2018 IMAR Test

Part of IMAR Test

Subcontests

(4)

1

IMAR Test 2018 P4

Prove that every non-negative integer

n

is expressible in the form

n=t^2+u^2+v^2+w^2

t,u,v,w

are integers such that

t+u+v+w

is a perfect square.
* * *

1

IMAR Test 2018 P3

S

be a finite set and let

\mathcal{P}(S)

be its power set, i.e., the set of all subsets of

S

, the empty set and

S

, inclusive. If

\mathcal{A}

\mathcal{B}

are non-empty subsets of

\mathcal{P}(S),

\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}.

Given a non-negative integer

n\leqslant |S|,

determine the minimal size

\mathcal{A}\vee \mathcal{B}

may have, where

\mathcal{A}

\mathcal{B}

are non-empty subsets of

\mathcal{P}(S)

|\mathcal{A}|+|\mathcal{B}|>2^n

.
Amer. Math. Monthly

1

IMAR Test 2018 P2

P

be a non-zero polynomial with non-negative real coefficients, let

N

be a positive integer, and let

\sigma

be a permutation of the set

\{1,2,...,n\}

. Determine the least value the sum

\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}

may achieve, as

x_1,x_2,...,x_n

run through the set of positive real numbers.
Fedor Petrov

1

IMAR Test 2018 P1

ABC

be a triangle whose angle at

A

is right, and let

D

be the foot of the altitude from

A

. A variable point

M

traces the interior of the minor arc

AB

ABC

. The internal bisector of the angle

DAM

CM

N

. The line through

N

and perpendicular to

CM

crosses the line

AD

P

. Determine the locus of the point where the line

BN

crosses the line

CP