MathDB

1

A rhombus' vertices on another rhombus' boundary

ABCD

be a rhombus where

\angle DAB = 60^\circ

, and

P

be the intersection between

AC

and

BD

. Let

Q,R,S

be three points on the boundary of

ABCD

such that

PQRS

is a rhombus. Prove that exactly one of

Q,R,S

lies on one of

A,B,C,D

.

6

1

Find all primes such that two other expressions are primes

Find all primes

p

such that

4p^2+1

and

6p^2+1

are both primes.

5

1

Solving a puzzle-like problem

Nine of the numbers

4, 5, 6, 7, 8, 12, 13, 16, 18, 19

are going to be inputted to the empty cells in the following table: \begin{array} {|c|c|c|} \cline{1-3} 10 & & \\ \cline{1-3} & & 9 \\ \cline{1-3} & 3 & \\ \cline{1-3} 11 & & 17 \\ \cline{1-3} & 20 & \\ \cline{1-3} \end{array} such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.

4

1

Prove that AE = EC + CB

Given a triangle

ABC

where

AC > BC

,

D

is located on the circumcircle of

ABC

such that

D

is the midpoint of the arc

AB

that contains

C

.

E

is a point on

AC

such that

DE

is perpendicular to

AC

. Prove that

AE = EC + CB

.

3

1

Solutions to three equations of three variables

Find all solutions (real and complex) for

x,y,z

, given that:

x+y+z = 6 \\ x^2+y^2+z^2 = 12 \\ x^3+y^3+z^3 = 24

2

1

Products of dice values

Five dice are rolled. The product of the faces are then computed. Which result has a larger probability of occurring;

180

or

144

?

1

Prove of divisibility

Prove that

n^4 - n^2

is divisible by

12

for all integers

n > 1

.

2002 Indonesia MO

Subcontests

A rhombus' vertices on another rhombus' boundary

Find all primes such that two other expressions are primes

Solving a puzzle-like problem

Prove that AE = EC + CB

Solutions to three equations of three variables

Products of dice values

Prove of divisibility