MathDB

1

These three integers cannot all be prime

a_1,a_2

and

a_3

are distinct positive integers, such that
(i)

a_1

is a divisor of

a_2+a_3+a_2a_3

;
(ii)

a_2

is a divisor of

a_3+a_1+a_3a_1

;
(iii)

a_3

is a divisor of

a_1+a_2+a_1a_2

.
Prove that

a_1,a_2

and

a_3

cannot all be prime.

5

1

How many square roots of 1 mod n?

For

n\ge 2

, find the number of integers

x

with

0\le x<n

, such that

x^2

leaves a remainder of

1

when divided by

n

.

4

1

Isosceles triangles and tangent circles

Let

A_1

and

B_1

be two points on the base

AB

of isosceles triangle

ABC

(with

\widehat{C}>60^\circ

) such that

\widehat{A_1CB_1}=\widehat{BAC}

. A circle externally tangent to the circumcircle of triangle

\triangle A_1B_1C

is tangent also to rays

CA

and

CB

at points

A_2

and

B_2

respectively. Prove that

A_2B_2=2AB

.

3

1

Equation involving floors

Find all real numbers

x

such that

x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88

. The notation

\lfloor x\rfloor

means the greatest integer less than or equal to

x

.

2

1

How many triangles can you find inside this polygon?

Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there?

1

GCD of two numbers consisting of ones

Let

a=1111\dots1111

and

b=1111\dots1111

where

a

has forty ones and

b

has twelve ones. Determine the greatest common divisor of

a

and

b

.

2004 South africa National Olympiad

Subcontests

These three integers cannot all be prime

How many square roots of 1 mod n?

Isosceles triangles and tangent circles

Equation involving floors

How many triangles can you find inside this polygon?

GCD of two numbers consisting of ones