MathDB

1

\Sigma_{j=0}^{k-1} a_j is divisible by a_k for all indices k < n, in integers

n \ge 3

and let

a_0 = n

. Does there exist a permutation

a_1, a_2,..., a_{n-1}

of the first

n-1

positive integers such that

\Sigma_{j=0}^{k-1} a_j

is divisible by

a_k

for all indices

k < n

?

4

1

n^n + 1 is a perfect number

A positive integer

m

is perfect if the sum of all its positive divisors,

1

and

m

inclusive, is equal to

2m

. Determine the positive integers

n

such that

n^n + 1

is a perfect number.

3

1

spanning tree has an an edge whose adjunction to T produces a simple cycle

Fix an integer

n \ge 2

, let

Q_n

be the graph consisting of all vertices and all edges of an

n

-cube, and let

T

be a spanning tree in

Q_n

. Show that

Q_n

has an edge whose adjunction to

T

produces a simple cycle of length at least

2n

.

2

1

every line through a point meets the boundary of a polygon at 2n points

Given a positive integer

n

, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly

2n

points?

2016 IMAR Test

Subcontests

\Sigma_{j=0}^{k-1} a_j is divisible by a_k for all indices k < n, in integers

n^n + 1 is a perfect number

spanning tree has an an edge whose adjunction to T produces a simple cycle

every line through a point meets the boundary of a polygon at 2n points

2016 IMAR Test

Subcontests

\Sigma_{j=0}^{k-1} a_j is divisible by a_k for all indices k &lt; n, in integers

n^n + 1 is a perfect number

spanning tree has an an edge whose adjunction to T produces a simple cycle

every line through a point meets the boundary of a polygon at 2n points

\Sigma_{j=0}^{k-1} a_j is divisible by a_k for all indices k < n, in integers