MathDB

Antitours in Mahishmati

Source: India TST 2023 Day 1 P1

7/9/2023

In the fictional country of Mahishmati, there are

50

cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city

A

, an ordered list of cities

C_1,\ldots, C_{50}

is called an antitour from

A

if
[*] every city (including

A

) appears in the list exactly once, and [*] for each

k\in \{1,2,\ldots, 50\}

, it is impossible to go from

A

to

C_k

by a sequence of exactly

k

(not necessarily distinct) flights.
Baahubali notices that there is an antitour from

A

for any city

A

. Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city.
Proposed by Sutanay Bhattacharya

combinatoricsgraph theory

Wait wasn&#039;t it the reciprocal in the paper?

Source: India TST 2023 Day 2 P1

7/9/2023

Let

\mathbb{Z}_{\ge 0}

be the set of non-negative integers and

\mathbb{R}^+

be the set of positive real numbers. Let

f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+

be a function such that

f(0, k) = 2^k

and

f(k, 0) = 1

for all integers

k \ge 0

, and

f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}

for all integers

m, n \ge 1

. Prove that

f(99, 99)<1.99

.
Proposed by Navilarekallu Tejaswi

algebrainequalities

Decimal functions in binary

Source: India TST 2023 Day 3 P1

7/9/2023

Let

\mathbb{N}

be the set of all positive integers. Find all functions

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

such that

f(x) + y

and

f(y) + x

have the same number of

1

's in their binary representations, for any

x,y \in \mathbb{N}

.

number theory

Slightly weird points which are not so weird

Source: India TST 2023 Day 4 P1

7/9/2023

Suppose an acute scalene triangle

ABC

has incentre

I

and incircle touching

BC

at

D

. Let

Z

be the antipode of

A

in the circumcircle of

ABC

. Point

L

is chosen on the internal angle bisector of

\angle BZC

such that

AL = LI

. Let

M

be the midpoint of arc

BZC

, and let

V

be the midpoint of

ID

. Prove that

\angle IML = \angle DVM

geometryTST

Odd points give concurrency of perpendicular bisectors

Source: India TST 2023 Practice Test 1 P1

7/9/2023

Let

ABC

be a triangle, and let

D

be the foot of the

A-

altitude. Points

P, Q

are chosen on

BC

such that

DP = DQ = DA

. Suppose

AP

and

AQ

intersect the circumcircle of

ABC

again at

X

and

Y

. Prove that the perpendicular bisectors of the lines

PX

,

QY

, and

BC

are concurrent.
Proposed by Pranjal Srivastava

geometryperpendicular bisector

Keep on dividing till you can&#039;t no more

Source: India TST 2023 Practice Test 2 P1

7/9/2023

The numbers

1,2,3,4,\ldots , 39

are written on a blackboard. In one step we are allowed to choose two numbers

a

and

b

on the blackboard such that

a

divides

b

, and replace

a

and

b

by the single number

\tfrac{b}{a}

. This process is continued till no number on the board divides any other number. Let

S

be the set of numbers which is left on the board at the end. What is the smallest possible value of

|S|

?
Proposed by B.J. Venkatachala

number theory

1

Problems(6)