2023 IMAR Test

Part of IMAR Test

Subcontests

(4)

1

Polynomial with squared binomial coefficients

n{}

be a non-negative integer and consider the standard power expansion of the following polynomial

\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.

The coefficients

a_{2k+1}

all vanish since the polynomial is invariant under the change

X\mapsto -X.

Prove that the coefficients

a_{2k}

are all positive.

1

Permutation NT

p{}

be an odd prime number. Determine whether there exists a permutation

a_1,\ldots,a_p

1,\ldots,p

(i-j)a_k+(j-k)a_i+(k-i)a_j\neq 0,

for all pairwise distinct

i,j,k.

1

Non-adjacent polygonal faces

n\geqslant 6

coplanar lines, no two parallel and no three concurrent. These lines split the plane into unbounded polygonal regions and polygons with pairwise disjoint interiors. Two polygons are non-adjacent if they do not share a side. Show that there are at least

(n-2)(n-3)/12

pairwise non-adjacent polygons with the same number of sides each.

1

Isosceles triangle geo

ABC

be an acute triangle, and let

D,E,F

be the feet of its altitudes from

A,B,C

respectively. The lines

AB{}

DE

K{}

AC

DF

L{}.

M

be the midpoint of the side

BC

and let the line

AM

cross the circle

(ABC)

N{}.

The parallel through

M{}

EF

crosses the line

KL

P{}.

Prove that the triangle

MNP