MathDB

1

2019 numbers adding up to n

N\in\mathbb{N}

so that for all integer

n > N

, one may find

2019

pairwise co-prime positive integers with

n=a_1+a_2+\cdots+a_{2019}

and

2019<a_1<a_2<\cdots<a_{2019}

N5

1

two operations on number form infinite sequence

Initially, Alice is given a positive integer

a_0

. At time

i

, Alice has two choices,

\begin{cases}a_i\mapsto\frac1{a_{i-1}}\\a_i\mapsto2a_{i-1}+1\end{cases}

Note that it is dangerous to perform the first operation, so Alice cannot choose this operation in two consecutive turns. However, if

x>8763

, then Alice could only perform the first operation. Determine all

a_0

so that

\{i\in\mathbb N\mid a_i\in\mathbb N\}

is an infinite set.

C3

1

walking around circle of two sexes

There are a total of

n

boys and girls sitting in a big circle. Now, Dave wants to walk around the circle. For a start point, if at any time, one of the following two conditions holds:
1. he doesn't see any girl 2. the number of boys he saw

\ge

the number of girls he saw

+k

Then we say this point is good. What is the maximum of

r

with the property that there is at least one good point whenever the number of girls is

r

?

C2

1

period of shuffling sequence

For

2n

numbers in a row, Bob could perform the following operation:

S_i=(a_1,a_2,\ldots,a_{2n})\mapsto S_{i+1}=(a_1,a_3,\ldots,a_{2n-1},a_2,a_4,\ldots,a_{2n}).

Let

T

be the order of this operation. In other words,

T

is the smallest positive integer such that

S_i=S_{i+T}

. Prove that

T<2n

.

C1

1

even partitions = odd partitions, sum of even numbers is n

Given a natural number

n

, if the tuple

(x_1,x_2,\ldots,x_k)

satisfies

2\mid x_1,x_2,\ldots,x_k

x_1+x_2+\ldots+x_k=n

then we say that it's an even partition. We define odd partition in a similar way. Determine all

n

such that the number of even partitions is equal to the number of odd partitions.

A4

1

iterated FE over N

Find all functions

f:\mathbb N\to\mathbb N

so that

f^{2f(b)}(2a)=f(f(a+b))+a+b

holds for all positive integers

a,b

.

A3

1

system of equations with 2019th root

Find all

3

-tuples of positive reals

(a,b,c)

such that

\begin{cases}a\sqrt[2019]b-c=a\\b\sqrt[2019]c-a=b\\c\sqrt[2019]a-b=c\end{cases}

A2

1

polynomial |f(k)-t^k)<1, bound degree

Given a real number

t\ge3

, suppose a polynomial

f\in\mathbb R[x]

satisfies

\left|f(k)-t^k\right|<1,\enspace\forall k=0,1,\ldots,n.

Prove that

\deg f\ge n

.

A5

1

f^{x+1}(y)+f^{y+1}(x)=2f(x+y)

Find all functions

f : \mathbb N \mapsto \mathbb N

such that the following identity

f^{x+1}(y)+f^{y+1}(x)=2f(x+y)

holds for all

x,y \in \mathbb N

N4

1

$p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$

Given a sequence of prime numbers

p_1, p_2,\cdots

, with the following property:

p_{n+2}

is the largest prime divisor of

p_n+p_{n+1}+2018

Show that the set

\{p_i\}_{i\in \mathbb{N}}

is finite.

G5

1

IMOC 2019 G5 (DH bisects <EDF, circumcircle, orthocenter related)

Given a scalene triangle

\vartriangle ABC

with orthocenter

H

and circumcenter

O

. The exterior angle bisector of

\angle BAC

intersects circumcircle of

\vartriangle ABC

at

N \ne A

. Let

D

be another intersection of

HN

and the circumcircle of

\vartriangle ABC

. The line passing through

O

, which is parallel to

AN

, intersects

AB,AC

at

E, F

, respectively. Prove that

DH

bisects the angle

\angle EDF

. https://3.bp.blogspot.com/-F1mFwojG_I0/XnYNR8ofqSI/AAAAAAAALeo/zge24WF0EO8umPAaXprKAeXJHAj7pr6tQCK4BGAYYCw/s1600/imoc2019g5.png

G4

1

IMOC 2019 G4 (tangent circumcircles)

\vartriangle ABC

is a scalene triangle with circumcircle

\Omega

. For a arbitrary

X

in the plane, define

D_x,E_x, F_x

to be the intersection of tangent line of

X

(with respect to

BXC

) and

BC,CA,AB

, respectively. Let the intersection of

AX

with

\Omega

be

S_x

and

T_x = D_xS_x \cap \Omega

. Show that

\Omega

and circumcircle of

\vartriangle T_xE_xF_x

are tangent to each other. https://2.bp.blogspot.com/-rTMODHbs5Ac/XnYNQYjYzBI/AAAAAAAALeg/576nGDQ6NDA0-W5XqiNczNtI07cEZxPeQCK4BGAYYCw/s1600/imoc2019g4.png

G3

1

IMOC 2019 G3 (PQ bisects segment BC, orthocenter, circumcircle related)

Given a scalene triangle

\vartriangle ABC

has orthocenter

H

and circumcircle

\Omega

. The tangent lines passing through

A,B,C

are

\ell_a,\ell_b,\ell_c

. Suppose that the intersection of

\ell_b

and

\ell_c

is

D

. The foots of

H

on

\ell_a,AD

are

P,Q

respectively. Prove that

PQ

bisects segment

BC

. https://4.bp.blogspot.com/-iiQoxMG8bEs/XnYNK7R8S3I/AAAAAAAALeY/FYvSuF6vQQsofASnXJUgKZ1T9oNnd-02ACK4BGAYYCw/s400/imoc2019g3.png

G2

1

IMOC 2019 G2 (4 midpoints are concyclic, orthocenter, symmetry wrt line)

Given a scalene triangle

\vartriangle ABC

with orthocenter

H

. The midpoint of

BC

is denoted by

M

.

AH

intersects the circumcircle at

D \ne A

and

DM

intersects circumcircle of

\vartriangle ABC

at

T\ne D

. Now, assume the reflection points of

M

with respect to

AB,AC,AH

are

F,E,S

. Show that the midpoints of

BE,CF,AM,TS

are concyclic. https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png

G1

1

IMOC 2019 G1 (DE pass through orthocenter of BIC, BD=CA,CE=BA, incenter)

Let

I

be the incenter of a scalene triangle

\vartriangle ABC

. In other words,

\overline{AB},\overline{BC},\overline{CA}

are distinct. Prove that if

D,E

are two points on rays

\overrightarrow{BA},\overrightarrow{CA}

, satisfying

\overline{BD}=\overline{CA},\overline{CE}=\overline{BA}

then line

DE

pass through the orthocenter of

\vartriangle BIC

. http://2.bp.blogspot.com/-aHCD5tL0FuA/XnYM1LoZjWI/AAAAAAAALeE/C6hO9W9FGhcuUP3MQ9aD7SNq5q7g_cY9QCK4BGAYYCw/s1600/imoc2019g1.png

C4

1

IMOC 2019 C4

Determine the largest

k

such that for all competitive graph with

2019

points, if the difference between in-degree and out-degree of any point is less than or equal to

k

, then this graph is strongly connected.

N2

1

IMOC 2019 N2

Find all pairs of positive integers

(m, n)

such that

m^n * n^m = m^m + n^n

N1

1

IMOC 2019 N1

Find all pairs of positive integers

(x, y)

so that

(xy - 6)^2 | x^2 + y^2

A1

1

Functional equation

Find all functions

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

such that for all

x,y\in\mathbb{R}

,

f(xy+f(x))=f(xf(y))+x

2019-IMOC

Subcontests

2019 numbers adding up to n

two operations on number form infinite sequence

walking around circle of two sexes

period of shuffling sequence

even partitions = odd partitions, sum of even numbers is n

iterated FE over N

system of equations with 2019th root

polynomial |f(k)-t^k)<1, bound degree

f^{x+1}(y)+f^{y+1}(x)=2f(x+y)

$p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$

IMOC 2019 G5 (DH bisects <EDF, circumcircle, orthocenter related)

IMOC 2019 G4 (tangent circumcircles)

IMOC 2019 G3 (PQ bisects segment BC, orthocenter, circumcircle related)

IMOC 2019 G2 (4 midpoints are concyclic, orthocenter, symmetry wrt line)

IMOC 2019 G1 (DE pass through orthocenter of BIC, BD=CA,CE=BA, incenter)

IMOC 2019 C4

IMOC 2019 N2

IMOC 2019 N1

Functional equation

2019-IMOC

Subcontests

2019 numbers adding up to n

two operations on number form infinite sequence

walking around circle of two sexes

period of shuffling sequence

even partitions = odd partitions, sum of even numbers is n

iterated FE over N

system of equations with 2019th root

polynomial |f(k)-t^k)&lt;1, bound degree

f^{x+1}(y)+f^{y+1}(x)=2f(x+y)

$p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$

IMOC 2019 G5 (DH bisects &lt;EDF, circumcircle, orthocenter related)

IMOC 2019 G4 (tangent circumcircles)

IMOC 2019 G3 (PQ bisects segment BC, orthocenter, circumcircle related)

IMOC 2019 G2 (4 midpoints are concyclic, orthocenter, symmetry wrt line)

IMOC 2019 G1 (DE pass through orthocenter of BIC, BD=CA,CE=BA, incenter)

IMOC 2019 C4

IMOC 2019 N2

IMOC 2019 N1

Functional equation

polynomial |f(k)-t^k)<1, bound degree

IMOC 2019 G5 (DH bisects <EDF, circumcircle, orthocenter related)