MathDB

\angle QXY = \angle ACP, \angle QYX = \angle ABP

Source: Iran MO Third Round 2021 G3

9/25/2021

Given triangle

ABC

variable points

X

and

Y

are chosen on segments

AB

and

AC

, respectively. Point

Z

on line

BC

is chosen such that

ZX=ZY

. The circumcircle of

XYZ

cuts the line

BC

for the second time at

T

. Point

P

is given on line

XY

such that

\angle PTZ = 90^ \circ

. Point

Q

is on the same side of line

XY

with

A

furthermore

\angle QXY = \angle ACP

and

\angle QYX = \angle ABP

. Prove that the circumcircle of triangle

QXY

passes through a fixed point (as

X

and

Y

vary).

geometrycircumcircle

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

Source: Iran MO Third Round A3

9/25/2021

Polynomial

P

with non-negative real coefficients and function

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

are given such that for all

x, y\in \mathbb{R}^+

we have

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

(a) Prove that

P

has degree at most 1. (b) Find all function

f

and non-constant polynomials

P

satisfying the equality.

algebrapolynomialfunctionfuctional equation

x_{n+1} = x_n^{s(n)} + 1, s(n)

Source: Iran MO Third Round 2021 N3

9/25/2021

x_1

is a natural constant. Prove that there does not exist any natural number

m> 2500

such that the recursive sequence

\{x_i\} _{i=1} ^ \infty

defined by

x_{n+1} = x_n^{s(n)} + 1

becomes eventually periodic modulo

m

. (That is there does not exist natural numbers

N

and

T

such that for each

n\geq N

,

m\mid x_n - x_{n+T}

). (

s(n)

is the sum of digits of

n

.)

number theory

f(P \cdot Q) = f(P) + f(Q) f(P(Q(x))) f(P \circ Q)

Source: Iran MO Third Round 2021 F3

9/25/2021

Find all functions

f: \mathbb{Q}[x] \to \mathbb{R}

such that: (a) for all

P, Q \in \mathbb{Q}[x]

,

f(P \circ Q) = f(Q \circ P);

(b) for all

P, Q \in \mathbb{Q}[x]

with

PQ \neq 0

,

f(P\cdot Q) = f(P) + f(Q).

(

P \circ Q

indicates

P(Q(x))

.)

functionalgebra

3

Problems(4)