MathDB

1

Regarding the incircle and Gergonne Point

ABC

, the incircle touches

BC

in

D

,

CA

in

E

and

AB

in

F

. The bisector of

\angle BAC

intersects

BC

in

G

. The lines

BE

and

CF

intersect in

J

. The line through

J

perpendicular to

EF

intersects

BC

in

K

. Prove that

\frac{GK}{DK}=\frac{AE}{CE}+\frac{AF}{BF}

5

1

Functional Equation

Let

\mathbb{N}_0

denote the set of all nonnegative integers. Determine all functions

f:\mathbb{N}_0\to\mathbb{N}_0

with the following two properties:
[*]

0\le f(x)\le x^2

for all

x\in\mathbb{N}_0

[*]

x-y

divides

f(x)-f(y)

for all

x,y\in\mathbb{N}_0

with

x>y

4

1

Number of feasible airport networks

An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network feasible if it satisfies the following conditions:
[*] All connections operate in both directions [*] If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.
Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?

3

1

SAMO 2011 Senior Round 3 Problem 3

We call a sequence of

m

consecutive integers a friendly sequence if its first term is divisible by

1

, the second by

2

, ..., the

(m-1)^{th}

by

m-1

, and in addition, the last term is divisible by

m^2

Does a friendly sequence exist for (a)

m=20

and (b)

m=11

?

2

1

SAMO 2011 Senior Round 3 Problem 2

Suppose that

x

and

y

are real numbers that satisfy the system of equations

2^x-2^y=1

4^x-4^y=\frac{5}{3}

Determine

x-y

1

SAMO 2011 Senior Round 3 Problem 1

Consider the sequence

2, 3, 5, 6, 7, 8, 10, ...

of all positive integers that are not perfect squares. Determine the

2011^{th}

term of the sequence.

2011 South africa National Olympiad

Subcontests

Regarding the incircle and Gergonne Point

Functional Equation

Number of feasible airport networks

SAMO 2011 Senior Round 3 Problem 3

SAMO 2011 Senior Round 3 Problem 2

SAMO 2011 Senior Round 3 Problem 1