2024 Indonesia TST

Part of Indonesia TST

Subcontests

(4)

2

Construction problem

Given a sequence of integers

A_1,A_2,\cdots A_{99}

such that for every sub-sequence that contains

m

consecutive elements, there exist not more than

max\{ \frac{m}{3} ,1\}

odd integers. Let

S=\{ (i,j) \ | i<j \}

A_i

A_j

max\{ |S|\}

"Family" of subsets

A

1000

\mathcal F =

A_1,A_2,\ldots,A_n

} a family of subsets of A such that (a) Each element in

\mathcal F

consists of 3 members (b) For every five elements in

\mathcal F

, the union of them all will have at least

12

members Find the largest value of

n

2

Weird configuration geometry

Given a concyclic quadrilateral

ABCD

with circumcenter

O

E

be the intersection of

AD

BC

F

be the intersection of

AC

BD

w

BD

AC

F

is the orthocenter of

\triangle QEP

PQ

is a diameter of

w

EO

passes through the center of

w

iff Iran Lemma

Given an acute triangle

ABC

. The incircle with center

I

BC,CA,AB

D,E,F

respectively. Let

M,N

be the midpoint of the minor arc of

AB

AC

respectively. Prove that

M,F,E,N

are collinear if and only if

\angle BAC =90

^{\circ}

1

Interesting NT function

Find all functions

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

such that for every prime number

p

and natural number

x

\{ x,f(x),\cdots f^{p-1}(x) \}

is a complete residue system modulo

p

f^{k+1}(x)=f(f^k(x))

for every natural number

k

f^1(x)=f(x)

.
Proposed by IndoMathXdZ

2

Absolute value, bash go brrr

Given real numbers

x,y,z

which satisfies

|x+y+z|+|xy+yz+zx|+|xyz| \le 1

max\{ |x|,|y|,|z|\} \le 1

Second degree polynomial

Find all second degree polynomials

P(x)

such that for all

a \in\mathbb{R} , a \geq 1

P(a^2+a) \geq a.P(a+1)