2016 USAMO

Part of USAMO

Subcontests

(4)

1

Angel Bisector and Equilateral Pentagon AMNPQ

An equilateral pentagon

AMNPQ

is inscribed in triangle

ABC

M\in\overline{AB}

Q\in\overline{AC}

N,P\in\overline{BC}

S

be the intersection of

\overleftrightarrow{MN}

\overleftrightarrow{PQ}

\ell

the angle bisector of

\angle MSQ

\overline{OI}

\ell

O

is the circumcenter of triangle

ABC

I

is the incenter of triangle

ABC

1

Wizard101

n

k

are given, with

n\ge k\ge2

. You play the following game against an evil wizard.
The wizard has

2n

cards; for each

i=1,\ldots,n

, there are two cards labeled

i

. Initially, the wizard places all cards face down in a row, in unknown order.
You may repeatedly make moves of the following form: you point to any

k

of the cards. The wizard then turns those cards face up. If any two of the cards match, the game is over and you win. Otherwise, you must look away, while the wizard arbitrarily permutes the

k

chosen cards and then turns them back face-down. Then, it is your turn again.
We say this game is winnable if there exist some positive integer

m

and some strategy that is guaranteed to win in at most

m

moves, no matter how the wizard responds.
For which values of

n

k

is the game winnable?

1

Ratio of Factorials

Prove that for any positive integer

k

(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}

1

Incenters, Excenters, and Feet, OH MY!

\triangle ABC

be an acute triangle, and let

I_B, I_C,

O

B

C

-excenter, and circumcenter, respectively. Points

E

Y

are selected on

\overline{AC}

\angle ABY=\angle CBY

\overline{BE}\perp\overline{AC}

. Similarly, points

F

Z

are selected on

\overline{AB}

\angle ACZ=\angle BCZ

\overline{CF}\perp\overline{AB}

\overleftrightarrow{I_BF}

\overleftrightarrow{I_CE}

P

\overline{PO}

\overline{YZ}

are perpendicular.
Proposed by Evan Chen and Telv Cohl