}{2}, \frac{\sqrt{13} + 5}{2}$ and $\frac{\sqrt{13} + 1}{2} + \frac{\sqrt{13} + 3}{2} > \frac{\sqrt{13} + 5}{2}$ ). However, they are not integer, so are not solutions to the problem.

The only solution to the problem is the triangle with sides $4, 5, 6$ from case 2.

[Solution by pf02, August 2024]

@@ Line 20: / Line 20: @@
+==Solution 2==
-{{alternate solutions}}
+(Note: this proof is an expansion by pf02 of an outline of a solution
+posted here before.)
-==Solution 2==
 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
@@ Line 37: / Line 38: @@
 <math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}</math>.
-Denoting this  ratio by <math>r</math>, we have <math>\sin A = ra, \sin B = rb, \sin C = rc</math>.
+Denote this ratio by <math>r</math>; we have <math>\sin A = ra, \sin B = rb, \sin C = rc</math>.
-Substituting in <math>\sin ^2 A = \sin B(\sin B + \sin C)</math> and simplifying by <math>r^2</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>.
+We get <math>(*)</math>.
 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
 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, we give a slightly
+For the sake of completeness, and for fun, I give a slightly
 different proof here.
@@ Line 54: / Line 56: @@
 numbers are ordered.  The six possibilities are:
-: <math>a = b - 2, c = b - 1, b</math>
+: <math>\ \ a = b - 2, c = b - 1, b</math>
-: <math>c = b - 2, a = b - 1, b</math>
+: <math>\ \ c = b - 2, a = b - 1, b</math>
-: <math>a = b - 1, b, c = b + 1</math>
+: <math>\ \ a = b - 1, b, c = b + 1</math>
-: <math>c = b - 1, b, a = b + 1</math>
+: <math>\ \ c = b - 1, b, a = b + 1</math>
-: <math>b, a = b + 1, c = b + 2</math>
+: <math>\ \ b, a = b + 1, c = b + 2</math>
-: <math>b, c = b + 1, a = b + 2</math>
+: <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.  As a
+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
@@ Line 85: / Line 87: @@
 <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>.
-If <math>b</math> is the shortest side, <math>(b+2)^2 = b^2 +b(b+1)</math>
-<math>\implies (b-4)(b+1)=0</math>,
-<math>\implies b=4, c=5, a=6</math>,
-No other permutation of <math>a</math>, <math>b</math> and <math>c</math> in terms of size gives integral values to <math>(*)</math> [show]. So there is only one such triangle.
 ==Solution 3==
@@ Line 99: / Line 97: @@
 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:
+: <math>\ \ \frac{x}{\sin \alpha} = \frac{x + 1}{\sin 2\alpha} = \frac{x + 2}{\sin (\pi - 3\alpha)}</math>
+: <math>\ \ \frac{x}{\sin \alpha} = \frac{x + 1}{\sin (\pi - 3\alpha)} = \frac{x + 2}{\sin 2\alpha}</math>
+: <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
+: <math>\ \ x = \frac{x + 1}{2\cos \alpha} = \frac{x + 2}{4\cos^2 \alpha - 1}</math>
+: <math>\ \ x = \frac{x + 1}{4\cos^2 \alpha - 1} = \frac{x + 2}{2\cos \alpha}</math>
+: <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==
 {{IMO box|year=1968|before=First Problem|num-a=2}}
-* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=361671&sid=6798c42a2ab57f3ca82ffba974ed589c#p361671 Discussion on AoPS/MathLinks]
 [[Category:Olympiad Geometry Problems]]

1968 IMO (Problems) • Resources
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