Difference between revisions of "1991 IMO Problems/Problem 5"
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<math>\sum_{i=1}^{3}\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}</math> | <math>\sum_{i=1}^{3}\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \sum_{i=1}^{3}\frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}</math> | ||
− | This implies that for at least one of the values of <math>i=1</math>,<math>2</math>,or <math>3</math> the following is true: | + | This implies that for at least one of the values of <math>i=1</math>,<math>2</math>,or <math>3</math>, the following is true: |
<math>\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}</math> | <math>\frac{sin(A_{i}-30^{\circ})}{sin(30^{\circ})}\le \frac{sin(A_{i}-\alpha_{i})}{sin(\alpha_{i})}</math> | ||
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<math>\frac{sin(\alpha_{i})}{sin(A_{i}-\alpha_{i})}\le \frac{sin(30^{\circ})}{sin(A_{i}-30^{\circ})}</math> | <math>\frac{sin(\alpha_{i})}{sin(A_{i}-\alpha_{i})}\le \frac{sin(30^{\circ})}{sin(A_{i}-30^{\circ})}</math> | ||
− | Which means that for at least one of the values of <math>i=1</math>,<math>2</math>,or <math>3</math> the following is true: | + | Which means that for at least one of the values of <math>i=1</math>,<math>2</math>,or <math>3</math>, the following is true: |
<math>\alpha_{i} \le 30^{\circ}</math> | <math>\alpha_{i} \le 30^{\circ}</math> | ||
{{alternate solutions}} | {{alternate solutions}} |
Revision as of 12:27, 12 November 2023
Problem
Let be a triangle and
an interior point of
. Show that at least one of the angles
is less than or equal to
.
Solution
Let ,
, and
be
,
,
, respcetively.
Let ,
, and
be
,
,
, respcetively.
Using law of sines on we get:
, therefore,
Using law of sines on we get:
, therefore,
Using law of sines on we get:
, therefore,
Multiply all three equations we get:
Using AM-GM we get:
. [Inequality 1]
Note that for ,
decreases with increasing
and fixed
Therefore, decreases with increasing
and fixed
From trigonometric identity:
,
since , then:
Therefore,
and also,
Adding these two inequalities we get:
.
. [Inequality 2]
Combining [Inequality 1] and [Inequality 2] we see the following:
This implies that for at least one of the values of ,
,or
, the following is true:
or
Which means that for at least one of the values of ,
,or
, the following is true:
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.