MathDB

1

divisibility with condition on lcm

n\ge 2

be an integer. Given numbers

a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}

such that

\operatorname{lcm}(a_i,a_j)>2n

for all

1\le i<j\le n

, prove that

a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).

18

1

Trivial diophantine

Find all integer triples

(a, b, c)

satisfying the equation

5 a^2 + 9 b^2 = 13 c^2.

19

1

Easy oly poly divisibility

Find all polynomials

p

with integer coefficients such that the number

p(a) - p(b)

is divisible by

a + b

for all integers

a, b

, provided that

a + b \neq 0

.

17

1

Largest number of primes

Distinct positive integers

a, b, c, d

satisfy

\begin{cases} a \mid b^2 + c^2 + d^2,\\ b\mid a^2 + c^2 + d^2,\\ c \mid a^2 + b^2 + d^2,\\ d \mid a^2 + b^2 + c^2,\end{cases}

and none of them is larger than the product of the three others. What is the largest possible number of primes among them?

16

1

No NoN-Null iNts

Show that no non-zero integers

a

,

b

,

x

,

y

satisfy

\begin{cases} a x - b y = 16,\\ a y + b x = 1. \end{cases}

15

1

Wild combo geo appeared!

For which positive integers

n\geq4

does there exist a convex

n

-gon with side lengths

1, 2, \dots, n

(in some order) and with all of its sides tangent to the same circle?

14

1

Invasion for the win

Let

ABC

be a triangle with circumcircle

\Gamma

and circumcentre

O

. Denote by

M

the midpoint of

BC

. The point

D

is the reflection of

A

over

BC

, and the point

E

is the intersection of

\Gamma

and the ray

MD

. Let

S

be the circumcentre of the triangle

ADE

. Prove that the points

A

,

E

,

M

,

O

, and

S

lie on the same circle.

13

1

Intersections on intersections

Let

D

be the foot of the

A

-altitude of an acute triangle

ABC

. The internal bisector of the angle

DAC

intersects

BC

at

K

. Let

L

be the projection of

K

onto

AC

. Let

M

be the intersection point of

BL

and

AD

. Let

P

be the intersection point of

MC

and

DL

. Prove that

PK \perp AB

.

12

1

Trivial geo from bw21

Let

I

be the incentre of a triangle

ABC

. Let

F

and

G

be the projections of

A

onto the lines

BI

and

CI

, respectively. Rays

AF

and

AG

intersect the circumcircles of the triangles

CFI

and

BGI

for the second time at points

K

and

L

, respectively. Prove that the line

AI

bisects the segment

KL

.

11

1

Isogonality of P&reflection

A point

P

lies inside a triangle

ABC

. The points

K

and

L

are the projections of

P

onto

AB

and

AC

, respectively. The point

M

lies on the line

BC

so that

KM = LM

, and the point

P'

is symmetric to

P

with respect to

M

. Prove that

\angle BAP = \angle P'AC

.

10

1

John and a string of paper

John has a string of paper where

n

real numbers

a_i \in [0, 1]

, for all

i \in \{1, \ldots, n\}

, are written in a row. Show that for any given

k < n

, he can cut the string of paper into non-empty

k

pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than

1

.

9

1

points on the same circle

We are given

2021

points on a plane, no three of which are collinear. Among any

5

of these points, at least

4

lie on the same circle. Is it necessarily true that at least

2020

of the points lie on the same circle?

8

1

Dogs and coins

We are given a collection of

2^{2^k}

coins, where

k

is a non-negative integer. Exactly one coin is fake. We have an unlimited number of service dogs. One dog is sick but we do not know which one. A test consists of three steps: select some coins from the collection of all coins; choose a service dog; the dog smells all of the selected coins at once. A healthy dog will bark if and only if the fake coin is amongst them. Whether the sick dog will bark or not is random. \\ Devise a strategy to find the fake coin, using at most

2^k+k+2

tests, and prove that it works.

7

1

Stone on hexagonal board

Let

n>2

be an integer. Anna, Edda and Magni play a game on a hexagonal board tiled with regular hexagons, with

n

tiles on each side. The figure shows a board with 5 tiles on each side. The central tile is marked. [asy]unitsize(.25cm); real s3=1.73205081; pair[] points={(-4,4*s3),(-2,4*s3),(0,4*s3),(2,4*s3),(4,4*s3),(-5,3*s3), (-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (5,3*s3), (-6,2*s3),(-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3),(6,2*s3),(-7,s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3),(7,s3),(-8,0), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (8,0),(-7,-s3),(-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (7,-s3), (-6,-2*s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (6,-2*s3), (-5,-3*s3), (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3), (5,-3*s3), (-4,-4*s3), (-2,-4*s3), (0,-4*s3), (2,-4*s3), (4,-4*s3)};
void draw_hexagon(pair p) { draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30))); } {for (int i=0;i<61;++i){draw_hexagon(points);}} label((0,0), "\Large

*

"); [/asy] The game begins with a stone on a tile in one corner of the board. Edda and Magni are on the same team, playing against Anna, and they win if the stone is on the central tile at the end of any player's turn. Anna, Edda and Magni take turns moving the stone: Anna begins, then Edda, then Magni, then Anna, and so on.
The rules for each player's turn are:
[*] Anna has to move the stone to an adjacent tile, in any direction. [*] Edda has to move the stone straight by two tiles in any of the

6

possible directions. [*] Magni has a choice of passing his turn, or moving the stone straight by three tiles in any of the

6

possible directions.
Find all

n

for which Edda and Magni have a winning strategy.

6

1

Existence of indices

Let

n

be a positive integer and

t

be a non-zero real number. Let

a_1, a_2, \ldots, a_{2n-1}

be real numbers (not necessarily distinct). Prove that there exist distinct indices

i_1, i_2, \ldots, i_n

such that, for all

1 \le k, l \le n

, we have

a_{i_k} - a_{i_l} \neq t

.

5

1

x+y from symmetric system

Let

x,y\in\mathbb{R}

be such that

x = y(3-y)^2

and

y = x(3-x)^2

. Find all possible values of

x+y

.

4

1

Mario and Luigi go kart racing

Let

\Gamma

be a circle in the plane and

S

be a point on

\Gamma

. Mario and Luigi drive around the circle

\Gamma

with their go-karts. They both start at

S

at the same time. They both drive for exactly

6

minutes at constant speed counterclockwise around the circle. During these

6

minutes, Luigi makes exactly one lap around

\Gamma

while Mario, who is three times as fast, makes three laps.
While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.
How many points in the plane, apart from

S

, are marked with a banana by the end of the

6

minutes.

3

1

Infinite sequence on positives

Determine all infinite sequences

(a_1,a_2,\dots)

of positive integers satisfying

a_{n+1}^2=1+(n+2021)a_n

for all

n \ge 1

.

2

1

Trivial inequality from bw21

Let

a

,

b

,

c

be the side lengths of a triangle. Prove that

\sqrt[3]{(a^2+bc)(b^2+ca)(c^2+ab)} > \frac{a^2+b^2+c^2}{2}.

1

fe with natural on reals

Let

n

be a positive integer. Find all functions

f\colon \mathbb{R}\rightarrow \mathbb{R}

that satisfy the equation

(f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y)

for all

x ,y \in \mathbb{R}

.

2021 Baltic Way

Subcontests

divisibility with condition on lcm

Trivial diophantine

Easy oly poly divisibility

Largest number of primes

No NoN-Null iNts

Wild combo geo appeared!

Invasion for the win

Intersections on intersections

Trivial geo from bw21

Isogonality of P&reflection

John and a string of paper

points on the same circle

Dogs and coins

Stone on hexagonal board

Existence of indices

x+y from symmetric system

Mario and Luigi go kart racing

Infinite sequence on positives

Trivial inequality from bw21

fe with natural on reals

2021 Baltic Way

Subcontests

divisibility with condition on lcm

Trivial diophantine

Easy oly poly divisibility

Largest number of primes

No NoN-Null iNts

Wild combo geo appeared!

Invasion for the win

Intersections on intersections

Trivial geo from bw21

Isogonality of P&amp;reflection

John and a string of paper

points on the same circle

Dogs and coins

Stone on hexagonal board

Existence of indices

x+y from symmetric system

Mario and Luigi go kart racing

Infinite sequence on positives

Trivial inequality from bw21

fe with natural on reals

Isogonality of P&reflection