2015 PAMO

Part of Pan African

Subcontests

(6)

1

Cyclic implies parallelogram - 2015 PAMO Problem 6

ABCD

be a quadrilateral (with non-perpendicular diagonals).
The perpendicular from

A

BC

CD

K

. The perpendicular from

A

CD

BC

L

. The perpendicular from

C

AB

AD

M

. The perpendicular from

C

AD

AB

N

.
1. Prove that

KL

MN

. 2. Prove that

KLMN

is a parallelogram if

ABCD

1

Most probable sum - 2015 PAMO Problem 5

There are seven cards in a hat, and on the card

k

there is a number

2^{k-1}

k=1,2,...,7

. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds

124

. What is the most probable sum he can get?

1

Sum of greatest odd divisors - 2015 PAMO Problem 4

For a positive integer

n

d(n)

its greatest odd divisor. Find the value of the sum

d(1008)+d(1009)+...+d(2015)

1

Some sum is divisible by 2015 - 2015 PAMO Problem 3

a_1,a_2,...,a_{11}

be integers. Prove that there are numbers

b_1,b_2,...,b_{11}

b_i

-1,0

1

, but not all being

0

, such that the number

N=a_1b_1+a_2b_2+...+a_{11}b_{11}

is divisible by

2015

1

Concurrent lines in hexagon - 2015 PAMO Problem 2

A convex hexagon

ABCDEF

is such that AB=BC CD=DE EF=FA and \angle ABC=2\angle AEC \angle CDE=2\angle CAE \angle EFA=2\angle ACE Show that

AD

CF

EB

are concurrent.

1

Inequality - 2015 PAMO Problem 1

\sqrt{x-1}+\sqrt{2x+9}+\sqrt{19-3x}<9

x

for which the left-hand side is well defined.