MathDB

1

Reposted

T

is chosen inside a triangle

ABC

. Let

A_1

,

B_1

, and

C_1

be the reflections of

T

in

BC

,

CA

, and

AB

, respectively. Let

\Omega

be the circumcircle of the triangle

A_1B_1C_1

. The lines

A_1T

,

B_1T

, and

C_1T

meet

\Omega

again at

A_2

,

B_2

, and

C_2

, respectively. Prove that the lines

AA_2

,

BB_2

, and

CC_2

are concurrent on

\Omega

.
Proposed by Mongolia

C3

1

Long game to go..

Let

n

be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of

n + 1

squares in a row, numbered

0

to

n

from left to right. Initially,

n

stones are put into square

0

, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with

k

stones, takes one of these stones and moves it to the right by at most

k

squares (the stone should say within the board). Sisyphus' aim is to move all

n

stones to square

n

. Prove that Sisyphus cannot reach the aim in less than

\left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil

turns. (As usual,

\lceil x \rceil

stands for the least integer not smaller than

x

. )

A6

1

A special polynomial condition

Let

m,n\geq 2

be integers. Let

f(x_1,\dots, x_n)

be a polynomial with real coefficients such that

f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.

Prove that the total degree of

f

is at least

n

.

A3

1

Showing properties about subsets of positive integers.

Given any set

S

of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets

F

and

G

of

S

such that

\sum_{x\in F}1/x=\sum_{x\in G}1/x

;
(2) There exists a positive rational number

r<1

such that

\sum_{x\in F}1/x\neq r

for all finite subsets

F

of

S

.

G7

1

Just messy figure geometry

Let

O

be the circumcentre, and

\Omega

be the circumcircle of an acute-angled triangle

ABC

. Let

P

be an arbitrary point on

\Omega

, distinct from

A

,

B

,

C

, and their antipodes in

\Omega

. Denote the circumcentres of the triangles

AOP

,

BOP

, and

COP

by

O_A

,

O_B

, and

O_C

, respectively. The lines

\ell_A

,

\ell_B

,

\ell_C

perpendicular to

BC

,

CA

, and

AB

pass through

O_A

,

O_B

, and

O_C

, respectively. Prove that the circumcircle of triangle formed by

\ell_A

,

\ell_B

, and

\ell_C

is tangent to the line

OP

.

N7

1

NT Arithmetic Progression

Let

n \ge 2018

be an integer, and let

a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n

be pairwise distinct positive integers not exceeding

5n

. Suppose that the sequence

\frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n}

forms an arithmetic progression. Prove that the terms of the sequence are equal.

G5

1

Triangle form by perpendicular bisector

Let

ABC

be a triangle with circumcircle

\Omega

and incentre

I

. A line

\ell

intersects the lines

AI

,

BI

, and

CI

at points

D

,

E

, and

F

, respectively, distinct from the points

A

,

B

,

C

, and

I

. The perpendicular bisectors

x

,

y

, and

z

of the segments

AD

,

BE

, and

CF

, respectively determine a triangle

\Theta

. Show that the circumcircle of the triangle

\Theta

is tangent to

\Omega

.

N1

1

Ez Number Theory

Determine all pairs

(n, k)

of distinct positive integers such that there exists a positive integer

s

for which the number of divisors of

sn

and of

sk

are equal.

C6

1

Infinite process on a board

Let

a

and

b

be distinct positive integers. The following infinite process takes place on an initially empty board.
[*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by

a

and the other by

b

. [*] If no such pair exists, we write two times the number

0

.
Prove that, no matter how we make the choices in

(i)

, operation

(ii)

will be performed only finitely many times.
Proposed by [I]Serbia[/I].

N6

1

Unique NT Function

Let

f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}

be a function such that

f(m + n) | f(m) + f(n)

for all pairs

m,n

of positive integers. Prove that there exists a positive integer

c > 1

which divides all values of

f

.

N3

1

Expressing a term of the sequence by the sum of previous terms

Define the sequence a_0,a_1,a_2,\hdots by

a_n=2^n+2^{\lfloor n/2\rfloor}

. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

C5

1

Tournament with lowest cost

Let

k

be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for

2k

players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay

1

coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

A4

1

Maximising a sequence-given value

Let

a_0,a_1,a_2,\dots

be a sequence of real numbers such that

a_0=0, a_1=1,

and for every

n\geq 2

there exists

1 \leq k \leq n

satisfying

a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}.

Find the maximum possible value of

a_{2018}-a_{2017}

.

N5

1

Nice NT here

Four positive integers

x,y,z

and

t

satisfy the relations

xy - zt = x + y = z + t.

Is it possible that both

xy

and

zt

are perfect squares?

G3

1

Good triangles in a circle

A circle

\omega

with radius

1

is given. A collection

T

of triangles is called good, if the following conditions hold:
[*] each triangle from

T

is inscribed in

\omega

; [*] no two triangles from

T

have a common interior point.
Determine all positive real numbers

t

such that, for each positive integer

n

, there exists a good collection of

n

triangles, each of perimeter greater than

t

.

C1

1

Partition set with equal sum and differnt cardinality

Let

n\geqslant 3

be an integer. Prove that there exists a set

S

of

2n

positive integers satisfying the following property: For every

m=2,3,...,n

the set

S

can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality

m

.

A1

1

Rational FEs

Let

\mathbb{Q}_{>0}

denote the set of all positive rational numbers. Determine all functions

f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}

satisfying

f(x^2f(y)^2)=f(x)^2f(y)

for all

x,y\in\mathbb{Q}_{>0}

C7

1

Crossing Circles

Consider

2018

pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular

edges

that meet at

vertices

. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice- once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least

2061

yellow points, then the vertices of some region are all yellow.
Proposed by India

G2

1

Concyclicity on some quadrilateral from isosceles triangle

Let

ABC

be a triangle with

AB=AC

, and let

M

be the midpoint of

BC

. Let

P

be a point such that

PB<PC

and

PA

is parallel to

BC

. Let

X

and

Y

be points on the lines

PB

and

PC

, respectively, so that

B

lies on the segment

PX

,

C

lies on the segment

PY

, and

\angle PXM=\angle PYM

. Prove that the quadrilateral

APXY

is cyclic.

A5

1

Fractions in an FE taking positive reals

Determine all functions

f:(0,\infty)\to\mathbb{R}

satisfying

\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)

for all

x,y>0

.

N2

1

Divisibility of numbers in the table

Let

n>1

be a positive integer. Each cell of an

n\times n

table contains an integer. Suppose that the following conditions are satisfied:
[*] Each number in the table is congruent to

1

modulo

n

. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to

n

modulo

n^2

.
Let

R_i

be the product of the numbers in the

i^{\text{th}}

row, and

C_j

be the product of the number in the

j^{\text{th}}

column. Prove that the sums R_1+\hdots R_n and C_1+\hdots C_n are congruent modulo

n^4

.

A7

1

inequality, asymmetric, nonhomogeneous, with a 7 and a radical!

Find the maximal value of

S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},

where

a

,

b

,

c

,

d

are nonnegative real numbers which satisfy

a+b+c+d = 100

.
Proposed by Evan Chen, Taiwan

2018 IMO Shortlist

Subcontests

Reposted

Long game to go..

A special polynomial condition

Showing properties about subsets of positive integers.

Just messy figure geometry

NT Arithmetic Progression

Triangle form by perpendicular bisector

Ez Number Theory

Infinite process on a board

Unique NT Function

Expressing a term of the sequence by the sum of previous terms

Tournament with lowest cost

Maximising a sequence-given value

Nice NT here

Good triangles in a circle

Partition set with equal sum and differnt cardinality

Rational FEs

Crossing Circles

Concyclicity on some quadrilateral from isosceles triangle

Fractions in an FE taking positive reals

Divisibility of numbers in the table

inequality, asymmetric, nonhomogeneous, with a 7 and a radical!