MathDB

1

An isosceles triangle is given, prove a right angle

ABC

be an isosceles triangle with

BC=CA

, and let

D

be a point inside side

AB

such that

AD< DB

. Let

P

and

Q

be two points inside sides

BC

and

CA

, respectively, such that

\angle DPB = \angle DQA = 90^{\circ}

. Let the perpendicular bisector of

PQ

meet line segment

CQ

at

E

, and let the circumcircles of triangles

ABC

and

CPQ

meet again at point

F

, different from

C

. Suppose that

P

,

E

,

F

are collinear. Prove that

\angle ACB = 90^{\circ}

.

N5

1

Integer FE Again

Determine all functions

f

defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[*]

(i)

f(n) \neq 0

for at least one

n

; [*]

(ii)

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

for every positive integers

x

and

y

; [*]

(iii)

there are infinitely many positive integers

n

such that

f(k)=f(n-k)

for all

k<n

.

N7

1

Some free permutation

Let

\mathcal{S}

be a set consisting of

n \ge 3

positive integers, none of which is a sum of two other distinct members of

\mathcal{S}

. Prove that the elements of

\mathcal{S}

may be ordered as

a_1, a_2, \dots, a_n

so that

a_i

does not divide

a_{i - 1} + a_{i + 1}

for all

i = 2, 3, \dots, n - 1

.

C4

1

Another SL problem about fibonacci numbers :3

The Fibonacci numbers

F_0, F_1, F_2, . . .

are defined inductively by

F_0=0, F_1=1

, and

F_{n+1}=F_n+F_{n-1}

for

n \ge 1

. Given an integer

n \ge 2

, determine the smallest size of a set

S

of integers such that for every

k=2, 3, . . . , n

there exist some

x, y \in S

such that

x-y=F_k

.
Proposed by Croatia

A1

1

Two bounds to find

Version 1. Let

n

be a positive integer, and set

N=2^{n}

. Determine the smallest real number

a_{n}

such that, for all real

x

,

\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .

Version 2. For every positive integer

N

, determine the smallest real number

b_{N}

such that, for all real

x

,

\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .

A2

1

Sets with Polynomials

Let

\mathcal{A}

denote the set of all polynomials in three variables

x, y, z

with integer coefficients. Let

\mathcal{B}

denote the subset of

\mathcal{A}

formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with

P, Q, R \in \mathcal{A}

. Find the smallest non-negative integer

n

such that

x^i y^j z^k \in \mathcal{B}

for all non-negative integers

i, j, k

satisfying

i + j + k \geq n

.

G4

1

Disk inequality

In the plane, there are

n \geqslant 6

pairwise disjoint disks

D_{1}, D_{2}, \ldots, D_{n}

with radii

R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}

. For every

i=1,2, \ldots, n

, a point

P_{i}

is chosen in disk

D_{i}

. Let

O

be an arbitrary point in the plane. Prove that

O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.

(A disk is assumed to contain its boundary.)

N1

1

a_ia_{i+1}a_{i+2}a_{i+3}=i(mod p)

Given a positive integer

k

show that there exists a prime

p

such that one can choose distinct integers

a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}

such that p divides

a_ia_{i+1}a_{i+2}a_{i+3}-i

for all

i= 1, 2, \cdots, k

.
South Africa

C1

1

Permutations of Integers from 1 to n

Let

n

be a positive integer. Find the number of permutations

a_1

,

a_2

,

\dots a_n

of the sequence

1

,

2

,

\dots

,

n

satisfying

a_1 \le 2a_2\le 3a_3 \le \dots \le na_n

.
Proposed by United Kingdom

G5

1

Rhombus and insimilicenters

Let

ABCD

be a cyclic quadrilateral. Points

K, L, M, N

are chosen on

AB, BC, CD, DA

such that

KLMN

is a rhombus with

KL \parallel AC

and

LM \parallel BD

. Let

\omega_A, \omega_B, \omega_C, \omega_D

be the incircles of

\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN

.
Prove that the common internal tangents to

\omega_A

, and

\omega_C

and the common internal tangents to

\omega_B

and

\omega_D

are concurrent.

C7

1

Saddle pairs in a grid

Consider any rectangular table having finitely many rows and columns, with a real number

a(r, c)

in the cell in row

r

and column

c

. A pair

(R, C)

, where

R

is a set of rows and

C

a set of columns, is called a saddle pair if the following two conditions are satisfied:
[*]

(i)

For each row

r^{\prime}

, there is

r \in R

such that

a(r, c) \geqslant a\left(r^{\prime}, c\right)

for all

c \in C

; [*]

(ii)

For each column

c^{\prime}

, there is

c \in C

such that

a(r, c) \leqslant a\left(r, c^{\prime}\right)

for all

r \in R

.
A saddle pair

(R, C)

is called a minimal pair if for each saddle pair

\left(R^{\prime}, C^{\prime}\right)

with

R^{\prime} \subseteq R

and

C^{\prime} \subseteq C

, we have

R^{\prime}=R

and

C^{\prime}=C

. Prove that any two minimal pairs contain the same number of rows.

G7

1

Circumcircles eat fish

Let

P

be a point on the circumcircle of acute triangle

ABC

. Let

D,E,F

be the reflections of

P

in the

A

-midline,

B

-midline, and

C

-midline. Let

\omega

be the circumcircle of the triangle formed by the perpendicular bisectors of

AD, BE, CF

.
Show that the circumcircles of

\triangle ADP, \triangle BEP, \triangle CFP,

and

\omega

share a common point.

C5

1

Sorry it's combo

Let

p

be an odd prime, and put

N=\frac{1}{4} (p^3 -p) -1.

The numbers

1,2, \dots, N

are painted arbitrarily in two colors, red and blue. For any positive integer

n \leqslant N,

denote

r(n)

the fraction of integers

\{ 1,2, \dots, n \}

that are red. Prove that there exists a positive integer

a \in \{ 1,2, \dots, p-1\}

such that

r(n) \neq a/p

for all

n = 1,2, \dots , N.

[I]Netherlands

G3

1

geometry with quadrilateral, tangent circles wanted

Let

ABCD

be a convex quadrilateral with

\angle ABC>90

,

CDA>90

and

\angle DAB=\angle BCD

. Denote by

E

and

F

the reflections of

A

in lines

BC

and

CD

, respectively. Suppose that the segments

AE

and

AF

meet the line

BD

at

K

and

L

, respectively. Prove that the circumcircles of triangles

BEK

and

DFL

are tangent to each other.
\emph{Slovakia}

G8

1

An I for an I

Let

ABC

be a triangle with incenter

I

and circumcircle

\Gamma

. Circles

\omega_{B}

passing through

B

and

\omega_{C}

passing through

C

are tangent at

I

. Let

\omega_{B}

meet minor arc

AB

of

\Gamma

at

P

and

AB

at

M\neq B

, and let

\omega_{C}

meet minor arc

AC

of

\Gamma

at

Q

and

AC

at

N\neq C

. Rays

PM

and

QN

meet at

X

. Let

Y

be a point such that

YB

is tangent to

\omega_{B}

and

YC

is tangent to

\omega_{C}

.
Show that

A,X,Y

are collinear.

A7

1

n variable inequalities where all variables are bounded

Let

n

and

k

be positive integers. Prove that for

a_1, \dots, a_n \in [1,2^k]

one has

\sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}.

A8

1

Hard functional equation

Let

R^+

be the set of positive real numbers. Determine all functions

f:R^+

\rightarrow

R^+

such that for all positive real numbers

x

and

y:

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

Ukraine

A5

1

Algebraic Magic Trick

A magician intends to perform the following trick. She announces a positive integer

n

, along with

2n

real numbers

x_1 < \dots < x_{2n}

, to the audience. A member of the audience then secretly chooses a polynomial

P(x)

of degree

n

with real coefficients, computes the

2n

values

P(x_1), \dots , P(x_{2n})

, and writes down these

2n

values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

G6

1

Computer too strong

Let

ABC

be a triangle with

AB < AC

, incenter

I

, and

A

excenter

I_{A}

. The incircle meets

BC

at

D

. Define

E = AD\cap BI_{A}

,

F = AD\cap CI_{A}

. Show that the circumcircle of

\triangle AID

and

\triangle I_{A}EF

are tangent to each other

C8

1

Bananalysis Board

Players

A

and

B

play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player

A

erases two numbers

x

and

y

from the blackboard, and then player

B

writes one of the numbers

x+y

and

|x-y|

on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds:
[*]

(1)

one of the numbers on the blackboard is larger than the sum of all other numbers; [*]

(2)

there are only zeros on the blackboard.
Player

B

must then give as many cookies to player

A

as there are numbers on the blackboard. Player

A

wants to get as many cookies as possible, whereas player

B

wants to give as few as possible. Determine the number of cookies that

A

receives if both players play optimally.

N2

1

A prime graph

For each prime

p

, construct a graph

G_p

on

\{1,2,\ldots p\}

, where

m\neq n

are adjacent if and only if

p

divides

(m^{2} + 1-n)(n^{2} + 1-m)

. Prove that

G_p

is disconnected for infinitely many

p

N6

1

Phinding constants is Diphicult

For a positive integer

n

, let

d(n)

be the number of positive divisors of

n

, and let

\varphi(n)

be the number of positive integers not exceeding

n

which are coprime to

n

. Does there exist a constant

C

such that

\frac {\varphi ( d(n))}{d(\varphi(n))}\le C

for all

n\ge 1

Cyprus

A6

1

Big function eats Eyed

Find all functions

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

satisfying

f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)

for all integers

a

and

b

N4

1

P-Dop P-Dop

For any odd prime

p

and any integer

n,

let

d_p (n) \in \{ 0,1, \dots, p-1 \}

denote the remainder when

n

is divided by

p.

We say that

(a_0, a_1, a_2, \dots)

is a p-sequence, if

a_0

is a positive integer coprime to

p,

and

a_{n+1} =a_n + d_p (a_n)

for

n \geqslant 0.

(a) Do there exist infinitely many primes

p

for which there exist

p

-sequences

(a_0, a_1, a_2, \dots)

and

(b_0, b_1, b_2, \dots)

such that

a_n >b_n

for infinitely many

n,

and

b_n > a_n

for infinitely many

n?

(b) Do there exist infinitely many primes

p

for which there exist

p

-sequences

(a_0, a_1, a_2, \dots)

and

(b_0, b_1, b_2, \dots)

such that

a_0 <b_0,

but

a_n >b_n

for all

n \geqslant 1?

[I]United Kingdom

C2

1

24 convex quadrilaterals

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals

Q_{1}, \ldots, Q_{24}

whose corners are vertices of the 100-gon, so that
[*] the quadrilaterals

Q_{1}, \ldots, Q_{24}

are pairwise disjoint, and [*] every quadrilateral

Q_{i}

has three corners of one color and one corner of the other color.

A3

1

Minimizing simple sum of fractions

Suppose that

a,b,c,d

are positive real numbers satisfying

(a+c)(b+d)=ac+bd

. Find the smallest possible value of

\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.

Israel

2020 IMO Shortlist

Subcontests

An isosceles triangle is given, prove a right angle

Integer FE Again

Some free permutation

Another SL problem about fibonacci numbers :3

Two bounds to find

Sets with Polynomials

Disk inequality

a_ia_{i+1}a_{i+2}a_{i+3}=i(mod p)

Permutations of Integers from 1 to n

Rhombus and insimilicenters

Saddle pairs in a grid

Circumcircles eat fish

Sorry it's combo

geometry with quadrilateral, tangent circles wanted

An I for an I

n variable inequalities where all variables are bounded

Hard functional equation

Algebraic Magic Trick

Computer too strong

Bananalysis Board

A prime graph

Phinding constants is Diphicult

Big function eats Eyed

P-Dop P-Dop

24 convex quadrilaterals

Minimizing simple sum of fractions