MathDB

1

Fractions does not add up to get integers

m, n

\frac 1m + \frac 1{m+1} + \dotsb + \frac 1 {m+n}

is not an integer.

C2

1

Large integer without repeating abab

Find the largest positive integer

n

such that no two adjacent digits are the same, and for any two distinct digits

0 \leq a,b \leq 9

, you can't get the string

abab

just by removing digits from

n

.

C1

1

String with As and Bs

There are

3n

A

s and

2n

B

s in a string, where

n

is a positive integer, prove that you can find a substring in this string that contains

3

A

s and

2

B

s.

N1

1

Mersenne primes

Prove: if

2^{2^n-1}-1

is a prime, then

n

is a prime.

A1

1

9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) >=9

Prove that if positive reals

x,y

satisfy

x+y= 3

,

x,y \ge 1

then

9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) \ge 9

A2

1

P(x)f(Q(x))=f(P(x)Q(x)) if f(x)+f(y)=f(x+y)

Find all functions

f:R \to R

, such that

f(x)+f(y)=f(x+y)

, and there exists non-constant polynomials

P(x)

,

Q(x)

such that

P(x)f(Q(x))=f(P(x)Q(x))

A3

1

f(n(f(m)) <= nm

Find all injective function

f: N \to N

satisfying that for all positive integers

m,n

, we have:

f(n(f(m)) \le nm

G2

1

DE _|_AC if DF _|_ AB

Given a triangle

ABC

,

D

is the reflection from the perpendicular foot from

A

to

BC

through the midpoint of

BC

.

E

is the reflection from the perpendicular foot from

B

to

CA

through the midpoint of

CA

.

F

is the reflection from the perpendicular foot from

C

to

AB

through the midpoint of

AB

. Prove:

DE \perp AC

if and only if

DF \perp AB

G1

1

AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2, projections , circumcenter, incenter

Let

O

be the circumcenter and

I

be the incenter of

\vartriangle

,

P

is the reflection from

I

through

O

, the foot of perpendicular from

P

to

BC,CA,AB

is

X,Y,Z

, respectively. Prove that

AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2

.

N0

1

remainder of 3^{2021} mod 15 2021 IMOC qualification problem, N0

Compute the remainder of

3^{2021}

mod

15

A0

1

f(x)=1/1+2^{1-2x} 2021 IMOC qualification problem, A0

Consider the following function

f(x)=\frac{1}{1+2^{1-2x}}

. Compute the value of

f\left(\frac{1}{10}\right)+f\left(\frac{2}{10}\right)+...+f\left(\frac{9}{10}\right).

C0

1

coins on vertices of 2021-gon 2021 IMOC qualification problem, C0

There is a regular

2021

-gon. We put a coin with heads up on every vertex of it. Every time, you can choose one vertex, and flip the coin on the vertices adjacent to it. Can you make all the coin tails up?

G0

1

area of triangle wanted 2021 IMOC qualification problem, G0

Let

D,E,F

be the midpoints of

BC

,

CA

,

AB

in

\vartriangle ABC

such that

AD= 9

,

BE= 12

,

CF= 15

. Calculate the area of

\vartriangle ABC

C3

1

Two player playing with cards numbered 1~n

There are n cards on a table numbered from

1

to

n

, where

n

is an even number. Two people take turns taking away the cards. The first player will always take the card with the largest number on it, but the second player will take a random card. Prove: the probability that the first player takes the card with the number

i

is

\frac{i-1}{n-1}

G3

1

Triangle with 45 degree angle and the Nine Point Center

Given a

\triangle ABC

,

\angle A=45^\circ

,

O

is the circumcenter and

H

is the orthocenter of

\triangle ABC

.

M

is the midpoint of

\overline{BC}

, and

N

is the midpoint of

\overline{OH}

. Prove that

\angle BAM=\angle CAN

.

N3

1

A number with 2021 Prime divisors

Prove: There exists a positive integer

n

with

2021

prime divisors, satisfying

n|2^n+1

.

2021-IMOC qualification

Subcontests

Fractions does not add up to get integers

Large integer without repeating abab

String with As and Bs

Mersenne primes

9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) >=9

P(x)f(Q(x))=f(P(x)Q(x)) if f(x)+f(y)=f(x+y)

f(n(f(m)) <= nm

DE _|_AC if DF _|_ AB

AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2, projections , circumcenter, incenter

remainder of 3^{2021} mod 15 2021 IMOC qualification problem, N0

f(x)=1/1+2^{1-2x} 2021 IMOC qualification problem, A0

coins on vertices of 2021-gon 2021 IMOC qualification problem, C0

area of triangle wanted 2021 IMOC qualification problem, G0

Two player playing with cards numbered 1~n

Triangle with 45 degree angle and the Nine Point Center

A number with 2021 Prime divisors

2021-IMOC qualification

Subcontests

Fractions does not add up to get integers

Large integer without repeating abab

String with As and Bs

Mersenne primes

9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) &gt;=9

P(x)f(Q(x))=f(P(x)Q(x)) if f(x)+f(y)=f(x+y)

f(n(f(m)) &lt;= nm

DE _|_AC if DF _|_ AB

AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2, projections , circumcenter, incenter

remainder of 3^{2021} mod 15 2021 IMOC qualification problem, N0

f(x)=1/1+2^{1-2x} 2021 IMOC qualification problem, A0

coins on vertices of 2021-gon 2021 IMOC qualification problem, C0

area of triangle wanted 2021 IMOC qualification problem, G0

Two player playing with cards numbered 1~n

Triangle with 45 degree angle and the Nine Point Center

A number with 2021 Prime divisors

9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) >=9

f(n(f(m)) <= nm