MathDB

1

I thought fairies were nice

2023

treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules:
[*]if more than one chests are unlocked, it locks one of them, or [*]if there is only one unlocked chest, it unlocks all the chests.
Given that this process goes on forever, prove that there is a constant

C

with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds

C

, regardless of how the fairy chooses the chests to unlock.

G1

1

this hAOpefully shoudn't BE weird

Let

ABCDE

be a convex pentagon such that

\angle ABC = \angle AED = 90^\circ

. Suppose that the midpoint of

CD

is the circumcenter of triangle

ABE

. Let

O

be the circumcenter of triangle

ACD

.
Prove that line

AO

passes through the midpoint of segment

BE

.

C2

1

ISL 2023 C2

Determine the maximal length

L

of a sequence

a_1,\dots,a_L

of positive integers satisfying both the following properties:
[*]every term in the sequence is less than or equal to

2^{2023}

, and [*]there does not exist a consecutive subsequence

a_i,a_{i+1},\dots,a_j

(where

1\le i\le j\le L

) with a choice of signs

s_i,s_{i+1},\dots,s_j\in\{1,-1\}

for which

s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.

C7

1

Coloring K_n so that all Hamiltonian paths contain all k colors

The Imomi archipelago consists of

n\geq 2

islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of

k

companies. It is known that if any one of the

k

companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at).
Determine the maximal possible value of

k

in terms of

n

.
Anton Trygub, Ukraine

C1

1

Flipping L's

Let

m

and

n

be positive integers greater than

1

. In each unit square of an

m\times n

grid lies a coin with its tail side up. A move consists of the following steps.
[*]select a

2\times 2

square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square.
Determine all pairs

(m,n)

for which it is possible that every coin shows head-side up after a finite number of moves.
Thanasin Nampaisarn, Thailand

C6

1

too many equality cases

Let

N

be a positive integer, and consider an

N \times N

grid. A right-down path is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A right-up path is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the

N \times N

grid cannot be partitioned into less than

N

right-down or right-up paths. For example, the following partition of the

5 \times 5

grid uses

5

paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] Proposed by Zixiang Zhou, Canada

N5

1

Every Integer is a divisor

Let

a_1<a_2<a_3<\dots

be positive integers such that

a_{k+1}

divides

2(a_1+a_2+\dots+a_k)

for every

k\geqslant 1

. Suppose that for infinitely many primes

p

, there exists

k

such that

p

divides

a_k

. Prove that for every positive integer

n

, there exists

k

such that

n

divides

a_k

.

A4

1

another functional inequality?

Let

\mathbb R_{>0}

be the set of positive real numbers. Determine all functions

f \colon \mathbb R_{>0} \to \mathbb R_{>0}

such that

x \left(f(x) + f(y)\right) \geqslant \left(f(f(x)) + y\right) f(y)

for every

x, y \in \mathbb R_{>0}

.

G3

1

Concurrence in Cyclic Quadrilateral

Let

ABCD

be a cyclic quadrilateral with

\angle BAD < \angle ADC

. Let

M

be the midpoint of the arc

CD

not containing

A

. Suppose there is a point

P

inside

ABCD

such that

\angle ADB = \angle CPD

and

\angle ADP = \angle PCB

.
Prove that lines

AD, PM

, and

BC

are concurrent.

G7

1

APMO 2010 P4 but you do it on all 3 sides

Let

ABC

be an acute, scalene triangle with orthocentre

H

. Let

\ell_a

be the line through the reflection of

B

with respect to

CH

and the reflection of

C

with respect to

BH

. Lines

\ell_b

and

\ell_c

are defined similarly. Suppose lines

\ell_a

,

\ell_b

, and

\ell_c

determine a triangle

\mathcal T

.
Prove that the orthocentre of

\mathcal T

, the circumcentre of

\mathcal T

, and

H

are collinear.
Fedir Yudin, Ukraine

N8

1

hi arnov

Determine all functions

f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}

such that, for all positive integers

a

and

b

,

f^{bf(a)}(a+1)=(a+1)f(b).

N2

1

How many cases did you check?

Determine all ordered pairs

(a,p)

of positive integers, with

p

prime, such that

p^a+a^4

is a perfect square.
Proposed by Tahjib Hossain Khan, Bangladesh

G6

1

Help my diagram has too many points

Let

ABC

be an acute-angled triangle with circumcircle

\omega

. A circle

\Gamma

is internally tangent to

\omega

at

A

and also tangent to

BC

at

D

. Let

AB

and

AC

intersect

\Gamma

at

P

and

Q

respectively. Let

M

and

N

be points on line

BC

such that

B

is the midpoint of

DM

and

C

is the midpoint of

DN

. Lines

MP

and

NQ

meet at

K

and intersect

\Gamma

again at

I

and

J

respectively. The ray

KA

meets the circumcircle of triangle

IJK

again at

X\neq K

.
Prove that

\angle BXP = \angle CXQ

.
Kian Moshiri, United Kingdom

G5

1

Guessing Point is Hard

Let

ABC

be an acute-angled triangle with circumcircle

\omega

and circumcentre

O

. Points

D\neq B

and

E\neq C

lie on

\omega

such that

BD\perp AC

and

CE\perp AB

. Let

CO

meet

AB

at

X

, and

BO

meet

AC

at

Y

.
Prove that the circumcircles of triangles

BXD

and

CYE

have an intersection lie on line

AO

.
Ivan Chan Kai Chin, Malaysia

N4

1

Arithmetic Sequence of Products

Let

a_1, \dots, a_n, b_1, \dots, b_n

be

2n

positive integers such that the

n+1

products

a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n

form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

G2

1

PQ = r and 6 more conditions

Let

ABC

be a triangle with

AC > BC,

let

\omega

be the circumcircle of

\triangle ABC,

and let

r

be its radius. Point

P

is chosen on

\overline{AC}

such taht

BC=CP,

and point

S

is the foot of the perpendicular from

P

to

\overline{AB}

. Ray

BP

mets

\omega

again at

D

. Point

Q

is chosen on line

SP

such that

PQ = r

and

S,P,Q

lie on a line in that order. Finally, let

E

be a point satisfying

\overline{AE} \perp \overline{CQ}

and

\overline{BE} \perp \overline{DQ}

. Prove that

E

lies on

\omega

.

A1

1

Feeding Pok&eacute;mon is hard

Professor Oak is feeding his

100

Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is

100

kilograms. Professor Oak distributes

100

kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The dissatisfaction level of a Pokémon who received

N

kilograms of food and whose bowl has a capacity of

C

kilograms is equal to

\lvert N-C\rvert

.
Find the smallest real number

D

such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the

100

Pokémon is at most

D

.
Oleksii Masalitin, Ukraine

N7

1

Hardest N7 in history

Let

a,b,c,d

be positive integers satisfying

\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.

Determine all possible values of

a+b+c+d

.

A7

1

The Return of Triangle Geometry

Let

N

be a positive integer. Prove that there exist three permutations

a_1,\dots,a_N

,

b_1,\dots,b_N

, and

c_1,\dots,c_N

of

1,\dots,N

such that

\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023

for every

k=1,2,\dots,N

.

N6

1

Hardest N6 in history

A sequence of integers

a_0, a_1 …

is called kawaii if

a_0 =0, a_1=1,

and

(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0

for all integers

n \geq 0

. An integer is called kawaii if it belongs to some kawaii sequence. Suppose that two consecutive integers

m

and

m+1

are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that

m

is divisible by

3,

and that

m/3

is also kawaii.

C4

1

Modular Strip

Let

n\geqslant 2

be a positive integer. Paul has a

1\times n^2

rectangular strip consisting of

n^2

unit squares, where the

i^{\text{th}}

square is labelled with

i

for all

1\leqslant i\leqslant n^2

. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an

n\times n

square satisfying the following property: if the unit square in the

i^{\text{th}}

row and

j^{\text{th}}

column is labelled with

a_{ij}

, then

a_{ij}-(i+j-1)

is divisible by

n

.
Determine the smallest number of pieces Paul needs to make in order to accomplish this.

A5

1

Permutations inequality

Let

a_1,a_2,\dots,a_{2023}

be positive integers such that
[*]

a_1,a_2,\dots,a_{2023}

is a permutation of

1,2,\dots,2023

, and [*]

|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|

is a permutation of

1,2,\dots,2022

.
Prove that

\max(a_1,a_{2023})\ge 507

.

A2

1

Functional Inequality Implies Uniform Sign

Let

\mathbb{R}

be the set of real numbers. Let

f:\mathbb{R}\rightarrow\mathbb{R}

be a function such that

f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2

for every

x,y\in\mathbb{R}

. Assume that the inequality is strict for some

x_0,y_0\in\mathbb{R}

.
Prove that either

f(x)\geqslant 0

for every

x\in\mathbb{R}

or

f(x)\leqslant 0

for every

x\in\mathbb{R}

.

N3

1

the epitome of olympiad nt

For positive integers

n

and

k \geq 2

, define

E_k(n)

as the greatest exponent

r

such that

k^r

divides

n!

. Prove that there are infinitely many

n

such that

E_{10}(n) > E_9(n)

and infinitely many

m

such that

E_{10}(m) < E_9(m)

.

2023 ISL

Subcontests

I thought fairies were nice

this hAOpefully shoudn't BE weird

ISL 2023 C2

Coloring K_n so that all Hamiltonian paths contain all k colors

Flipping L's

too many equality cases

Every Integer is a divisor

another functional inequality?

Concurrence in Cyclic Quadrilateral

APMO 2010 P4 but you do it on all 3 sides

hi arnov

How many cases did you check?

Help my diagram has too many points

Guessing Point is Hard

Arithmetic Sequence of Products

PQ = r and 6 more conditions

Feeding Pok&eacute;mon is hard

Hardest N7 in history

The Return of Triangle Geometry

Hardest N6 in history

Modular Strip

Permutations inequality

Functional Inequality Implies Uniform Sign

the epitome of olympiad nt

2023 ISL

Subcontests

I thought fairies were nice

this hAOpefully shoudn&#039;t BE weird

ISL 2023 C2

Coloring K_n so that all Hamiltonian paths contain all k colors

Flipping L&#039;s

too many equality cases

Every Integer is a divisor

another functional inequality?

Concurrence in Cyclic Quadrilateral

APMO 2010 P4 but you do it on all 3 sides

hi arnov

How many cases did you check?

Help my diagram has too many points

Guessing Point is Hard

Arithmetic Sequence of Products

PQ = r and 6 more conditions

Feeding Pok&amp;eacute;mon is hard

Hardest N7 in history

The Return of Triangle Geometry

Hardest N6 in history

Modular Strip

Permutations inequality

Functional Inequality Implies Uniform Sign

the epitome of olympiad nt

this hAOpefully shoudn't BE weird

Flipping L's

Feeding Pokémon is hard