2017 IMAR Test

Part of IMAR Test

Subcontests

(4)

1

least cardinal a cover of class of n-gons by triangles

n

be an integer greater than or equal to

3

P_n

be the collection of all planar (simple)

n

-gons no two distinct sides of which are parallel or lie along some line. For each member

P

P_n

f_n(P)

be the least cardinal a cover of

P

by triangles formed by lines of support of sides of

P

may have. Determine the largest value

f_n(P)

may achieve, as

P

P_n

1

Beautiful set

S

a set of odd positive interger numbers with

n\geq 3

elements such that no element divides another element. We say that a set

S

beautiful

if for any 3 elements from

S

, there is one the divides the sum of the other 2. We call a beautiful set

S

maximal

if we can't add another number to the set such that

S

will still be beautiful. Find the values of

n

for which there exists a

maximal

1

Concyclic points

P

be a point in the interior

\triangle ABC

AD,BE,CF

3 concurrent cevians through

P

D,E,F

BC,CA,AB

. The circle with the diameter

BC

intersects the circle with the diameter

AD

D_1,D_2

. Analogously we define

E_1,E_2

F_1,F_2

D_1,D_2,E_1,E_2,F_1,F_2

1

Sum of residues

k\leq n

r_k

2^n

k

\sum r_i> \frac{n*log_2(\frac{n}{3})}{2}-n

n\geq 2