MathDB

1

Finding three elements of the product set with certain divisibility

A

and

B

be sets of positive integers with

|A|\ge 2

and

|B|\ge 2

. Let

S

be a set consisting of

|A|+|B|-1

numbers of the form

ab

where

a\in A

and

b\in B

. Prove that there exist pairwise distinct

x,y,z\in S

such that

x

is a divisor of

yz

.

19

1

A rounded multiple of the divisor function divides the number itself

Denote by

d(n)

the number of positive divisors of a positive integer

n

. Prove that there are infinitely many positive integers

n

such that

\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor

divides

n

.

18

1

Finding the number of solutions of a congruence

Let

n\geq 1

be a positive integer. We say that an integer

k

is a fan of

n

if

0\leq k\leq n-1

and there exist integers

x,y,z\in\mathbb{Z}

such that \begin{align*} x^2+y^2+z^2 &\equiv 0 \pmod n;\\ xyz &\equiv k \pmod n. \end{align*} Let

f(n)

be the number of fans of

n

. Determine

f(2020)

.

17

1

v_p(n!) hits each residue class infinitely often

For a prime number

p

and a positive integer

n

, denote by

f(p, n)

the largest integer

k

such that

p^k \mid n!

. Let

p

be a given prime number and let

m

and

c

be given positive integers. Prove that there exist infinitely many positive integers

n

such that

f(p, n) \equiv c \pmod m

.

16

1

A game with Richard and Kaarel, forming sums of products modulo p

Richard and Kaarel are taking turns to choose numbers from the set

\{1,\dots,p-1\}

where

p > 3

is a prime. Richard is the first one to choose. A number which has been chosen by one of the players cannot be chosen again by either of the players. Every number chosen by Richard is multiplied with the next number chosen by Kaarel.
Kaarel wins the game if at any moment after his turn the sum of all of the products calculated so far is divisible by

p

. Richard wins if this does not happen, i.e. the players run out of numbers before any of the sums is divisible by

p

. Can either of the players guarantee their victory regardless of their opponent's moves and if so, which one?

15

1

Bob drawing three triangles, wants to control their perimeter

On a plane, Bob chooses 3 points

A_0

,

B_0

,

C_0

(not necessarily distinct) such that

A_0B_0+B_0C_0+C_0A_0=1

. Then he chooses points

A_1

,

B_1

,

C_1

(not necessarily distinct) in such a way that

A_1B_1=A_0B_0

and

B_1C_1=B_0C_0

. Next he chooses points

A_2

,

B_2

,

C_2

as a permutation of points

A_1

,

B_1

,

C_1

. Finally, Bob chooses points

A_3

,

B_3

,

C_3

(not necessarily distinct) in such a way that

A_3B_3=A_2B_2

and

B_3C_3=B_2C_2

. What are the smallest and the greatest possible values of

A_3B_3+B_3C_3+C_3A_3

Bob can obtain?

14

1

Reflecting a circle, you find another one

An acute triangle

ABC

is given and let

H

be its orthocenter. Let

\omega

be the circle through

B

,

C

and

H

, and let

\Gamma

be the circle with diameter

AH

. Let

X\neq H

be the other intersection point of

\omega

and

\Gamma

, and let

\gamma

be the reflection of

\Gamma

over

AX

.
Suppose

\gamma

and

\omega

intersect again at

Y\neq X

, and line

AH

and

\omega

intersect again at

Z \neq H

. Show that the circle through

A,Y,Z

passes through the midpoint of segment

BC

.

13

1

A line passing through the orthocenter, creating an angular bisector

Let

ABC

be an acute triangle with circumcircle

\omega

. Let

\ell

be the tangent line to

\omega

at

A

. Let

X

and

Y

be the projections of

B

onto lines

\ell

and

AC

, respectively. Let

H

be the orthocenter of

BXY

. Let

CH

intersect

\ell

at

D

. Prove that

BA

bisects angle

CBD

.

12

1

Midpoints of arcs and a bunch of right angles

Let

ABC

be a triangle with circumcircle

\omega

. The internal angle bisectors of

\angle ABC

and

\angle ACB

intersect

\omega

at

X\neq B

and

Y\neq C

, respectively. Let

K

be a point on

CX

such that

\angle KAC = 90^\circ

. Similarly, let

L

be a point on

BY

such that

\angle LAB = 90^\circ

. Let

S

be the midpoint of arc

CAB

of

\omega

. Prove that

SK=SL

.

11

1

Two Thales circles and a tangent line

Let

ABC

be a triangle with

AB > AC

. The internal angle bisector of

\angle BAC

intersects the side

BC

at

D

. The circles with diameters

BD

and

CD

intersect the circumcircle of

\triangle ABC

a second time at

P \not= B

and

Q \not= C

, respectively. The lines

PQ

and

BC

intersect at

X

. Prove that

AX

is tangent to the circumcircle of

\triangle ABC

.

10

1

Playing hide and seek on the unit square

Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point

B

in the unit square. Then Alice chooses a sequence of points

P_0, P_1, \ldots, P_N

in the plane. After choosing

P_k

(but before choosing

P_{k+1}

) for

k \geq 1

, Bob tells "warmer'' if

P_k

is closer to

B

than

P_{k-1}

, otherwise he says "colder''. After Alice has chosen

P_N

and heard Bob's answer, Alice chooses a final point

A

. Alice wins if the distance

AB

is at most

\frac 1 {2020}

, otherwise Bob wins. Show that if

N=18

, Alice cannot guarantee a win.

9

1

Is there a cool assignment for each graph?

Each vertex

v

and each edge

e

of a graph

G

are assigned numbers

f(v)\in\{1,2\}

and

f(e)\in\{1,2,3\}

, respectively. Let

S(v)

be the sum of numbers assigned to the edges incident to

v

plus the number

f(v)

. We say that an assignment

f

is cool if

S(u) \ne S(v)

for every pair

(u,v)

of adjacent (i.e. connected by an edge) vertices in

G

. Prove that for every graph there exists a cool assignment.

8

1

How many guests can have distinguished orders?

Let

n

be a given positive integer. A restaurant offers a choice of

n

starters,

n

main dishes,

n

desserts and

n

wines. A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine. No two people place exactly the same order. It turns out that there is no collection of

n

guests such that their orders coincide in three of these aspects, but in the fourth one they all differ. (For example, there are no

n

people that order exactly the same three courses of food, but

n

different wines.) What is the maximal number of guests?

7

1

Packing bricks into a box, but exactly how many?

A mason has bricks with dimensions

2\times5\times8

and other bricks with dimensions

2\times3\times7

. She also has a box with dimensions

10\times11\times14

. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out. Find all possible values of the total number of bricks that she can pack.

6

1

A party game on a graph

Let

n>2

be a given positive integer. There are

n

guests at Georg's bachelor party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. All guests now proceed simultaneously as follows. Every guest takes one cup for each of his friends at the party and distributes all the water from his jug evenly in the cups. He then passes a cup to each of his friends. Each guest having received a cup of water from each of his friends pours the water he has received into his jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?

5

1

Funny system of equations in three variables

Find all real numbers

x,y,z

so that \begin{align*} x^2 y + y^2 z + z^2 &= 0 \\ z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4). \end{align*}

4

1

Good old functional equation on the reals

Find all functions

f:\mathbb{R} \to \mathbb{R}

so that

f(f(x)+x+y) = f(x+y) + y f(y)

for all real numbers

x, y

.

3

1

Two sequences with powers of 2 related to sqrt(3)

A real sequence

(a_n)_{n=0}^\infty

is defined recursively by

a_0 = 2

and the recursion formula

a_{n} = \begin{dcases} a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.} \end{dcases}

Another real sequence

(b_n)_{n=1}^\infty

is defined in terms of the first by the formula

b_{n} = \begin{dcases} 0 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,} \end{dcases}

valid for each

n\geq 1

. Prove that

b_1 + b_2 + \cdots + b_{2020} < \frac23.

2

1

Cyclic inequality with fractions and roots

Let

a, b, c

be positive real numbers such that

abc = 1

. Prove that

\frac{1}{a\sqrt{c^2 + 1}} + \frac{1}{b\sqrt{a^2 + 1}} + \frac{1}{c\sqrt{b^2+1}} > 2.

1

Recursive sequence with square roots

Let

a_0>0

be a real number, and let a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \textrm{for } n=1,2,\ldots ,2020. Show that

a_{2020}<\frac1{2020}

.

2020 Baltic Way

Subcontests

Finding three elements of the product set with certain divisibility

A rounded multiple of the divisor function divides the number itself

Finding the number of solutions of a congruence

v_p(n!) hits each residue class infinitely often

A game with Richard and Kaarel, forming sums of products modulo p

Bob drawing three triangles, wants to control their perimeter

Reflecting a circle, you find another one

A line passing through the orthocenter, creating an angular bisector

Midpoints of arcs and a bunch of right angles

Two Thales circles and a tangent line

Playing hide and seek on the unit square

Is there a cool assignment for each graph?

How many guests can have distinguished orders?

Packing bricks into a box, but exactly how many?

A party game on a graph

Funny system of equations in three variables

Good old functional equation on the reals

Two sequences with powers of 2 related to sqrt(3)

Cyclic inequality with fractions and roots

Recursive sequence with square roots