India EGMO 2022 TST

Part of India EGMO TST

Subcontests

(6)

1

Bob writes polynomials

P(x)

is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial

P(x)

on a board. At every step, if the polynomial on the board is

f(x)

, he can replace it with
1.

f(x)+c

for a real number

c

, or 2. the polynomial

P(f(x))

.
Can he always find a finite sequence of steps so the final polynomial on the board has exactly

2020

real roots? What about

2021

?
~Sutanay Bhattacharya

1

Interesting tangency Geometry

I

I_A

denote the incentre and excentre opposite to

A

\triangle ABC

respectively. Let

A'

be the antipode of

A

\odot (ABC)

L

be the midpoint of arc

(BAC)

LB

LC

AI

Y

Z

respectively. Prove that

\odot (LYZ)

\odot (A'II_A)

.
~Mahavir Gandhi

1

Colouring digits to make a rational Number

N

be a positive integer. Suppose given any real

x\in (0,1)

with decimal representation

0.a_1a_2a_3a_4\cdots

, one can color the digits

a_1,a_2,\cdots

N

colors so that the following hold: 1. each color is used at least once; 2. for any color, if we delete all the digits in

x

except those of this color, the resulting decimal number is rational. Find the least possible value of

N

.
~Sutanay Bhattacharya

1

Magical conditional geometry

I

be incentre of scalene

\triangle ABC

L

be midpoint of arc

BAC

M

BC

and let the line through

M

AI

LI

P

Q

BC

PQ\perp LI

S

AM

T

LI

IS\perp BC

AQ\perp ST

.
~Mahavir Gandhi

1

Alice keeps writing numbers

a,b

be arbitrary co-prime natural numbers. Alice writes the natural number

t < b

on a blackboard. Every second she replaces the number on the blackboard, say

x

, with the smallest natural number in

\{x \pm a, x \pm b \}

that she has not yet ever written. She keeps doing this as long as possible. Prove that this process goes on indefinitely and that Alice will write down every natural number.
~Pranjal Srivastava and Rohan Goyal

1

Interesting inequality in n variables

n\ge 3

be an integer, and suppose

x_1,x_2,\cdots ,x_n

are positive real numbers such that

x_1+x_2+\cdots +x_n=1.

x_1^{1-x_2}+x_2^{1-x_3}\cdots+x_{n-1}^{1-x_n}+x_n^{1-x_1}<2.

~Sutanay Bhattacharya