MathDB

1

2018 PAMO Shortlist: Colouring regions created by lines in a plane

n

lines are said to be in standard form if no two are parallel and no three are concurrent. Does there exist a value of

k

such that given any

n

lines in standard form, it is possible to colour the regions bounded by the

n

lines using

k

colours in such a way that no two regions of the same colour share a common intersection point of the

n

lines?

C4

1

2018 PAMO Shortlist: Permutations avoiding consecutive numbers

Seven cyclists follow one another, in a line, on a narrow way. By the end of the training, each cyclist complains about the style of driving of the one in front of him. They agree to rearrange themselves the next day, so that no cyclist would follow the same cyclist he follows the first day. How many such rearrangements are possible?

C3

1

Players take turns moving coin on a chessboard to an unvisited square

A game is played on an

m \times n

chessboard. At the beginning, there is a coin on one of the squares. Two players take turns to move the coin to an adjacent square (horizontally or vertically). The coin may never be moved to a square that has been occupied before. If a player cannot move any more, he loses. Prove:
[*] If the size (number of squares) of the board is even, then the player to move first has a winning strategy, regardless of the initial position. [*] If the size of the board is odd, then the player to move first has a winning strategy if and only if the coin is initially placed on a square whose colour is not the same as the colour of the corners.

C2

1

2018 PAMO Shortlist: Players take turns choosing coefficients of a polynomial

Adamu and Afaafa choose, each in his turn, positive integers as coefficients of a polynomial of degree

n

. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if

n

is odd; otherwise Adamu plays first. Prove that:
[*] Adamu has a winning strategy if

n

is odd. [*] Afaafa has a winning strategy if

n

is even.

G6

1

Arcs are equal in construction involving altitude and angle bisector

Let

\Gamma

be the circumcircle of an acute triangle

ABC

. The perpendicular line to

AB

passing through

C

cuts

AB

in

D

and

\Gamma

again in

E

. The bisector of the angle

C

cuts

AB

in

F

and

\Gamma

again in

G

. The line

GD

meets

\Gamma

again at

H

and the line

HF

meets

\Gamma

again at

I

. Prove that

AI = EB

.

G5

1

2018 PAMO Shortlist: Lenths of lines related to tangency points of incircle

Let

ABC

be a triangle with

AB \neq AC

. The incircle of

ABC

touches the sides

BC

,

CA

,

AB

at

X

,

Y

,

Z

respectively. The line through

Z

and

Y

intersects

BC

extended in

X^\prime

. The lines through

B

that are parallel to

AX

and

AC

intersect

AX^\prime

in

K

and

L

respectively. Prove that

AK = KL

.

G4

1

Intersections of angle bisectors at A and B and the circle with diameter AB

Let

ABC

be a triangle and

\Gamma

be the circle with diameter

[AB]

. The bisectors of

\angle BAC

and

\angle ABC

cut the circle

\Gamma

again at

D

and

E

, respectively. The incicrcle of the triangle

ABC

cuts the lines

BC

and

AC

in

F

and

G

, respectively. Show that the points

D, E, F

and

G

lie on the same line.

G2

1

Intersection of tangents to circles passing through two vertices of a triangle

Let

P

be a point on the median

AM

of a triangle

ABC

. Suppose that the tangents to the circumcircles of

ABP

and

ACP

at

B

and

C

respectively meet at

Q

. Show that

\angle PAB = \angle CAQ

.

G1

1

2018 PAMO Shortlist: Circle with diameter AC, angle bisector, midsegment concur

In a triangle

ABC

, let

D

and

E

be the midpoints of

AB

and

AC

, respectively, and let

F

be the foot of the altitude through

A

. Show that the line

DE

, the angle bisector of

\angle ACB

and the circumcircle of

ACF

pass through a common point.
Alternate version: In a triangle

ABC

, let

D

and

E

be the midpoints of

AB

and

AC

, respectively. The line

DE

and the angle bisector of

\angle ACB

meet at

G

. Show that

\angle AGC

is a right angle.

N7

1

2018 PAMO Shortlist: Diophantine equation $(x^2 + y^2)^n = (xy)^{2018}$

Find all non-negative integers

n

for which the equation

{\left( x^2 + y^2 \right)}^n = {(xy)}^{2018}

admits positive integral solutions.

N6

1

2018 PAMO Shortlist: Arithmetic and geometric mean of divisors is an integer

Prove that there are infinitely many integers

n

such that both the arithmetic mean of its divisors and the geometric mean of its divisors are integers.
(Recall that for

k

positive real numbers,

a_1, a_2, \dotsc, a_k

, the arithmetic mean is

\frac{a_1 +a_2 +\dotsb +a_k}{k}

, and the geometric mean is

\sqrt[k]{a_1 a_2\dotsb a_k}

.)

N5

1

2018 PAMO Shortlist: Diophantine equation

Find all quadruplets

(a, b, c, d)

of positive integers such that

\left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4.

N4

1

Divisibility of natural numbers containing each digit from $1$ to $7$

Let

S

be a set of

49

-digit numbers

n

, with the property that each of the digits

1, 2, 3, \dots, 7

appears in the decimal expansion of

n

seven times (and

8, 9

and

0

do not appear). Show that no two distinct elements of

S

divide each other.

N2

1

2018 PAMO Shortlist: Digits rearranged to form a multiple of $4$

A positive integer is called special if its digits can be arranged to form an integer divisible by

4

. How many of the integers from

1

to

2018

are special?

N1

1

2018 PAMO Shortlist: Are there solutions to $4(ab - a - c^2) = b$?

Does there exist positive integers

a, b, c

such that

4(ab - a - c^2) = b

?

A7

1

2018 PAMO Shortlist: Sequence eventually contains a square

Let

f(n) = n + \lfloor \sqrt{n} \rfloor

. Prove that for every positive integer

m

, the integer sequence

m, f(m), f(f(m)), \dots

contains at least one square of an integer.

A6

1

2018 PAMO Shortlist: Inequality with condition $a^3 + b^3 + c^3 = 5abc$

Let

a, b, c

be positive real numbers such that

a^3 + b^3 + c^3 = 5abc

.
Show that

\left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9.

A5

1

2018 PAMO Shortlist: Completely multiplicative involution on the natural numbers

Let

g : \mathbb{N} \to \mathbb{N}

be a function satisfying:
[*]

g(xy) = g(x)g(y)

for all

x, y \in \mathbb{N}

, [*]

g(g(x)) = x

for all

x \in \mathbb{N}

, and [*]

g(x) \neq x

for

2 \leq x \leq 2018

.
Find the minimum possible value of

g(2)

.

A3

1

Minimum sum of ratios in square divided into red and white rectangles

Akello divides a square up into finitely many white and red rectangles, each (rectangle) with sides parallel to the sides of the parent square. Within each white rectangle, she writes down the value of its width divided by its height, while within each red rectangle, she writes down the value of its height divided by its width. Finally, she calculates

x

, the sum of these numbers. If the total area of the white rectangles equals the total area of the red rectangles, what is the least possible value of

x

she can get?

A2

1

2018 PAMO Shortlist: Bivariate polynomial vanishes at floor function inputs

Find a non-zero polynomial

f(x, y)

such that

f(\lfloor 3t \rfloor, \lfloor 5t \rfloor) = 0

for all real numbers

t

.

A1

1

PAMO Problem 1: Functional Equation

Find all functions

f : \mathbb{Z} \to \mathbb{Z}

such that

(f(x + y))^2 = f(x^2) + f(y^2)

for all

x, y \in \mathbb{Z}

.

2018 Pan-African Shortlist

Subcontests

2018 PAMO Shortlist: Colouring regions created by lines in a plane

2018 PAMO Shortlist: Permutations avoiding consecutive numbers

Players take turns moving coin on a chessboard to an unvisited square

2018 PAMO Shortlist: Players take turns choosing coefficients of a polynomial

Arcs are equal in construction involving altitude and angle bisector

2018 PAMO Shortlist: Lenths of lines related to tangency points of incircle

Intersections of angle bisectors at A and B and the circle with diameter AB

Intersection of tangents to circles passing through two vertices of a triangle

2018 PAMO Shortlist: Circle with diameter AC, angle bisector, midsegment concur

2018 PAMO Shortlist: Diophantine equation $(x^2 + y^2)^n = (xy)^{2018}$

2018 PAMO Shortlist: Arithmetic and geometric mean of divisors is an integer

2018 PAMO Shortlist: Diophantine equation

Divisibility of natural numbers containing each digit from $1$ to $7$

2018 PAMO Shortlist: Digits rearranged to form a multiple of $4$

2018 PAMO Shortlist: Are there solutions to $4(ab - a - c^2) = b$?

2018 PAMO Shortlist: Sequence eventually contains a square

2018 PAMO Shortlist: Inequality with condition $a^3 + b^3 + c^3 = 5abc$

2018 PAMO Shortlist: Completely multiplicative involution on the natural numbers

Minimum sum of ratios in square divided into red and white rectangles

2018 PAMO Shortlist: Bivariate polynomial vanishes at floor function inputs

PAMO Problem 1: Functional Equation