1982 IMO Problems/Problem 5
Problem
The diagonals and
of the regular hexagon
are divided by inner points
and
respectively, so that
Determine
if
and
are collinear.
Solution 1
O is the center of the regular hexagon. Then we clearly have . And therefore we have also obviously
, as
.
So we have
and
. Because of
the quadrilateral
is cyclic.
. And as we also have
we get
.
. And as
we get
.
This solution was posted and copyrighted by Kathi. The original thread for this problem can be found here: [1]
Solution 2
Let be the intersection of
and
.
is the mid-point of
.
Since
,
, and
are collinear, then by Menelaus Theorem,
.
Let the sidelength of the hexagon be
. Then
.
Substituting them into the first equation yields
This solution was posted and copyrighted by leepakhin. The original thread for this problem can be found here: [2]
Solution 3
Note . From the relation
results
, i.e.
. Thus,
Therefore, , i.e.
This solution was posted and copyrighted by Virgil Nicula. The original thread for this problem can be found here: [3]
Solution 4
File:IMO1982P5.png
, consider
unit. Now,
(after all the simplifying, and substituting
=
).
Now indicates
. So let's go ahead and write
and
. Applying the cosine rule, we get:
.
$MN = \sqrt{CM^2 + CN^2 - 2 \dot CM \dot CN \dot cos \angle MCN} = \sqrt{3 + a^{2} - 2\sqrt{3}\dot a + a^{2} - a\sqrt{3} + a^{2}} = \sqrt{3a^{2} - 3\sqrt{3}\dot a + 3}
This means$ (Error compiling LaTeX. Unknown error_msg)MN = BM \dot \sqrt{3} $.
B, M, and N are collinear if and only if$ (Error compiling LaTeX. Unknown error_msg)\angle CMN = \angle AMB \implies sin\angle CMN = sin\angle AMB
\frac{a}{sin\angle CMN} = \frac{\sqrt{3}\dot BM}{\frac{\sqrt{3}}{2}} = 2BM $.
Now$ (Error compiling LaTeX. Unknown error_msg)\frac{1}{sin\angle AMB} = \frac{BM}{sin 30^{circ}} = 2BM $.
So,$ (Error compiling LaTeX. Unknown error_msg)\frac{1}{sin\angle AMB} = \frac{a}{sin\angle CMN $$ (Error compiling LaTeX. Unknown error_msg)\implies a = 1 = AM$.
So since$ (Error compiling LaTeX. Unknown error_msg)r = \frac{AM}{AC} = \frac{1}{\sqrt{3}} $.
So the answer is$ (Error compiling LaTeX. Unknown error_msg)r = \frac{1}{\sqrt{3}} $.
See Also
1982 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |