MathDB

1

nice diophantine with existence

N

such that the following holds: There exist pairwise coprime positive integers

a,b,c

with

\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.

C5

1

game on graph involving connectedness

Alice and Bob are playing a game on a graph with

n\ge3

vertices. At each moment, Alice needs to choose two vertices so that the graph is connected even if one of them (along with the edges incident to it) is removed. Each turn, Bob removes one edge in the graph, and upon the removal, Alice needs to re-select the two vertices if necessary. However, Bob has to guarantee that after each removal, any two vertices in the graph are still connected via at most

k

intermediate vertices. Here

0\le k\le n-2

is some given integer. Suppose that Bob always knows which two vertices Alice chooses, and that initially, the graph is a complete graph. Alice's objective is to change her choice of the two vertices as few times as possible, and Bob's objective is to make Alive re-select as many times as possible. If both Alice and Bob are sufficiently smart, how many times will Alice change her choice of the two vertices? (usjl)

C3

1

sending secret message with three digits

Sunny wants to send some secret message to usjl. The secret message is a three digit number, where each digit is one digit from

0

to

9

(so

000

is also possibly the secret message). However, when Sunny sends the message to usjl, at most one digit might be altered. Therefore, Sunny decides to send usjl a longer message so that usjl can decipher the message to get the original secret message Sunny wants to send. Sunny and usjl can communicate the strategy beforehand. Show that sending a

4

-digit message does not suffice. Also show that sending a

6

-digit message suffices. If it is deduced that sending a

c

-digit message suffices for some

c>6

, then partial credits may be awarded.

C2

1

changing letters around circle if neighbors are the same

There are

N\ge3

letters arranged in a circle, and each letter is one of

L

,

T

and

F

. For a letter, we can do the following operation: if its neighbors are the same, then change it to the same letter too; otherwise, change it so that it is different from both its neighbors. Show that for any initial state, one can perform finitely many operations to achieve a stable state. Here, a stable state means that any operation does not change any of the

N

letters. (ltf0501)

A4

1

polynomial sequence, writing numbers never stops

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers

a_{-1},\ldots,a_{-k}

on a blackboard, so he decided to do the following process just for fun: if there are real numbers

a_{-k},\ldots,a_{n-1}

on the blackboard, then he computes the polynomial

P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t).

He then writes a real number

a_n

, where

a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}.

If

a_n

is undefined (that is,

P_n(i)+P_n(-i)=0

), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists

m>n\ge0

such that

am=an

), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)

C6

1

Arrangement of shy boys and girls

\definecolor{A}{RGB}{70,255,50}\color{A}\fbox{C6.} There are

n

\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys} and

n

\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls} in a club. Some of them are friends with each other. The \definecolor{A}{RGB}{0,0,255}\color{A}\text{boys} want to get into a relationship, so some subset of them wants to ask some \definecolor{A}{RGB}{255,0,255}\color{A}\text{girls} out for a trip. Because the \definecolor{A}{RGB}{0,0,255}\color{A}\text{boys} are shy, for a nonempty set

B

of \definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}, they want to make sure that each of the girl they ask out is friend with one of the \definecolor{A}{RGB}{0,0,255}\color{A}\text{boys} in

B

. If the number of \definecolor{A}{RGB}{255,0,255}\color{A}\text{girls} they are able to ask out is smaller than the number of the \definecolor{A}{RGB}{0,0,255}\color{A}\text{boys} in

B

, then the nonempty set

B

of those \definecolor{A}{RGB}{0,0,255}\color{A}\text{boys} is called a group of complete losers. Show that for any

0 \le k < 2n

, there exists an arrangement of the friendships among those

2n

people so that there are exactly

k

groups of complete losers.
Proposed by [color=#419DAB]ltf0501. [color=#3D9186]#1737

C4

1

explosive C4 from IMOC - three squares is a square

\definecolor{A}{RGB}{70,80,0}\color{A}\fbox{C4.} Show that for any positive integer

n \ge 3

and some subset of

\lbrace{1, 2, . . . , n}\rbrace

with size more than

\frac{n}2 + 1

, there exist three distinct elements

a, b, c

in the subset such that \definecolor{A}{RGB}{255,70,255}\color{A} (ab)^2 + (bc)^2 + (ca)^2is a perfect square.
Proposed by [color=#419DAB]ltf0501. [color=#3D9186]#1736

A6

1

f(P(x)) = P(f(x)) - range of polynomial and f

\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.} Let

P (x)

be a polynomial with real coefficients such that

\deg P \ge 3

is an odd integer. Let

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

be a function such that \definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)). \definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)} Prove that the range of

f

is finite. \definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)} Show that for any positive integer

n

, there exist

P

,

f

that satisfies the above condition and also that the range of

f

has cardinality

n

.
Proposed by [color=#419DAB]ltf0501. [color=#3D9186]#1735

A3

1

Quadratic in denominator sym. ineq.

\definecolor{A}{RGB}{250,120,0}\color{A}\fbox{A3.} Assume that

a, b, c

are positive reals such that

a + b + c = 3

. Prove that \definecolor{A}{RGB}{200,0,200}\color{A} \frac{1}{8a^2-18a+11}+\frac{1}{8b^2-18b+11}+\frac{1}{8c^2-18c+11}\le 3.
Proposed by [color=#419DAB]ltf0501. [color=#3D9186]#1734

A1

1

AM-GM functional inequality 2020

\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.} Find all

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

such that \definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)holds for all

a,b,c\in\mathbb{R}

.
Proposed by [color=#FFFF00]usjl. [color=#B6D7A8]#1733

N1

1

Integer equation

<span class='latex-bold'>N1.</span>

Find all nonnegative integers

a,b,c

such that \begin{align*} a^2+b^2+c^2-ab-bc-ca = a+b+c \end{align*} Proposed by usjl

N6

1

gcd + fibonacci

<span class='latex-bold'>N6.</span>

Let

a,b

be positive integers. If

a,b

satisfy that \begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*} is also a positive integer, show that \begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*} is a Fibonacci number. Proposed by usjl

N3

1

sequence involving rad(x)

<span class='latex-bold'>N3:</span>

For any positive integer

n

, define

rad(n)

to be the product of all prime divisors of

n

(without multiplicities), and in particular

rad(1)=1

. Consider an infinite sequence of positive integers

\{a_n\}_{n=1}^{\infty}

satisfying that \begin{align*} a_{n+1} = a_n + rad(a_n), \: \forall n \in \mathbb{N} \end{align*} Show that there exist positive integers

t,s

such that

a_t

is the product of the

s

smallest primes. Proposed by ltf0501

N4

1

a^i+b^i divisiblity

<span class='latex-bold'>N4:</span>

Let

a,b

be two positive integers such that for all positive integer

n>2020^{2020}

, there exists a positive integer

m

coprime to

n

with \begin{align*} \text{

a^n+b^n \mid a^m+b^m

} \end{align*} Show that

a=b

Proposed by ltf0501

C1

1

N black triangles, One big triangle wanted

Find all positive integer

N

such that for any infinite triangular grid with exactly

N

black unit equilateral triangles, there exists an equilateral triangle

S

whose sides align with grid lines such that there is exactly one black unit equilateral triangle outside of

S.

(ltf0501)

N5

1

FE on Famous Identity

<span class='latex-bold'>N5.</span>

Find all

f: \mathbb{N} \rightarrow \mathbb{N}

such that for all

a,b,c \in \mathbb{N}

f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc

A2

1

A nice fe

Find all function

f:\mathbb{R}^+

\rightarrow \mathbb{R}^+

such that:

f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+

G6

1

IMOC 2020 G6 2 tangents of different circle concurrent with a line

Let

ABC

be a triangle, and

M_a, M_b, M_c

be the midpoints of

BC, CA, AB

, respectively. Extend

M_bM_c

so that it intersects

\odot (ABC)

at

P

. Let

AP

and

BC

intersect at

Q

. Prove that the tangent at

A

to

\odot(ABC)

and the tangent at

P

to

\odot (P QM_a)

intersect on line

BC

.
(Li4)

G5

1

IMOC 2020 G5 concyclic wanted, parallelogram and concurrent related

Let

O, H

be the circumcentor and the orthocenter of a scalene triangle

ABC

. Let

P

be the reflection of

A

w.r.t.

OH

, and

Q

is a point on

\odot (ABC)

such that

AQ, OH, BC

are concurrent. Let

A'

be a points such that

ABA'C

is a parallelogram. Show that

A', H, P, Q

are concylic.
(ltf0501).

G4

1

IMOC 2020 G4 concyclic wanted, incenter related

Let

I

be the incenter of triangle

ABC

. Let

BI

and

AC

intersect at

E

, and

CI

and

AB

intersect at

F

. Suppose that

R

is another intersection of

\odot (ABC)

and

\odot (AEF)

. Let

M

be the midpoint of

BC

, and

P, Q

are the intersections of

AI, MI

and

EF

, respectively. Show that

A, P, Q, R

are concyclic.
(ltf0501).

G3

1

IMOC 2020 G3 (circumcenter of OM_BM_C lies on Euler line of BIC)

Triangle

ABC

has incenter

I

and circumcenter

O

.

AI, BI, CI

intersect the circumcircle of

ABC

again at

M_A, M_B, M_C

, respectively. Show that the Euler line of

BIC

passes through the circumcenter of

OM_BM_C

.
(houkai)

G2

1

IMOC 2020 G2 concurency of AO_{An}, BO_{Bn}, CO_{Cn}

Let

O

be the circumcenter of triangle

ABC

. Define

O_{A0} = O_{B0} = O_{C0} = O

. Recursively, define

O_{An}

to be the circumcenter of

\vartriangle BO_{A(n-1)}C

. Similarly define

O_{Bn}, O_{Cn}

. Find all

n \ge 1

so that for any triangle

ABC

such that

O_{An}, O_{Bn}, O_{Cn}

all exist, it is true that

AO_{An}, BO_{Bn}, CO_{Cn}

are concurrent.
(Li4)

G1

1

IMOC 2020 G1 (tangent circles wanted)

Let

O

be the circumcenter of triangle

ABC

. Choose a point

X

on the circumcircle

\odot (ABC)

such that

OX\parallel BC

. Assume that

\odot(AXO)

intersects

AB, AC

at

E, F

, respectively, and

OE, OF

intersect

BC

at

P, Q

, respectively. Furthermore, assume that

\odot(XP Q)

and

\odot (ABC)

intersect at

R

. Prove that

OR

and

\odot (XP Q)

are tangent to each other.
(ltf0501)

A5

1

Inequalities with Sequences

Let

0<c<1

be a given real number. Determine the least constant

K

such that the following holds: For all positive real

M

that is greater than

1

, there exists a strictly increasing sequence

x_0, x_1, \ldots, x_n

(of arbitrary length) such that

x_0=1, x_n\geq M

and

\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\right)^c}{x_i^{c+1}}\leq K.

(From 2020 IMOCSL A5. I think this problem is particularly beautiful so I want to make a separate thread for it :D )

2020-IMOC

Subcontests

nice diophantine with existence

game on graph involving connectedness

sending secret message with three digits

changing letters around circle if neighbors are the same

polynomial sequence, writing numbers never stops

Arrangement of shy boys and girls

explosive C4 from IMOC - three squares is a square

f(P(x)) = P(f(x)) - range of polynomial and f

Quadratic in denominator sym. ineq.

AM-GM functional inequality 2020

Integer equation

gcd + fibonacci

sequence involving rad(x)

a^i+b^i divisiblity

N black triangles, One big triangle wanted

FE on Famous Identity

A nice fe

IMOC 2020 G6 2 tangents of different circle concurrent with a line

IMOC 2020 G5 concyclic wanted, parallelogram and concurrent related

IMOC 2020 G4 concyclic wanted, incenter related

IMOC 2020 G3 (circumcenter of OM_BM_C lies on Euler line of BIC)

IMOC 2020 G2 concurency of AO_{An}, BO_{Bn}, CO_{Cn}

IMOC 2020 G1 (tangent circles wanted)

Inequalities with Sequences