MathDB

1

Z->Z polynomial, summation with +-1 coefficients

f

is a polynomial sends

\mathbb Z

to

\mathbb Z

and for

n\in\mathbb N_{\ge2}

, there exists

x\in\mathbb Z

so that

n\nmid f(x)

, show that for every

k\in\mathbb Z

, there is a non-negative integer

t

and

a_1,\ldots,a_t\in\{-1,1\}

such that

a_1f(1)+\ldots+a_tf(t)=k.

N5

1

sum of squares of divisors of k,n is n,k for all n

Find all positive integers

k

such that for every

n\in\mathbb N

, if there are

k

factors (not necessarily distinct) of

n

so that the sum of their squares is

n

, then there are

k

factors (not necessarily distinct) of

n

so that their sum is exactly

n

.

N3

1

xy|(x^2-x+1)(y^2-y+1)

Find all pairs of positive integers

(x,y)

so that

\frac{(x^2-x+1)(y^2-y+1)}{xy}\in\mathbb N.

N2

1

lcm and gcd FE

Find all functions

f:\mathbb N\to\mathbb N

satisfying

\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))

for all

x,y\in\mathbb N

.

N1

1

iterated divisibility FE

Find all functions

f:\mathbb N\to\mathbb N

satisfying

x+f^{f(x)}(y)\mid2(x+y)

for all

x,y\in\mathbb N

.

C6

1

sub-decks of cards, sum of cards divisible by n

In a deck of cards, there are

kn

cards numbered from

1

to

n

and there are

k

cards of each number. Now, divide this deck into

k

sub-decks with equal sizes. Prove that if

\gcd(k,n)=1

, then one could always pick

n

cards, one from each sub-deck, such that the sum of those cards is divisible by

n

.

C5

1

game, coloring grids in chessboard ft. Alice and Bob

Alice and Bob are playing the following game: They have an

8\times8

chessboard. Initially, all grids are white. Each round, Alice chooses a white grid and paints it black. Then Bob chooses one of the neighbors of that grid and paints it black. Or he does nothing. After that, Alice may decide to continue the game or not. The goal of Alice is to maximize the number of connected components of black grids, on the other hand, Bob wants to minimize that number. If both of them are extremely smart, how many connected components will be in the end of the game?

C4

1

set of primes is infinite

For a sequence

\{a_i\}_{i\ge1}

consisting of only positive integers, prove that if for all different positive integers

i

and

j

, we have

a_i\nmid a_j

, then

\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}

is a infinite set.

C3

1

L-tiling axb chessboard, # of grid covered even times

Given an

a\times b

chessboard where

a,b\ge3

, alice wants to use only

L

-dominoes (as the figure shows) to cover this chessboard. How many grids, at least, are covered even times? https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi82LzhmZDkwMGQzZjU3M2QxMzk4Y2NjNDg5ZTMwM2ZmYjJiMWU3MmUwLnBuZw==&rn=MjAxOC1DMy5wbmc=

C2

1

grids covered by one square in chessboard

Given an odd

n\in\mathbb N

. In an

n\times n

chessboard, you may place many

2\times2

squares. How many grids, at most, are covered by exactly one square?

C1

1

map is 3-colorable

IMOC is a small country without any lake. One day, the king decides to divide IMOC into many regions so that each region borders the sea. Prove that the map is

3

-colorable.

A7

1

system 3=ab+bc+ca=de+ef+fd=..., prove cf+fi+ic=3

If the reals

a,b,c,d,e,f,g,h,i

satisfy

\begin{cases}ab+bc+ca=3\\de+ef+fd=3\\gh+hi+ig=3\\ad+dg+ga=3\\be+eh+hb=3\end{cases}

show that

cf+fi+ic=3

holds as well.

A5

1

(x^2+3)(y^2+3)(z^2+3)>=(xyz+x+y+z+4)^2 over R

Show that for all reals

x,y,z

, we have

\left(x^2+3\right)\left(y^2+3\right)\left(z^2+3\right)\ge(xyz+x+y+z+4)^2.

A4

1

f(x^2+f(y))-y=(f(x+y)-y)^2 over R

Find all functions

f:\mathbb R\to\mathbb R

such that

f\left(x^2+f(y)\right)-y=(f(x+y)-y)^2

holds for all

x,y\in\mathbb R

.

A3

1

f(xf(y)+y)=yf(x)+f(y) over R

Find all functions

f:\mathbb R\to\mathbb R

such that for reals

x,y

,

f(xf(y)+y)=yf(x)+f(y).

A2

1

f1 o g, f2 o g, ... reducible

For arbitrary non-constant polynomials

f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]

, is it always possible to find a polynomial

g(x)\in\mathbb Z[x]

such that

f_1(g(x)),\ldots,f_{2018}(g(x))

are all reducible.

A1

1

rational FE in four variables

Find all functions

f:\mathbb Q\to\mathbb Q

such that for all

x,y,z,w\in\mathbb Q

,

f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).

G3

1

IMOC 2018 G3 (CY is tangent to circumcircle of BCC', orthocenter related)

Given an acute

\vartriangle ABC

whose orthocenter is denoted by

H

. A line

\ell

passes

H

and intersects

AB,AC

at

P ,Q

such that

H

is the mid-point of

P,Q

. Assume the other intersection of the circumcircle of

\vartriangle ABC

with the circumcircle of

\vartriangle APQ

is

X

. Let

C'

is the symmetric point of

C

with respect to

X

and

Y

is the another intersection of the circumcircle of

\vartriangle ABC

and

AO

, where O is the circumcenter of

\vartriangle APQ

. Show that

CY

is tangent to circumcircle of

\vartriangle BCC'

. https://1.bp.blogspot.com/-itG6m1ipAfE/XndLDUtSf7I/AAAAAAAALfc/iZahX6yNItItRSXkDYNofR5hKApyFH84gCK4BGAYYCw/s1600/2018%2Bimoc%2Bg3.png

G2

1

IMOC 2018 G2 (collinearity given + wanted, 4 tangent circles)

Given

\vartriangle ABC

with circumcircle

\Omega

. Assume

\omega_a, \omega_b, \omega_c

are circles which tangent internally to

\Omega

at

T_a,T_b, T_c

and tangent to

BC,CA,AB

at

P_a, P_b, P_c

, respectively. If

AT_a,BT_b,CT_c

are collinear, prove that

AP_a,BP_b,CP_c

are collinear.

G4

1

IMOC 2018 G4 (parallel wanted, circumcircles, incenter, symmetrics related)

Given an acute

\vartriangle ABC

with incenter

I

. Let

I'

be the symmetric point

I

with respect to the midpoint of

B,C

and

D

is the foot of

A

. If

DI

and the circumcircle of vartriangle

BI'C

intersect at

T

and

TI'

intersects the circumcircle of

\vartriangle ATI

at

X

. Furthermore,

E,F

are tangent points of the incircle and

AB,AC, P

is the another intersection of the circumcircles of

\vartriangle ABC, \vartriangle AEF

. Show that

AX \parallel PI

. https://3.bp.blogspot.com/-tj9A8HIR6Vw/XndLEPMRvnI/AAAAAAAALfk/2vw_pZbhpnkTKIc1BcKf4K7SNZ11vu4TACK4BGAYYCw/s1600/2018%2Bimoc%2Bg4.png

G5

1

IMOC 2018 G5 (IX//OH, incenter, circumcenter, orthocenter, euler line)

Suppose

I,O,H

are incenter, circumcenter, orthocenter of

\vartriangle ABC

respectively. Let

D = AI \cap BC

,

E = BI \cap CA

,

F = CI \cap AB

and

X

be the orthocenter of

\vartriangle DEF

. Prove that

IX \parallel OH

.

G1

1

IMOC 2018 G1 (k ways to draw a non-intersecting n-gon)

Given an integer

n \ge 3

. Find the largest positive integer

k

with the following property: For

n

points in general position, there exists

k

ways to draw a non-intersecting polygon with those

n

points as it’s vertices.
[hide=Different wording]Given

n

, find the maximum

k

so that for every general position of

n

points , there are at least

k

ways of connecting the points to form a polygon.

A6

1

nhocnhoc139

Given

a, b, c > 0

. Prove that: (1\plus{}a\plus{}b\plus{}c)(1\plus{}ab\plus{}bc\plus{}ca) \ge 4\sqrt{2}\sqrt{(a\plus{}bc)(b\plus{}ca)(c\plus{}ab)} :)

N4

1

0 < a_{n+1} - a_n \leq 2001

Let a sequence

\{a_n\}

,

n \in \mathbb{N}^{*}

given, satisfying the condition

0 < a_{n+1} - a_n \leq 2001

for all

n \in \mathbb{N}^{*}

Show that there are infinitely many pairs of positive integers

(p, q)

such that

p < q

and

a_p

is divisor of

a_q

.

2018-IMOC

Subcontests

Z->Z polynomial, summation with +-1 coefficients

sum of squares of divisors of k,n is n,k for all n

xy|(x^2-x+1)(y^2-y+1)

lcm and gcd FE

iterated divisibility FE

sub-decks of cards, sum of cards divisible by n

game, coloring grids in chessboard ft. Alice and Bob

set of primes is infinite

L-tiling axb chessboard, # of grid covered even times

grids covered by one square in chessboard

map is 3-colorable

system 3=ab+bc+ca=de+ef+fd=..., prove cf+fi+ic=3

(x^2+3)(y^2+3)(z^2+3)>=(xyz+x+y+z+4)^2 over R

f(x^2+f(y))-y=(f(x+y)-y)^2 over R

f(xf(y)+y)=yf(x)+f(y) over R

f1 o g, f2 o g, ... reducible

rational FE in four variables

IMOC 2018 G3 (CY is tangent to circumcircle of BCC', orthocenter related)

IMOC 2018 G2 (collinearity given + wanted, 4 tangent circles)

IMOC 2018 G4 (parallel wanted, circumcircles, incenter, symmetrics related)

IMOC 2018 G5 (IX//OH, incenter, circumcenter, orthocenter, euler line)

IMOC 2018 G1 (k ways to draw a non-intersecting n-gon)

nhocnhoc139

0 < a_{n+1} - a_n \leq 2001

2018-IMOC

Subcontests

Z-&gt;Z polynomial, summation with +-1 coefficients

sum of squares of divisors of k,n is n,k for all n

xy|(x^2-x+1)(y^2-y+1)

lcm and gcd FE

iterated divisibility FE

sub-decks of cards, sum of cards divisible by n

game, coloring grids in chessboard ft. Alice and Bob

set of primes is infinite

L-tiling axb chessboard, # of grid covered even times

grids covered by one square in chessboard

map is 3-colorable

system 3=ab+bc+ca=de+ef+fd=..., prove cf+fi+ic=3

(x^2+3)(y^2+3)(z^2+3)&gt;=(xyz+x+y+z+4)^2 over R

f(x^2+f(y))-y=(f(x+y)-y)^2 over R

f(xf(y)+y)=yf(x)+f(y) over R

f1 o g, f2 o g, ... reducible

rational FE in four variables

IMOC 2018 G3 (CY is tangent to circumcircle of BCC', orthocenter related)

IMOC 2018 G2 (collinearity given + wanted, 4 tangent circles)

IMOC 2018 G4 (parallel wanted, circumcircles, incenter, symmetrics related)

IMOC 2018 G5 (IX//OH, incenter, circumcenter, orthocenter, euler line)

IMOC 2018 G1 (k ways to draw a non-intersecting n-gon)

nhocnhoc139

0 &lt; a_{n+1} - a_n \leq 2001

Z->Z polynomial, summation with +-1 coefficients

(x^2+3)(y^2+3)(z^2+3)>=(xyz+x+y+z+4)^2 over R

0 < a_{n+1} - a_n \leq 2001