MathDB

1

Hard algebra with permutations from KoMaL

(x_1,\ldots,x_n)

, denote by

N(x_1,\ldots,x_n)

the number of ordered index pairs

(i,j)

for which

1 \le i<j\le n

and

x_i=x_j

. Let

p

be an odd prime,

1 \le n<p

, and let

a_1,a_2,\ldots,a_n

and

b_1,b_2,\ldots,b_n

be arbitrary residue classes modulo

p

. Prove that there exists a permutation

\pi

of the indices

1,2,\ldots,n

for which

N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).

A. 714

1

Fixed points of composition of inversions

Consider

n \ge 2

pairwise disjoint disks

D_1,D_2,\ldots,D_n

on the Euclidean plane. For each

k=1,2,\ldots,n

, denote by

f_k

the inversion with respect to the boundary circle of

D_k

. (Here,

f_k

is defined at every point of the plane, except for the center of

D_k

.) How many fixed points can the transformation

f_n\circ f_{n-1}\circ\ldots\circ f_1

have, if it is defined on the largest possible subset of the plane?

A. 711

1

Image of regular m-gon is a regular n-gon

For which pairs

(m,n)

does there exist an injective function

f:\mathbb{R}^2\to\mathbb{R}^2

under which the image of every regular

m

-gon is a regular

n

-gon. (Note that

m,n\geq 3

, and that by a regular

N

-gon we mean the union of the boundary segments, not the closed polygonal region.)
Proposed by Sutanay Bhattacharya, Bishnupur, India

A. 710

1

Partitioning a regular n-gon in triangles

For which

n{}

can we partition a regular

n{}

-gon into finitely many triangles such that no two triangles share a side?
Based on a problem of the 2017 Miklós Schweitzer competition

A. 708

1

A very strange binary number is rational

Let

S

be a finite set of rational numbers. For each positive integer

k

, let

b_k=0

if we can select

k

(not necessarily distinct) numbers in

S

whose sum is

0

, and

b_k=1

otherwise. Prove that the binary number

0.b_1b_2b_3…

is a rational number. Would this statement remain true if we allowed

S

to be infinite?

A. 707

1

Number a betyars a betyar sees

100

betyárs stand on the Hortobágy plains. Every betyár's field of vision is a

100

degree angle. After each of them announces the number of other betyárs they see, we compute the sum of these

100

numbers. What is the largest value this sum can attain?

A. 704

1

Partitioning triangular board into disjoint unit triangles

A regular triangle has side length

n{}

. We divided its sides into

n{}

equal parts and drew a line segment parallel with each side through the dividing points. A lattice of

1+2+\ldots+(n+1)

intersection points is thus formed. For which positive integers

n{}

can this lattice be partitioned into triplets of points which are the vertices of a regular triangle of side length

1

?
Proposed by Alexander Gunning, Cambridge, UK

A. 703

1

Generalisation of IMO 2017 P6

Let

n\ge 2

be an integer. We call an ordered

n

-tuple of integers primitive if the greatest common divisor of its components is

1

. Prove that for every finite set

H

of primitive

n

-tuples, there exists a non-constant homogenous polynomial

f(x_1,x_2,\ldots,x_n)

with integer coefficients whose value is

1

at every

n

-tuple in

H

.
Based on the sixth problem of the 58th IMO, Brazil

A. 725

1

Functional challenge 4

Let

\mathbb R^+

denote the set of positive real numbers.Find all functions

f:\mathbb R^+\rightarrow \mathbb R^+

satisfying the following equation for all

x,y\in \mathbb R^+

:

f(xy+f(y)^2)=f(x)f(y)+yf(y)

A. 724

1

Real IMO Shortlist 2018 G5(FL passes through centre of insphere of tetrahedron)

A sphere

S

lies within tetrahedron

ABCD

, touching faces

ABD, ACD

, and

BCD

, but having no point in common with plane

ABC

. Let

E

be the point in the interior of the tetrahedron for which

S

touches planes

ABE

,

ACE

, and

BCE

as well. Suppose the line

DE

meets face

ABC

at

F

, and let

L

be the point of

S

nearest to plane

ABC

. Show that segment

FL

passes through the centre of the inscribed sphere of tetrahedron

ABCE

.
KöMaL A.723. (April 2018), G. Kós

A. 701

1

Anna and Bella's flights in EU

An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions. Furthermore, the flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour?
(Based on a Soviet problem)

A. 726

1

A and P equidistant from incenter

In triangle

ABC

with incenter

I

, line

AI

intersects the circumcircle of

ABC

at

S\ne A

. Let the reflection of

I

with respect to

BC

be

J

, and suppose that line

SJ

intersects the circumcircle of

ABC

for the second time at point

P\ne S

. Show that

AI=PI.

József Mészáros

A. 723

1

KöMaL A. 723

Let

f:\mathbb{R}\rightarrow \mathbb{R}

be a continuous function such that the limit

g(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-2f(x)+f(x-h)}{h^2}}

exists for all real

x

. Prove that

g(x)

is constant if and only if

f(x)

is a polynomial function whose degree is at most

2

.

A. 722

1

The Hawking space

The Hawking Space Agency operates

n-1

space flights between the

n

habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet.
In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely

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

monetary units in some order. prove that

n

or

n-2

is a square number.

A. 721

1

Inequality with positive reals summing to 1

Let

n\ge 2

be a positive integer, and suppose

a_1,a_2,\cdots ,a_n

are positive real numbers whose sum is

1

and whose squares add up to

S

. Prove that if

b_i=\tfrac{a^2_i}{S} \;(i=1,\cdots ,n)

, then for every

r>0

, we have

\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.

A. 720

1

Infinite set of lively integers

We call a positive integer lively if it has a prime divisor greater than

10^{10^{100}}

. Prove that if

S

is an infinite set of lively positive integers, then it has an infinite subset

T

with the property that the sum of the elements in any finite nonempty subset of

T

is a lively number.

A. 719

1

HM is parallel to OI

Let

ABC

be a scalene triangle with circumcenter

O

and incenter

I

. The

A

-excircle,

B

-excircle, and

C

-excircle of triangle

ABC

touch

BC

,

CA

, and

AB

at points

A_1

,

B_1

, and

C_1

, respectively. Let

P

be the orthocenter of

AB_1C_1

and

H

be the orthocenter of

ABC

. Show that if

M

is the midpoint of

PA_1

, then lines

HM

and

OI

are parallel.
Michael Ren

A. 716

1

OD perpendicular to BC

Let

ABC

be a triangle and let

D

be a point in the interior of the triangle which lies on the angle bisector of

\angle BAC

. Suppose that lines

BD

and

AC

meet at

E

, and that lines

CD

and

AB

meet at

F

. The circumcircle of

ABC

intersects line

EF

at points

P

and

Q

. Show that if

O

is the circumcenter of

DPQ

, then

OD

is perpendicular to

BC

.
Michael Ren

A. 718

1

common zeros of polynomials

Let

\mathbb{R}[x,y]

denote the set of two-variable polynomials with real coefficients. We say that the pair

(a,b)

is a zero of the polynomial

f \in \mathbb{R}[x,y]

if

f(a,b)=0

.
If polynomials

p,q \in \mathbb{R}[x,y]

have infinitely many common zeros, does it follow that there exists a non-constant polynomial

r \in \mathbb{R}[x,y]

which is a factor of both

p

and

q

?

A. 717

1

between two perfect squares

Let's call a positive integer

n

special, if there exist two nonnegativ integers (

a, b

), such that

n=2^a\times 3^b

.
Prove that if

k

is a positive integer, then there are at most two special numbers greater then

k^2

and less than

k^2+2k+1

.

A. 715

1

Minimal square tiles and powers of 2

Let

a

and

b

be positive integers. We tile a rectangle with dimensions

a

and

b

using squares whose side-length is a power of

2

, i.e. the tiling may include squares of dimensions

1\times 1, 2\times 2, 4\times 4

etc. Denote by

M

the minimal number of squares in such a tiling. Numbers

a

and

b

can be uniquely represented as the sum of distinct powers of

2

:

a=2^{a_1}+\cdots+2^{a_k},\; b=2^{b_1}+\cdots +2^{b_\ell}.

Show that

M=\sum_{i=1}^k \;\sum_{j=1}^{\ell} 2^{|a_i-b_j|}.

A. 713

1

Bounded expansive sequences

We say that a sequence

a_1,a_2,\cdots

is expansive if for all positive integers

j,\; i<j

implies

|a_i-a_j|\ge \tfrac 1j

. Find all positive real numbers

C

for which one can find an expansive sequence in the interval

[0,C]

.

A. 712

1

Hatters gonna hat

We say that a strictly increasing positive real sequence

a_1,a_2,\cdots

is an elf sequence if for any

c>0

we can find an

N

such that

a_n<cn

for

n=N,N+1,\cdots

. Furthermore, we say that

a_n

is a hat if

a_{n-i}+a_{n+i}<2a_n

for

\displaystyle 1\le i\le n-1

. Is it true that every elf sequence has infinitely many hats?

A. 709

1

Minimal C_a so that inequality holds

Let

a>0

be a real number. Find the minimal constant

C_a

for which the inequality

\displaystyle C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}

holds for any positive integer

n

and any sequence

0=x_0<x_1<\cdots <x_n

of real numbers.

A. 706

1

From Komal A.706

Find all positive integer

k

s for which such

f

exists and unique:

f(mn)=f(n)f(m)

for

n, m \in \mathbb{Z^+}

f^{n^k}(n)=n

for all

n \in \mathbb{Z^+}

for which

f^x (n)

means the n times operation of function

f

(i.e.

f(f(...f(n))...)

)

A. 702

1

Elegant triangle with minimal perimeter

Fix a triangle

ABC

. We say that triangle

XYZ

is elegant if

X

lies on segment

BC

,

Y

lies on segment

CA

,

Z

lies on segment

AB

, and

XYZ

is similar to

ABC

(i.e.,

\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z

). Of all the elegant triangles, which one has the smallest perimeter?

A. 705

1

Reflection of D moves on a line

Triangle

ABC

has orthocenter

H

. Let

D

be a point distinct from the vertices on the circumcircle of

ABC

. Suppose that circle

BHD

meets

AB

at

P\ne B

, and circle

CHD

meets

AC

at

Q\ne C

. Prove that as

D

moves on the circumcircle, the reflection of

D

across line

PQ

also moves on a fixed circle.
Michael Ren

KoMaL A Problems 2017/2018

Subcontests

Hard algebra with permutations from KoMaL

Fixed points of composition of inversions

Image of regular m-gon is a regular n-gon

Partitioning a regular n-gon in triangles

A very strange binary number is rational

Number a betyars a betyar sees

Partitioning triangular board into disjoint unit triangles

Generalisation of IMO 2017 P6

Functional challenge 4

Real IMO Shortlist 2018 G5(FL passes through centre of insphere of tetrahedron)

Anna and Bella's flights in EU

A and P equidistant from incenter

KöMaL A. 723

The Hawking space

Inequality with positive reals summing to 1

Infinite set of lively integers

HM is parallel to OI

OD perpendicular to BC

common zeros of polynomials

between two perfect squares

Minimal square tiles and powers of 2

Bounded expansive sequences

Hatters gonna hat

Minimal C_a so that inequality holds

From Komal A.706

Elegant triangle with minimal perimeter

Reflection of D moves on a line

KoMaL A Problems 2017/2018

Subcontests

Hard algebra with permutations from KoMaL

Fixed points of composition of inversions

Image of regular m-gon is a regular n-gon

Partitioning a regular n-gon in triangles

A very strange binary number is rational

Number a betyars a betyar sees

Partitioning triangular board into disjoint unit triangles

Generalisation of IMO 2017 P6

Functional challenge 4

Real IMO Shortlist 2018 G5(FL passes through centre of insphere of tetrahedron)

Anna and Bella's flights in EU

A and P equidistant from incenter

K&ouml;MaL A. 723

The Hawking space

Inequality with positive reals summing to 1

Infinite set of lively integers

HM is parallel to OI

OD perpendicular to BC

common zeros of polynomials

between two perfect squares

Minimal square tiles and powers of 2

Bounded expansive sequences

Hatters gonna hat

Minimal C_a so that inequality holds

From Komal A.706

Elegant triangle with minimal perimeter

Reflection of D moves on a line

KöMaL A. 723