MathDB

1

Good subsets of the natural numbers (BxMO 2022, Problem 4)

A

of the natural numbers

\mathbb{N} = \{0, 1, 2,\dots\}

is called good if every integer

n>0

has at most one prime divisor

p

such that

n-p\in A

. (a) Show that the set

S = \{0, 1, 4, 9,\dots\}

of perfect squares is good. (b) Find an infinite good set disjoint from

S

. (Two sets are disjoint if they have no common elements.)

3

1

Four concyclic points (BxMO 2022, Problem 3)

Let

ABC

be a scalene acute triangle. Let

B_1

be the point on ray

[AC

such that

|AB_1|=|BB_1|

. Let

C_1

be the point on ray

[AB

such that

|AC_1|=|CC_1|

. Let

B_2

and

C_2

be the points on line

BC

such that

|AB_2|=|CB_2|

and

|BC_2|=|AC_2|

. Prove that

B_1

,

C_1

,

B_2

,

C_2

are concyclic.

2

1

Ants colliding on a line (BxMO 2022, Problem 2)

Let

n

be a positive integer. There are

n

ants walking along a line at constant nonzero speeds. Different ants need not walk at the same speed or walk in the same direction. Whenever two or more ants collide, all the ants involved in this collision instantly change directions. (Different ants need not be moving in opposite directions when they collide, since a faster ant may catch up with a slower one that is moving in the same direction.) The ants keep walking indefinitely. Assuming that the total number of collisions is finite, determine the largest possible number of collisions in terms of

n

.

1

Polynomial bounded by sum of coefficients (BxMO 2022, Problem 1)

Let

n\geqslant 0

be an integer, and let

a_0,a_1,\dots,a_n

be real numbers. Show that there exists

k\in\{0,1,\dots,n\}

such that

a_0+a_1x+a_2x^2+\cdots+a_nx^n\leqslant a_0+a_1+\cdots+a_k

for all real numbers

x\in[0,1]

.

2022 Benelux

Subcontests

Good subsets of the natural numbers (BxMO 2022, Problem 4)

Four concyclic points (BxMO 2022, Problem 3)

Ants colliding on a line (BxMO 2022, Problem 2)

Polynomial bounded by sum of coefficients (BxMO 2022, Problem 1)