2023 Pan-African

Part of Pan African

Subcontests

(6)

1

Orthocenter config

ABC

be an acute triangle with

AB<AC

D, E,

F

be the feet of the perpendiculars from

A, B,

C

to the opposite sides, respectively. Let

P

be the foot of the perpendicular from

F

DE

FP

and the circumcircle of triangle

BDF

Q

\angle PBQ = \angle PAD

1

4 numbers with sum more than n/2

n

stamps and an album with

10

pages. He distributes the

n

stamps in the album such that each page has a distinct number of stamps. He finds that, no matter how he does this, there is always a set of

4

pages such that the total number of stamps in these

4

pages is at least

\frac{n}{2}

. Determine the maximum possible value of

n

1

Polynomial has real and positive roots iff a=b

a, b

a \neq 0

P(x)=ax^4-4ax^3+(5a+b)x^2-4bx+b.

Show that all roots of

P(x)

are real and positive if and only if

a=b

1

PAMO 2023 Problem 3

Consider a sequence of real numbers defined by: \begin{align*} x_{1} & = c \\ x_{n+1} & = cx_{n} + \sqrt{c^{2} - 1}\sqrt{x_{n}^{2} - 1} \text{for all } n \geq 1. \end{align*} Show that if

c

is a positive integer, then

x_{n}

is an integer for all

n \geq 1

. (South Africa)

1

PAMO 2023 Problem 2

Find all positive integers

m

n

with no common divisor greater than 1 such that

m^3 + n^3

m^2 + 20mn + n^2

. (Professor Yongjin Song)

1

PAMO 2023 Problem 1

ABC

AB<AC

D

is a point on segment

AC

BD = CD

. A line parallel to

BD

BC

E

AB

F

G

is the intersection of

AE

BD

\angle BCG = \angle BCF

. (Côte d’Ivoire)