MathDB

1

Old problem in IMOC

a

,

b

,

c

are three distinct real numbers, given

\lambda >0

. Proof that

\frac{1+ \lambda ^2a^2b^2}{(a-b)^2}+\frac{1+ \lambda ^2b^2c^2}{(b-c)^2}+\frac{1+ \lambda ^2c^2a^2}{(c-a)^2} \geq \frac 32 \lambda.

Old problem, can be found [url=https://artofproblemsolving.com/community/c6h588854p3487434]here. Double post to have a cleaner thread for collection (as the original one contains a messy quote)

G1

1

Easy Geometry in 2024 IMOC

Given quadrilateral

ABCD

.

AC

and

BD

meets at

E

, and

M, N

are the midpoints of

AC, BD

, respectively. Let the circumcircles of

ABE

and

CDE

meets again at

X\neq E

. Prove that

E, M, N, X

are concyclic.
Proposed by chengbilly

G2

1

Quadrilateral Practice!

Triangle

ABC

has circumcenter

O

.

D

is an arbitrary point on

BC

, and

AD

intersects

\odot(ABC)

at

E

.

S

is a point on

\odot(ABC)

such that

D, O, E, S

are colinear.

AS

intersects

BC

at

P

.

Q

is a point on

BC

such that

D, O, A, Q

are concylic. Prove that

\odot(ABC)

is tangent to

\odot (APQ)

.
Proposed by chengbilly

G3

1

Cross-ratio Practice!

Triangle

ABC

has circumcircle

\Omega

and incircle

\omega

, where

\omega

is tangent to

BC, CA, AB

at

D,E,F

, respectively.

T

is an arbitrary point on

\omega

.

EF

meets

BC

at

K

,

AT

meets

\Omega

again at

P

,

PK

meets

\Omega

again at

S

.

X

is a point on

\Omega

such that

S, D, X

are colinear. Let

Y

be the intersection of

AX

and

EF

, prove that

YT

is tangent to

\omega

.
Proposed by chengbilly

G4

1

APMO intersects on Luna Cabrera

Given triangle

ABC

with

AB<AC

and its circumcircle

\Omega

. Let

I

be the incenter of

ABC

, and the feet from

I

to

BC

is

D

. The circle with center

A

and radius

AI

intersects

\Omega

at

E

and

F

.

P

is a point on

EF

such that

DP

is parallel to

AI

. Prove that

AP

and

MI

intersects on

\Omega

where

M

is the midpoint of arc

BAC

. [hide = Remark] In the test, the incenter called

O

and the circumcircle called

Luna

Cabrera

You have to prove

AP \cap MO \in Luna

Cabrera

Proposed by BlessingOfHeaven

G5

1

Intermediate Geometry in 2024 IMOC

Triangle

ABC

satisfying

AB<AC

has circumcircle

\Omega

.

E, F

lies on

AC, AB

, respectively, such that

BCEF

is cyclic.

T

lies on

EF

such that

\odot(TEF)

is tangent to

BC

at

T

.

A'

is the antipode of

A

on

\Omega

.

TA', TA

intersects

\Omega

again at

X, Y

, respectively, and

EF

intersects

\odot(TXY)

again at

W

. Prove that

\measuredangle WBA=\measuredangle ACW

Proposed by BlessingOfHeaven

G6

1

Difficult Geometry in 2024 IMOC

ABCD

is a cyclic quadrilateral and

AC

intersects

BD

at

E

.

M, N

are the midpoints of

AB, CD

, respectively.

\odot(AMN)

meets

\odot(ABCD)

again at

P

.

\odot(CMN)

meets

\odot(ABCD)

again at

Q

.

\odot(PEQ)

meets

BD

again at

T

. Prove that

M,N,T

are colinear.
Proposed by chengbilly

G7

1

Hong Kong Jane Street

Triangle

ABC

has circumcenter

O

and incenter

I

. The incircle is tangent to

AC, AB

at

E, F

, respectively.

H

is the orthocenter of

\triangle BIC

.

\odot(AEF)

and

\odot(ABC)

intersects again at

S

.

BC, AH

intersects

OI

again at

J, K

, respectively. Prove that

H, K, J, S

are concyclic.
Proposed by chengbilly

C7

1

Another Invisible Rabbit Appeared in 2024 imoc

On a plane there is an invisible [color=#eee]rabbit (rabbit) hiding on a lattice point. We want to put

n

hunters on some lattice points to catch the rabbit. In a turn each hunter can choose to shoot to left/right or top/bottom direction. On the

i

th turn there will be these steps in order 1. The rabbit jumps to an adjacent lattice point on the top, bottom, left, or right. 2. item Each hunter moves to an adjacent lattice point on the top, bottom, left or right (each hunter can move to different direction). Then they shoot a bullet which travels

\frac{334111214}{334111213}i

units on the directions they chose. If a bullet hits the rabbit then it is caught. Find the smallest number

n

such that the rabbit would definitely be caught in a finite number of turns.
Proposed by tob8y

C6

1

The Strangest Problem in 2024 IMOC

On an

m\times n

grid there's a snail in each cell. Each round, the snail army can choose four snail and make them perform the complete Quadrilateral Dance, which is rotating the four snails clockwise by

90

degrees, moving each one to an adjacent cell. Find all pairs of positive integers

(m,n)

such that the snails can achieve any permutation by performing a finite number of times of Quadrilateral Dance.
Proposed by chengbilly

C5

1

You Deer

Given integer

n\geq 2

, two invisible [color=#eee]rabbits (rabbits) discussed their strategy and was sent to two points

A, B

with distance

n

units on a plane, respectively. However, they do not know whether they are on the same or different side of the plane (when facing each other, the might view the left/right direction as the same or different). They both can see points

A,B

, and they need to hop to each other's starting point. Each hop would measure

1

unit in distance, and they would jump and land at the same time for each round. However, if at any time they landed no more than

1

unit away, they'll both turn into deer. Find the minimum number of round they need to reach their destiny while ensuring they won't turn into deer.
Proposed by redshrimp

N6

1

Rational NT FE

Find all functions

f:\mathbb{Q}^+\to\mathbb{Q}^+

such that

xy(f(x)-f(y))|x-f(f(y))

holds for all positive rationals

x

,

y

(we define that

a|b

if and only if exist

n \in \mathbb{Z}

such that

b=an

) Proposed by supercarry & windleaf1A

C3

1

The Attack and Defense of Snails

There are

n

snails on the plane where the

i

snail has

a_i

attack and

d_i

defense, where

a_i, d_i\in \mathbb{R}

and each snail has a distinct attack and a distinct defense. We said a 4-tuple of subsets of snails

(S_1, S_2, S_3, S_4)

is balanced if

|S_1|+|S_3|

is either

\lceil n/2\rceil

or

\lfloor n/2\rfloor

and there exist real numbers

A, D

such that \begin{align*} S_1=\{i\ |\ a_i\geq A\text{ and } d_i\geq D, 1\leq i\leq n\}\\ S_2=\{i\ |\ a_i

k

such that there is always at least

k

balanced 4-tuples of subsets.
Proposed by redshrimp

C2

1

Inspired by the logo of Jane Street

Given integer

n \geq 3

. There are

n

dots marked

1

to

n

clockwise on a big circle. And between every two neighboring dots, there is a light. At first, every light were dark. A and B are playing a game, A pick up

n

pairs from

\{ (i,j)|1 \leq i < j \leq n \}

and for every pairs

(i,j)

. B starts from the point marked

i

and choose to walk clockwise or counterclockwise to the point marked

j

. And B invert the status of all passing lights (bright

\leftrightarrow

dark) A hopes the number of dark light can be as much as possible while B hopes the number of bright light can be as much as possible. Suppose A, B are both smart, how many lights are bright in the end?
Proposed by BlessingOfHeaven
https://pbs.twimg.com/profile_images/1014932415201120256/u9KAaMZ4_400x400.jpg

N1

1

Large primes powers does not make square

Proof that for every primes

p

,

q

p^{q^2-q+1}+q^{p^2-p+1}-p-q

is never a perfect square. Proposed by chengbilly

C4

1

Bombing the Tree!

The REAL country has

n

islands, and there are

n-1

two-way bridges connecting these islands. Any two islands can be reached through a series of bridges. Arctan, the king of the REAL country, found that it is too difficult to manage

n

islands, so he wants to bomb some islands and their connecting bridges to divide the country into multiple small areas. Arctan wants the number of connected islands in each group is less than

\delta n

after bombing these islands, and the island he bomb must be a connected area. Besides, Arctan wants the number of islands to be bombed to be as less as possible. Find all real numbers

\delta

so that for any positive integer

n

and the layout of the bridge, the method of bombing the islands is the only one. Proposed by chengbilly

C1

1

Equal or Neq on the edges; Black or White in the cells

On a

n \times n

grid, each edge are written with

=

or

\neq

. We need to filled every cells with color black or white. Find the largest constant

k

, such that for every

n>777771449

and any layout of

=

and

\neq

, we can always find a way to colored every cells, such that at least

k \cdot 2n(n-1)

neighboring cells, there colors conform to the symbols on the edge. (Namely, two cells are filled with the same color if

=

was written on their edge; two cells are filled with different colors if

\neq

was written on their edge)
Proposed by chengbilly & sn6dh

A2

1

Minimum value

Given integer

n \geq 3

and

x_1

,

x_2

, …,

x_n

be

n

real numbers satisfying

|x_1|+|x_2|+…+|x_n|=1

. Find the minimum of

|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.

Proposed by snap7822

A4

1

Third degree FE

find all function

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

such that

f(x^3-xf(y)^2)=xf(x+y)f(x-y)

holds for all real number

x

,

y

.
Proposed by chengbilly

A7

1

REAL functions

Given positive integers

n

,

P_1

,

P_2

, …

P_n

and two sets

B=\{ (a_1,a_2,…,a_n)|a_i=0 \vee 1,\ \forall i \in \mathbb{N} \}, S=\{ (x_1,x_2,…,x_n)|1 \leq x_i \leq P_i \wedge x_i \in \mathbb{N} ,\ \forall i \in \mathbb{N} \}

A function

f:S \to \mathbb{Z}

is called Real, if and only if for any positive integers

(y_1,y_2,…,y_n)

and positive integer

a

which satisfied

1 \leq y_i \leq P_i-a

\forall i \in \mathbb{N}

, we always have: \begin{align*} \sum_{(a_1,a_2,…,a_n) \in B \wedge 2| \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&>\\ \sum_{(a_1,a_2,…,a_n) \in B \wedge 2 \nmid \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&. \end{align*} Find the minimum of

\sum_{i_1=1}^{P_1}\sum_{i_2=1}^{P_2}....\sum_{i_n=1}^{P_n}|f(i_1,i_2,...,i_n)|

, where

f

is a Real function.
Proposed by tob8y

A1

1

Infinite Series with Fraction

Given a positive integer

N

. Prove that

\sum_{m=1}^N \sum_{n=1}^N \frac{1}{mn^2+m^2n+2mn}<\frac{7}{4}.

Proposed by tan-1

N8

1

a^k+b^k-1|a^n+b^n-1

Find all integers

(a,b)

satisfying: there is an integer

k>1

such that

a^k+b^k-1\ |\ a^n+b^n-1

holds for all integer

n\geq k

(we define that

0|0

)

A3

1

An easy sequence but casework

Find all infinite integer sequences

a_1,a_2,\ldots

satisfying

a_{n+2}^{a_{n+1}}=a_{n+1}+a_n

holds for all

n\geq 1

. Define

0^0=1

A6

1

A strange inequality in Taiwan

Given positive real

a,b,c

satisfying

\frac{1}{\sqrt{a+1}}+\frac{3}{\sqrt{b+3}}+\frac{3}{\sqrt{c+3}}=\frac72

Prove that

abc\leq 3

.\\ I was asked to propose a inequality for the condition of

abc<3

inequality since <3 looks like a heart shape, then I construct a equality and with the help of wolfram, I gave the birth of this bad-looking inequality, I’m glad to see any method besides calculus.

N5

1

2^n+15|3^n+200

Find all positive integers

n

such that

2^n+15|3^n+200

N3

1

n(2^n-1) is a perfect square

Find all positive integers

n

such that

n(2^n-1)

is a perfect square

N4

1

a,b,c in S then ab+c in S

Given a set of integers

S

satisfies that: for any

a,b,c\in S

(

a,b,c

can be the same),

ab+c\in S

\\ Find all pairs of integers

(x,y)

such that if

x,y\in S

, then

S=\mathbb{Z}

.

N2

1

11^n+2^n+6=m^3

Find all positive integers

(m,n)

such that

11^n+2^n+6=m^3

N7

1

|xf(y)-yf(x)| is a perfect square

Find all functions

f:\mathbb{N}\to\mathbb{N}

such that

|xf(y)-yf(x)|

is a perfect square for every

x,y \in \mathbb{N}

2024-IMOC

Subcontests

Old problem in IMOC

Easy Geometry in 2024 IMOC

Quadrilateral Practice!

Cross-ratio Practice!

APMO intersects on Luna Cabrera

Intermediate Geometry in 2024 IMOC

Difficult Geometry in 2024 IMOC

Hong Kong Jane Street

Another Invisible Rabbit Appeared in 2024 imoc

The Strangest Problem in 2024 IMOC

You Deer

Rational NT FE

The Attack and Defense of Snails

Inspired by the logo of Jane Street

Large primes powers does not make square

Bombing the Tree!

Equal or Neq on the edges; Black or White in the cells

Minimum value

Third degree FE

REAL functions

Infinite Series with Fraction

a^k+b^k-1|a^n+b^n-1

An easy sequence but casework

A strange inequality in Taiwan

2^n+15|3^n+200

n(2^n-1) is a perfect square

a,b,c in S then ab+c in S

11^n+2^n+6=m^3

|xf(y)-yf(x)| is a perfect square