Problem 3

Part of 2002 Croatia National Olympiad

Problems(4)

2&Sigma;x^2y^2-&Sigma;x^4=576

Source: Croatia MO 2002 1st Grade P3

4/12/2021

Find all triples

(x,y,z)

of natural numbers that verify the equation

2x^2y^2+2y^2z^2+2z^2x^2-x^4-y^4-z^4=576.

Diophantine equationnumber theory

two triangles with angle relationships

Source: Croatia MO 2002 2nd Grade P3

4/12/2021

If two triangles with side lengths

a,b,c

and

a',b',c'

and the corresponding angle

\alpha,\beta,\gamma

and

\alpha',\beta',\gamma'

satisfy

\alpha+\alpha'=\pi

and

\beta=\beta'

, prove that

aa'=bb'+cc'

.

geometryTriangles

iterated function produces pairwise coprime numbers

Source: Croatia MO 2002 4th Grade P3

4/12/2021

Let

f(x)=x^{2002}-x^{2001}+1

. Prove that for every positive integer

m

, the numbers

m,f(m),f(f(m)),\ldots

are pairwise coprime.

number theoryalgebra

find length ratio in triangular prism

Source: Croatia MO 2002 3rd Grade P3

4/12/2021

Points

E

and

F

are taken on the diagonals

AB_1

and

CA_1

of the lateral faces

ABB_1A_1

and

CAA_1C_1

of a triangular prism

ABCA_1B_1C_1

so that

EF\parallel BC_1

. Find the ratio of the lengths of

EF

and

BC_1

.

ratio3D geometrygeometryprism