Difference between revisions of "1974 IMO Problems/Problem 2"

Revision as of 16:47, 6 January 2019

In the triangle ABC; prove that there is a point D on side AB such that CD is the geometric mean of AD and DB if and only if $\sin{A}\sin{B} \leq \sin^2 (\frac{C}{2})$ .

Solution

Since this is an "if and only if" statement, we will prove it in two parts.

Before we begin, note a few basic but important facts.

1. When two variables $x$ and $y$ are restricted by an equation $x+y=k$ for some constant $k$ , the maximum of their product occurs when $x=y=\frac{k}{2}$ .

2. The triangle inequality states that for a triangle with sides $a$ , $b$ , and $c$ fulfills $a + b > c$ , meaning that $a > \frac{c}{2}$ or $b > \frac{c}{2}$ , which is equivalent to saying that $a$ and $b$ both cannot be less than or equal to $\frac{c}{2}$ .

Part 1:

@@ Line 6: / Line 6: @@
 Since this is an "if and only if" statement, we will prove it in two parts.
+Before we begin, note a few basic but important facts.
+.  When two variables <math>x</math> and <math>y</math> are restricted by an equation <math>x+y=k</math> for some constant <math>k</math>, the maximum of their product occurs when <math>x=y=\frac{k}{2}</math>.
+.  The triangle inequality states that for a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> fulfills <math>a + b > c</math>, meaning that <math>a > \frac{c}{2}</math> or <math>b > \frac{c}{2}</math>, which is equivalent to saying that <math>a</math> and <math>b</math> both cannot be less than or equal to <math>\frac{c}{2}</math>.
 Part 1:

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Difference between revisions of "1974 IMO Problems/Problem 2"

Revision as of 16:47, 6 January 2019

Solution