MathDB

1

Sunny and Ming playing with strings

k\geq4

be an integer. Sunny and Ming play a game with strings. A string is a sequence that every element of it is an integer between

1

and

k

, inclusive. At first, Sunny chooses two positive integers

N,L\geq2

and write down

N

strings, each having length

L

. Then Ming mark at most

\frac{N}{2}

strings. Then Sunny chooses an unmarked string

s

and calculate the biggest integer

n

such that there exists another string satisfying its first

n

element is the same as the first

n

element of

s

. Then Sunny burn down all strings which first

n

element if different from the first

n

element of

s

, leaving only the ones which have the same first

n

element of

s

. Finally, Ming chooses an integer

d

between

1

and

k

, inclusive, and remove all strings which

(n+1)

th element is

d

. Sunny's score would be the number of strings left. Find the maximum score that Sunny can guarantee to get.
Proposed by USJL

C5

1

I have 0 feature :(

Define a "ternary sequence" is a sequence that every number is

0,1

or

2

. ternary sequence

(x_1,x_2,x_3,\cdots,x_n)

, define its difference to be

(|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|)

A difference will make the length of the sequence decrease by

1

, so we define the "feature value" of a ternary sequence with length

n

is the number left after

n-1

differences. How many ternary sequences has length

2023

and feature value

0

?
Proposed by CSJL

C4

1

SUPER BALANCED PERMUTATIONS

Let

N

be a given positive integer. Consider a permutation of

1,2,3,\cdots,N

, denoted as

p_1,p_2,\cdots,p_N

. For a section

p_l, p_{l+1},\cdots, p_r

, we call it "extreme" if

p_l

and

p_r

are the maximum and minimum value of that section. We say a permutation

p_1,p_2,\cdots,p_N

is "super balanced" if there isn't an "extreme" section with a length at least

3

. For example,

1,4,2,3

is "super balanced", but

3,1,2,4

isn't. Please answer the following questions: 1. How many "super balanced" permutations are there? 2. For each integer

M\leq N

. How many "super balanced" permutations are there such that

p_1=M

?
Proposed by ltf0501

C3

1

Cover a board with three types of dominoes

There are three types of piece shown as below. Today Alice wants to cover a

100 \times 101

board with these pieces without gaps and overlaps. Determine the minimum number of

1\times 1

pieces should be used to cover the whole board and not exceed the board. (There are an infinite number of these three types of pieces.) [asy] size(9cm,0); defaultpen(fontsize(12pt)); draw((9,10) -- (59,10) -- (59,60) -- (9,60) -- cycle); draw((59,10) -- (109,10) -- (109,60) -- (59,60) -- cycle); draw((9,60) -- (59,60) -- (59,110) -- (9,110) -- cycle); draw((9,110) -- (59,110) -- (59,160) -- (9,160) -- cycle); draw((109,10) -- (159,10) -- (159,60) -- (109,60) -- cycle); draw((180,11) -- (230,11) -- (230,61) -- (180,61) -- cycle); draw((180,61) -- (230,61) -- (230,111) -- (180,111) -- cycle); draw((230,11) -- (280,11) -- (280,61) -- (230,61) -- cycle); draw((230,61) -- (280,61) -- (280,111) -- (230,111) -- cycle); draw((280,11) -- (330,11) -- (330,61) -- (280,61) -- cycle); draw((280,61) -- (330,61) -- (330,111) -- (280,111) -- cycle); draw((330,11) -- (380,11) -- (380,61) -- (330,61) -- cycle); draw((330,61) -- (380,61) -- (380,111) -- (330,111) -- cycle); draw((401,11) -- (451,11) -- (451,61) -- (401,61) -- cycle); [/asy] Proposed by amano_hina

N6

1

NT random problem with tau function

Find all integer coefficient polynomial

P(x)

such that for all positive integer

x

, we have

\tau(P(x))\geq\tau(x)

Where

\tau(n)

denotes the number of divisors of

n

. Define

\tau(0)=\infty

. Note: you can use this conclusion. For all

\epsilon\geq0

, there exists a positive constant

C_\epsilon

such that for all positive integer

n

, the

n

th smallest prime is at most

C_\epsilon n^{1+\epsilon}

.
Proposed by USJL

N4

1

4 is not square, so (1,1) works?

Find all pair of positive integers

(m,n)

such that

mn(m^2+6mn+n^2)

is a perfect square.
Proposed by Li4 and Untro368

N3

1

2n+3|n!-1

Find all positive integer

n

satifying

2n+3|n!-1

Proposed by ltf0501

N2

1

A function that is O(n)

For a positive integer

n

, define

f(x)

to be the smallest positive integer

x

satisfying the following conditions: there exists a positive integer

k

and

k

distinct positive integers

n=a_0<a_1<a_2<\cdots<a_{k-1}=x

such that

a_0a_1\cdots a_{k-1}

is a perfect square. Find the smallest real number

c

such that there exists a positive integer

N

such that for all

n>N

we have

f(n)\leq cn

.
Proposed by Fysty and amano_hina

N1

1

Something with a lot of ! is integer

Find all positive integer

n

such that for all

i=1,2,\cdots,n

,

\frac{n!}{i!(n-i+1)!}

is an integer.
Proposed by ckliao914

A2

1

Maximal value

Given positive integer

n>2,

consider real numbers

a_1,a_2,\dots, a_n

satisfying

a^{2}_1+a^2_2+\dots a^2_n=1.

Find the maximal value of

|a_1-a_2|+|a_2-a_3| +\dots +|a_n-a_1|.

Proposed by ltf0501

A5

1

Large 3 var real functional equation

Find all functions

f:\mathbb R\to \mathbb R

such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all

x,y,z\in \mathbb R.

N5

1

Pythagorean-Euler Theorem?

Find all solution

(p,r)

of the "Pythagorean-Euler Theorem"

p^p+(p+1)^p+\cdots+(p+r)^p=(p+r+1)^p

Where

p

is a prime and

r

is a positive integer.
Proposed by Li4 and Untro368

G6

1

BE \cap CF lies on (ADEF) or EF //BC , if DM , BE , CF are concurrent

Let

D

be a point on the circumcircle of some triangle

ABC

. Let

E, F

be points on

AC

,

AB

, respectively, such that

A,D,E,F

are concyclic. Let

M

be the midpoint of

BC

. Show that if

DM

,

BE

,

CF

are concurrent, then either

BE \cap CF

is on the circle

ADEF

, or

EF

is parallel to

BC

.
proposed by USJL

A4

1

Mapping from the set of bijective functions

Let the set of all bijective functions taking positive integers to positive integers be

\mathcal B.

Find all functions

\mathbf F:\mathcal B\to \mathbb R

such that

(\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q)

for all

p,q \in \mathcal B.

Proposed by ckliao914

G5

1

(STX) bisects BC

P

is a point inside

ABC

.

BP

,

CP

intersect

AC, AB

at

E, F

, respectively.

AP

intersect

\odot (ABC)

again at X.

\odot (ABC)

and

\odot (AEF)

intersect again at

S

.

T

is a point on

BC

such that

P T \parallel EF

. Prove that

\odot (ST X)

passes through the midpoint of

BC

.
proposed by chengbilly

G4

1

collinear wanted, 2 circles and circumucircle related

Let

\vartriangle ABC

be an acute triangle with circumcircle

\Omega

. A line passing through

A

perpendicular to

BC

meets

\Omega

again at

D

. Draw two circles

\omega_b

,

\omega_c

with

B, C

as centers and

BD

,

CD

as radii, respectively, and they intersect

AB

,

AC

at

E, F,

respectively. Let

K\ne A

be the second intersection of

(AEF)

and

\Omega

, and let

\omega_b

,

\omega_c

intersect

KB

,

KC

at

P, Q

, respectively. The circumcenter of triangle

DP Q

is

O

, prove that

K, O, D

are collinear.
proposed by Li4

A6

1

Hard positive real FE

Find all functions

f:\mathbb R^+\to \mathbb R^+

such that

f(x+y)f(f(x))=f(1+yf(x))

for all

x,y\in \mathbb R^+.

Proposed by Ming Hsiao

A3

1

xy(f(x+y)-f(x)-f(y))=2f(xy)

Find all functions

f:\mathbb R\to \mathbb R

such that

xy(f(x+y)-f(x)-f(y))=2f(xy)

for all

x,y\in \mathbb R.

Proposed by USJL

G3

1

A, O_P, O_Q, R are concylic

Let

\vartriangle ABC

be an acute triangle.

R

is a point on arc

BC

. Choose two points

P, Q

on

AR

such that

B,P,C,Q

are concyclic. Let the second intersection of

BP

,

CP

,

BQ

,

CQ

and the circumcircle of

\vartriangle ABC

is

P_B

,

P_C

,

Q_B

,

Q_C

, respectively. Let the circumcenter of

\vartriangle P P_BP_C

and

\vartriangle QQ_BQ_C

are

O_P

and

O_Q

, respectively. Prove that

A,O_P,O_Q,R

are concylic.
proposed by andychang

G2

1

L,T,F are collinear iff B,E,A,R are concyclic.

The incenter of triangle

ABC

is

I

. the circumcircle of

ABC

is tangent to

BC

,

CA

,

AB

at

T, E, F

.

R

is a point on

BC

. Let the

C

-excenter of

\vartriangle CER

be

L

. Prove that points

L,T,F

are collinear if and only if

B,E,A,R

are concyclic.
proposed by kyou46

A1

1

Strict Inequality with cyclic sums

If positive real numbers

x,y,z

satisfies

x+y+z=3,

prove that

\sum_{\text{cyc}} y^2z^2<3+\sum_{\text{cyc}} yz.

Proposed by Li4 and Untro368.

G1

1

OX , BC concurrent with perp. bisector of AH, if XACH #, OH//BC

The circumcenter and orthocenter of

ABC

are

O

and

H

, respectively. Let

XACH

be a parallelogram. Show that if

OH

is parallel to

BC

, then

OX

and

AB

intersect at some point on the perpendicular bisector of

AH

.
proposed by USJL

C2

1

LTF and Ming play with stones

There are

2022

stones on a table. At the start of the game, Teacher Tseng will choose a positive integer

m

and let Ming and LTF play a game. LTF is the first to move, and he can remove at most

m

stones on his round. Then the two people take turns removing stone, each round they must remove at least one stone, and they cannot remove more than twice the amount of stones the last person removed. The player unable to move loses. Find the smallest positive integer

m

such that LTF has a winning strategy.
Proposed by ltf0501

C1

1

poor pigeon's egg got eaten by seagull

Given a positive integer

k

, a pigeon and a seagull play a game on an

n\times n

board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose

m

grids and lay an egg in each grid he chooses. The seagull will choose a

k\times k

grids and eat all the eggs inside them. If at any point every grid in the

n\times n

board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer

n\geq k

, find all

m

such that the pigeon wins.
Proposed by amano_hina

2022-IMOC

Subcontests

Sunny and Ming playing with strings

I have 0 feature :(

SUPER BALANCED PERMUTATIONS

Cover a board with three types of dominoes

NT random problem with tau function

4 is not square, so (1,1) works?

2n+3|n!-1

A function that is O(n)

Something with a lot of ! is integer

Maximal value

Large 3 var real functional equation

Pythagorean-Euler Theorem?

BE \cap CF lies on (ADEF) or EF //BC , if DM , BE , CF are concurrent

Mapping from the set of bijective functions

(STX) bisects BC

collinear wanted, 2 circles and circumucircle related

Hard positive real FE

xy(f(x+y)-f(x)-f(y))=2f(xy)

A, O_P, O_Q, R are concylic

L,T,F are collinear iff B,E,A,R are concyclic.

Strict Inequality with cyclic sums

OX , BC concurrent with perp. bisector of AH, if XACH #, OH//BC

LTF and Ming play with stones

poor pigeon's egg got eaten by seagull