MathDB

1

Permutation of the set gives all the residues modulo n

n\ge 4

beautiful if there exists some permutation

\{x_1,x_2,\dots ,x_{n-1}\}

of

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

such that

\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}

gives all the residues

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

modulo

n

. Prove that if

n

is beautiful then

n=2p,

for some prime number

p.

5

1

af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac)

Find all functions

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

such that for all integers

a, b, c

we have

af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac).

Note. The set

\mathbb{Z^+}

refers to the set of positive integers. Proposed by Mojtaba Zare, Iran

4

1

Easy combinatorics problem

Given positive integers

a,b,

find the least positive integer

m

such that among any

m

distinct integers in the interval

[-a,b]

there are three pair-wise distinct numbers that their sum is zero. Proposed by Marian Tetiva, Romania

3

1

Combo problem about regular polygons

Abdulqodir

cut out

2024

congruent regular

n-

gons from a sheet of paper and placed these

n-

gons on the table such that some parts of each of these

n-

gons may be covered by others. We say that a vertex of one of the afore-mentioned

n-

gons is

visible

if it is not in the interior of another

n-

gon that is placed on top of it. For any

n>2

determine the minimum possible number of visible vertices. \\ Proposed by David Hrushka, Slovakia

2

1

Sequence of positive integers

Find all positive integers

(r,s)

such that there is a non-constant sequence

a_n

os positive integers such that for all

n=1,2,\dots

a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).

Proposed by Navid Safaei, Iran

1

Nice geometry problem about incenter

Let

ABC

be a triangle with

AB<AC

and incenter

I.

A point

D

lies on segment

AC

such that

AB=AD,

and the line

BI

intersects

AC

at

E.

Suppose the line

CI

intersects

BD

at

F,

and

G

lies on segment

DI

such that

FD=FG.

Prove that the lines

AG

and

EF

intersect on the circumcircle of triangle

CEI.

\\ Proposed by Avan Lim Zenn Ee, Malaysia

2024 TASIMO

Subcontests

Permutation of the set gives all the residues modulo n

af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac)

Easy combinatorics problem

Combo problem about regular polygons

Sequence of positive integers

Nice geometry problem about incenter