MathDB

1

Find possible prime pairs

(p, q)

, is

p^2 + 2pq^2 + 1

also a prime?

7

1

Coin Toss Computational Probablity

Suppose a biased coin gives head with probability

\dfrac{2}{3}

. The coin is tossed repeatedly, if it shows heads then player

A

rolls a fair die, otherwise player

B

rolls the same die. The process ends when one of the players get a

6

, and that player is declared the winner.
If the probability that

A

will win is given by

\dfrac{m}{n}

where

m,n

are coprime, then what is the value of

m^2n

?

6

2

Computational Graph Theory Tree

There are

5

vertices labelled

1,2,3,4,5

. For any two pairs of vertices

u, v

, the edge

uv

is drawn with probability

1/2

. If the probability that the resulting graph is a tree is given by

\dfrac{p}{q}

where

p, q

are coprime, then find the value of

q^{1/10} + p

.

Old ISL resurrected

Define a positive integer

n

to be a fake square if either

n = 1

or

n

can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer

k \geqslant 2

, there exist distinct positive integers

a_1

,

a_2, \cdots, a_k

such that the polynomial

(x+a_1)(x+a_2) \cdots (x+a_k)

takes ‘fake square’ values for all

x = 1,2,\cdots,2023

.
Proposed by Prof. Aditya Karnataki

5

2

Computational polynomial

Consider a polynomial

P(x) \in \mathbb{R}[x]

, with degree

2023

, such that

P(\sin^2(x))+P(\cos^2(x)) =1

for all

x \in \mathbb{R}

. If the sum of all roots of

P

is equal to

\dfrac{p}{q}

with

p, q

coprime, then what is the product

pq

?

Radax on STEMS go brrr...

A convex quadrilateral

ABCD

is such that

\angle B = \angle D

and are both acute angles.

E

is on

AB

such that

CB = CE

and

F

is on

AD

such that

CF = CD

. If the circumcenter of

CEF

is

O_1

and the circumcenter of

ABD

is

O_2

. Prove that

C,O_1,O_2

are collinear.
Proposed by Kapil Pause

4

2

Sum of possible function values

Let

f : \mathbb{N} \to \mathbb{N}

be a function such that the following conditions hold:

\qquad\ (1) \; f(1) = 1.

\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.

\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.

\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.

Find the sum of all possible values of

f(2023)

.

Alice and Bob playing with polynomials

Alice has

n > 1

one variable quadratic polynomials written on paper she keeps secret from Bob. On each move, Bob announces a real number and Alice tells him the value of one of her polynomials at this number. Prove that there exists a constant

C > 0

such that after

Cn^5

questions, Bob can determine one of Alice’s polynomials.
Proposed by Rohan Goyal and Anant Mudgal

3

2

Computational Geometric Ratio

Given a triangle

ABC

with angles

\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}

, let

H

be its orthocentre, and

O

be its circumcenter. Let

F

be the midpoint of side

AB

, and

Q

be the foot of the perpendicular from

B

onto

AC

. Denote by

X

the intersection point of the lines

FH

and

QO

. Suppose the ratio of the length of

FX

and the circumradius of the triangle is given by

\dfrac{a + b \sqrt{c}}{d}

, then find the value of

1000a + 100b + 10c + d

.

Infinite primes dividing polynomial sequence

Suppose

f

is a nonconstant polynomial with integer coefficients with the following property:
[*]

f(0)

and

f(1)

are both odd. [*]Define a sequence of integers with

a_k = f(1)f(2) \cdots f(k)+1

Prove that there are infinitely many prime numbers dividing at least one element of the sequence.
Proposed by Sayandeep Shee

2

Computational Set Construction

Consider the set

S

of permutations of

1, 2, \dots, 2022

such that for all numbers

k

in the permutation, the number of numbers less than

k

that follow

k

is even.
For example, for

n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}

If

|S| = (a!)^b

where

a, b \in \mathbb{N}

, then find the product

ab

.

STEMS 2023 Mathematics Category A Q2

Given a complete bipartite graph on

n,n

vertices (call this

K_{n,n}

), we colour all its edges with

2

colours , red and blue . What is the least value of

n

such that for any colouring of the edges of the graph , there will exist at least one monochromatic

4

cycle ?

1

2

Computational board game

The following

100

numbers are written on the board:

2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.

Alice chooses two numbers

a,b,

erases them and writes the number

\dfrac{ab - 1}{a+b+2}

on the board. She keeps doing this until a single number remains on the board.
If the sum of all possible numbers she can end up with is

\dfrac{p}{q}

where

p, q

are coprime, then what is the value of

\log_{2}(p+q)

?

STEMS 2023 Mathematics Category B Q1

If in triangle

ABC

,

AC

=

15

,

BC

=

13

and

IG||AB

where

I

is the incentre and

G

is the centroid , what is the area of triangle

ABC

?

STEMS 2023 Math Cat A

Subcontests

Find possible prime pairs

Coin Toss Computational Probablity

Computational Graph Theory Tree

Old ISL resurrected

Computational polynomial

Radax on STEMS go brrr...

Sum of possible function values

Alice and Bob playing with polynomials

Computational Geometric Ratio

Infinite primes dividing polynomial sequence

Computational Set Construction

STEMS 2023 Mathematics Category A Q2

Computational board game

STEMS 2023 Mathematics Category B Q1