MathDB

2x^2-x-36=p^2

Source: Croatia MO 2001 1st Grade P1

4/13/2021

Find all integers

x

for which

2x^2-x-36

is the square of a prime number.

Diophantine equationnumber theoryPreSchool

complex equation (Croatia MO 2001 2nd Grade P1)

Source:

4/22/2021

Let

z\ne0

be a complex number such that

z^8=\overline z

. What are the possible values of

z^{2001}

?

complex numbersalgebra

lines through parallelogram independent of choice of parallelogram

Source: Croatia MO 2001 3rd Grade P1

4/22/2021

Let

O

and

P

be fixed points on a plane, and let

ABCD

be any parallelogram with center

O

. Let

M

and

N

be the midpoints of

AP

and

BP

respectively. Lines

MC

and

ND

meet at

Q

. Prove that the point

Q

lies on the lines

OP

, and show that it is independent of the choice of the parallelogram

ABCD

.

geometryparallelogram

sequence of circles, find radius

Source: Croatia MO 2001 4th Grade P1

4/22/2021

On the unit circle

k

with center

O

, points

A

and

B

with

AB=1

are chosen and unit circles

k_1

and

k_2

with centers

A

and

B

are drawn. A sequence of circles

(l_n)

is defined as follows: circle

l_1

is tangent to

k

internally at

D_1

and to

k_1,k_2

externally, and for

n>1

circle

l_n

is tangent to

k_1

and

k_2

and to

l_{n-1}

at

D_n

. For each

n

, compute

d_n=OD_n

and the radius

r_n

of

l_n

.

geometrycircles

Problem 1

Problems(4)