MathDB

2

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

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

such that

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

for all integers

x

and

y

?

BM=CF and <MBC=<FCA means ABC is equilateral

In an acute-angled triangle

ABC

,

CF

is an altitude, with

F

on

AB

, and

BM

is a median, with

M

on

CA

. Given that

BM=CF

and

\angle MBC=\angle FCA

, prove that triangle

ABC

is equilateral.

2

19 girls in line, no girl is shorter behind/in front of her

How many ways are there to line up

19

girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

Choosing a_0 so that sequence is infinite

A sequence

a_0,a_1,a_2,\ldots ,a_n,\ldots

of positive integers is constructed as follows: [*]if the last digit of

a_n

is less than or equal to

5

then this digit is deleted and

a_{n+1}

is the number consisting of the remaining digits. (If

a_{n+1}

contains no digits the process stops.) [*]otherwise

a_{n+1}=9a_n

. Can one choose

a_0

so that an infinite sequence is obtained?

1

2

Pairing off so that sum is a prime

a) Show that it is possible to pair off the numbers

1,2,3,\ldots ,10

so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers

1,2,3,\ldots ,20

so that the sums of each of the ten pairs are ten different prime numbers?

At least one of the 7 arcs created is at most 1/24

Seven distinct points are marked on a circle of circumference

c

. Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal

\frac{c}{24}

.

2010 Pan African

Subcontests

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

BM=CF and <MBC=<FCA means ABC is equilateral

19 girls in line, no girl is shorter behind/in front of her

Choosing a_0 so that sequence is infinite

Pairing off so that sum is a prime

At least one of the 7 arcs created is at most 1/24

2010 Pan African

Subcontests

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

BM=CF and &lt;MBC=&lt;FCA means ABC is equilateral

19 girls in line, no girl is shorter behind/in front of her

Choosing a_0 so that sequence is infinite

Pairing off so that sum is a prime

At least one of the 7 arcs created is at most 1/24

BM=CF and <MBC=<FCA means ABC is equilateral