Putnam Problem of the Day

[http://math.harvard.edu/putnam(...) !!84u21X0KjW+ No.10623607]

[math]
\text{Let } ABC \text{ be an acute-angled triangle whose side lengths satisfy the inequalities}
\\
AB < AC < BC \text{. If point } I \text{ is the center of the inscribed circle of triangle}
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ABC \text{ and point } O \text{ is the center of the circumscribed circle, prove that line } IO
\\
\text{intersects segments } AB \text{ and } BC \text{.}
[/math]

[http://math.harvard.edu/putnam(...) !!84u21X0KjW+ No.10623612]

Previous Thread >>10621806

[Anonymous No.10623697]

Can I have the raw Cirno pic please?

[Anonymous No.10623706]

Nice problem, but what's with the pedophile cartoon?

>>10623697
What does this post have to do with science and/or mathematics?

[Anonymous No.10623726]

>>10623706

How do you even browse this place if a pic as innocent as that triggers you?

[Anonymous No.10623781]

>>10623607
OP i just wanna say I love you for posting these, gonna work on this problem for an hour or so and see what progress I make

[Anonymous No.10624100]

>>10623607
This is supposed to be read with a picture, please draw one yourself.

A solution would be like so:
Take [math]\alpha[/math] the angle ABC.
[math]\beta[/math] the angle BAC.
[math]\gamma[/math] the angle BCA.
AB<AC<BC is the same as saying
[math]\alpha>\beta>\gamma[/math].

The angle AOC is [math]2\alpha[/math] If P
is the midpoint of AC (so that APC is perpendicular), then OAC is [math]90-\alpha[/math].
On the other hand angle IAC is [math]\beta/2[/math],
Using [math]\alpha+\beta+\gamma=180[/math] you see OAC<IAC
Similarly IBA<OBA, so I is inside the triangle
ABO. So line OI intersects segment AB.

BC is similar.

[Anonymous No.10624334]

>>10624100
>AB<AC<BC is the same as saying
>α > β > γ.
nah

[Anonymous No.10625595]

>>10624334
Thanks for the drawing mate, I don't know how to make one. If you could add O and I that would be better.
That's the problem with the geometry problems, the drawing I have has AC>BC>AB I fucked up the labellings. Anyway, what happens is that the bigger angles are opposite to the bigger sides.

[Anonymous No.10627337]

Where's the new thread?