MathDB

1

PAMO 2022 Problem 6 - Product is a power of 11

n_1, n_2, \dots, n_{2022}

such that the number

\left( n_1^{2020} + n_2^{2019} \right)\left( n_2^{2020} + n_3^{2019} \right) \cdots \left( n_{2021}^{2020} + n_{2022}^{2019} \right)\left( n_{2022}^{2020} + n_1^{2019} \right)

is a power of

11

?

5

1

PAMO 2022 Problem 5 - Partition of set of natural numbers

Let

r

be a positive integer. Find the smallest positive integer

m

satisfying the condition: For all sets

A_1, A_2, \dots, A_r

with

A_i \cap A_j = \emptyset

, for all

i \neq j

, and

\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}

, there exists

a, b \in A_k

for some

k

such that

1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}

.

4

1

PAMO 2022 Problem 4 - System of Functional Equations

Find all functions

f

and

g

defined from

\mathbb{R}_{>0}

to

\mathbb{R}_{>0}

such that for all

x, y > 0

the two equations hold

(f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2

(-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1)

Note:

\mathbb{R}_{>0}

denotes the set of positive real numbers.

3

1

PAMO 2022 Problem 3 - Exactly half of the value are smaller than $\sqrt{2} - 1$

Let

n

be a positive integer, and

a_1, a_2, \dots, a_{2n}

be a sequence of positive real numbers whose product is equal to

2

. For

k = 1, 2, \dots, 2n

, set

a_{2n + k} = a_k

, and define

A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}.

Suppose that

A_1, A_2, \dots, A_{2n}

are pairwise distinct; show that exactly half of them are less than

\sqrt{2} - 1

.

2

1

PAMO 2022 Problem 2 - Three expressions are all squares

Find all

3

-tuples

(a, b, c)

of positive integers, with

a \geq b \geq c

, such that

a^2 + 3b

,

b^2 + 3c

, and

c^2 + 3a

are all squares.

1

PAMO 2022 Problem 1 - Line Tangent to Circle Through Orthocenter

Let

ABC

be a triangle with

\angle ABC \neq 90^\circ

, and

AB

its shortest side. Let

H

be the orthocenter of

ABC

. Let

\Gamma

be the circle with center

B

and radius

BA

. Let

D

be the second point where the line

CA

meets

\Gamma

. Let

E

be the second point where

\Gamma

meets the circumcircle of the triangle

BCD

. Let

F

be the intersection point of the lines

DE

and

BH

.
Prove that the line

BD

is tangent to the circumcircle of the triangle

DFH

.

2022 Pan-African

Subcontests

PAMO 2022 Problem 6 - Product is a power of 11

PAMO 2022 Problem 5 - Partition of set of natural numbers

PAMO 2022 Problem 4 - System of Functional Equations

PAMO 2022 Problem 3 - Exactly half of the value are smaller than $\sqrt{2} - 1$

PAMO 2022 Problem 2 - Three expressions are all squares

PAMO 2022 Problem 1 - Line Tangent to Circle Through Orthocenter