1999 BAMO

Part of BAMO Contests

Subcontests

(5)

1

BAMO Geo

O = (0,0), A = (0,a), and B = (0,b)

0<b<a

\Gamma

be a circle with diameter

\overline{AB}

P

be any other point on

\Gamma

PA

meets the x-axis again at

Q

. Prove that angle

\angle BQP = \angle BOP

1

Just angle-chasing

ABCD

be a cyclic quadrilateral (a quadrilateral which can be inscribed in a circle). Let

E

F

be variable points on the sides

AB

CD

, respectively, such that

\frac{AE}{EB} = \frac{C}{FD}

P

be the point on the segment

EF

\frac{PE}{PF} = \frac{AB}{CD}

. Prove that the ratio between the areas of triangle

APD

BPC

does not depend on the choice of

E

F

1

BAMO 1999

Finitely many cards are placed in two stacks, with more cards in the left stack than the right. Each card has one or more distinct names written on it, although different cards may share some names. For each name, we define a “shuffle” by moving every card that has this name written on it to the opposite stack. Prove that it is always possible to end up with more cards in the right stack by picking several distinct names, and doing in turn the shuffle corresponding to each name.

1

Don't overthink it (BAMO)

Prove that among any

12

consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers other than

1

n

n

. For example, the proper divisors of

14

2

7

1

Locker problem

16

keys arranged in a 4 \times 4 array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that no matter what the starting positions are, it is always possible to open this lock. (Only one key at a time can be switched.)