MathDB

1

$n! + a$ divides $(2n)!$

a

is an integer and that

n! + a

divides

(2n)!

for infinitely many positive integers

n

. Prove that

a = 0

.

5

1

Bridges between villages across a bank

Several small villages are situated on the banks of a straight river. On one side, there are 20 villages in a row, and on the other there are 15 villages in a row. We would like to build bridges, each of which connects a village on the one side with a village on the other side. The bridges must not cross, and it should be possible to get from any village to any other village using only those bridges (and not any roads that might exist between villages on the same side of the river). How many different ways are there to build the bridges.

4

1

Let $ABC$ be an acute-angled triangle with $AB < AC$ ...

Let

ABC

be an acute-angled triangle with

AB < AC

, and let points

D

and

E

be chosen on the side

AC

and

BC

respectively in such a way that

AD = AE = AB

. The circumcircle of

ABE

intersects the line

AC

at

A

and

F

and the line

DE

at

E

and

P

. Prove that

P

is the circumcentre of

BDF

.

3

1

Special divisors of $n$

We call a divisor

d

of a positive integer

n

special if

d + 1

is also a divisor of

n

. Prove: at most half the positive divisors of a positive integer can be special. Determine all positive integers for which exactly half the positive divisors are special.

2

1

Solutions to $5x^3 + ax^2 + 8 = 5x^3 + 8x^2 + a = 0$.

Determine all pairs of real numbers

a

and

x

that satisfy the simultaneous equations

5x^3 + ax^2 + 8 = 0

and

5x^3 + 8x^2 + a = 0.

1

Points $E$ and $F$ inside a square $ABCD$

Points

E

and

F

lie inside a square

ABCD

such that the two triangles

ABF

and

BCE

are equilateral. Show that

DEF

is an equilateral triangle.

2015 South Africa National Olympiad

Subcontests

$n! + a$ divides $(2n)!$

Bridges between villages across a bank

Let $ABC$ be an acute-angled triangle with $AB < AC$ ...

Special divisors of $n$

Solutions to $5x^3 + ax^2 + 8 = 5x^3 + 8x^2 + a = 0$.

Points $E$ and $F$ inside a square $ABCD$

2015 South Africa National Olympiad

Subcontests

$n! + a$ divides $(2n)!$

Bridges between villages across a bank

Let $ABC$ be an acute-angled triangle with $AB &lt; AC$ ...

Special divisors of $n$

Solutions to $5x^3 + ax^2 + 8 = 5x^3 + 8x^2 + a = 0$.

Points $E$ and $F$ inside a square $ABCD$

Let $ABC$ be an acute-angled triangle with $AB < AC$ ...