India EGMO 2023 TST

Part of India EGMO TST

Subcontests

(6)

1

What are those points?

ABC

be an isosceles triangle with

AB = AC

P,Q,R

are points on segments

AC, AB, BC

respectively such that

AP = QB

\angle PBC = 90^\circ - \angle BAC

RP = RQ

O_1, O_2

be the circumcenters of

\triangle APQ

\triangle CRP

BR = O_1O_2

.
Proposed by Atul Shatavart Nadig

1

K-pop sequences

k

be a positive integer. A sequence of integers

a_1, a_2, \cdots

k

-pop if the following holds: for every

n \in \mathbb{N}

a_n

is equal to the number of distinct elements in the set

\{a_1, \cdots , a_{n+k} \}

. Determine, as a function of

k

k

-pop sequences there are.
Proposed by Sutanay Bhattacharya

1

Identical or Periodic?

f, g

\mathbb{R} \rightarrow \mathbb{R}

such that for all reals

x,y

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

Prove that either

f

is the identity function or

g

is periodic.
Proposed by Pranjal Srivastava

1

Drones by the Sibyl System

N \geqslant 3

be an integer. In the country of Sibyl, there are

N^2

towns arranged as the vertices of an

N \times N

grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.
Proposed by Sutanay Bhattacharya and Anant Mudgal

1

Alice takes beet, no, cube roots!

Alice has an integer

N > 1

on the blackboard. Each minute, she deletes the current number

x

on the blackboard and writes

2x+1

x

is not the cube of an integer, or the cube root of

x

otherwise. Prove that at some point of time, she writes a number larger than

10^{100}

.
Proposed by Anant Mudgal and Rohan Goyal

1

Colorful Discs

r > 0

be a real number. All the interior points of the disc

D(r)

r

are colored with one of two colors, red or blue.
[*]If

r > \frac{\pi}{\sqrt{3}}

, show that we can find two points

A

B

in the interior of the disc such that

AB = \pi

A,B

have the same color [*]Does the conclusion in (a) hold if

r > \frac{\pi}{2}

? Proposed by S Muralidharan