2024 Romania Team Selection Tests

Part of Romania Team Selection Test

Subcontests

(3)

1

Do the degrees of freedom matter?

A{}

be a point in the Cartesian plane. At each step, Ann tells Bob a number

0\leqslant a\leqslant 1

and he then moves

A{}

in one of the four cardinal directions, at his choice, by a distance of

a{}.

This process cotinues as long as Ann wishes. Amongst every 100 consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than 100 from the initial position of

A.{}

Can Ann achieve her goal?
Selected from an Argentine Olympiad

1

Nice NT with powers of two

n{}

be a positive integer and let

a{}

b{}

be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer

k{}

such that at least one of the numbers

a^k-b

b^k-a

is divisible by

2^n.

Cătălin Liviu Gherghe

1

Bashy min/max algebra

n\geqslant 2

be a fixed integer. Consider

n

a_1,a_2,\ldots,a_n

not all equal and let

d:=\max_{1\leqslant i<j\leqslant n}|a_i-a_j|;\qquad s=\sum_{1\leqslant i<j\leqslant n}|a_i-a_j|.

Determine in terms of

n{}

the smalest and largest values the quotient

s/d

may achieve.
Selected from the Kvant Magazine