1968 Yugoslav Team Selection Test

Part of Serbia Team Selection Test

Subcontests

(5)

1

tetrahedron, incenter=circumcenter

Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.

1

a summation over pairs of integers, polynomial

n

be an integer greater than

1

x\in\mathbb R

S(x,n)=\sum(x+p)(x+q)

, where the summation is over all pairs

(p,q)

of different numbers from

\{1,2,\ldots,n\}

. (b) Do there exist integers

x,n

S(x,n)=0

1

polynomial [n]->Z implies it's Z->Z

If a polynomial of degree n has integer values when evaluated in each of

k,k+1,\ldots,k+n

k

is an integer, prove that the polynomial has integer values when evaluated at each integer

x

1

triangle transformation

Each side of a triangle

ABC

is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of

\triangle ABC

three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by

A',B',C'

the vertices of these new equilateral triangles that don’t belong to the edges of

\triangle ABC

, respectively. Let

A'',B'',C''

be the points symmetric to

A',B',C'

with respect to

BC,CA,AB

.
(a) Prove that

\triangle A'B'C'

\triangle A''B''C''

are equilateral. (b) Prove that

ABC,A'B'C'

A''B''C''

have a common centroid.

1

(n-2)!=an+1 iff n prime

n>3

be a positive integer. Prove that

n

is prime if and only if there exists a positive integer

\alpha

n!=n(n-1)(\alpha n+1)