MathDB

1

Good functions from integers to reals

\mathbb{S}

be the set of all functions

f:\mathbb{Z}\rightarrow \mathbb{R}

. Now, consider the function

g:\mathbb{S} \rightarrow \mathbb{S} ,g(f(x)) = f(x + 1)-f(x)

. Now, we call a function

f \in \mathbb{S}

good if

g^n(f(x))=0

for some natural

n

. Prove that if

s \not = t \in S

are good functions then

s(m)-t(m)

is 0 for only finitely many

m \in \mathbb{Z}

.

A2 B4 C1

1

Another strange representation of a prime

If there are integers

a,b,c

such that

a^2+b^2+c^2-ab-bc-ca

is divisible by a prime

p

such that

\text{gcd}(p,\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1

, then prove that there are integers

x,y,z

such that

p=x^2+y^2+z^2-xy-yz-zx

.

A1

1

Bound on the complex roots of an irreducible monic polynomial.

Let

f

be an irreducible monic polynomial with integer coefficients such that

f(0)

is not equal to

1

. Let

z

be a complex number that is a root of

f

. Show that if

w

is another complex root of

f

, then

\frac{z}{w}

cannot be a positive integer greater than

1

.

A4 B3

1

Finally!A chess game in a mathematical Olympiad

Consider the starting position in a game of bughouse. Exhibit a sequence of moves on both boards, indicating the chronology, such that at the end: (a) The positions on both boards are the same as the original positions.
(b) It is White to play on one board, but Black to play on the other.
(c) All four players still have the right to castle subsequently (equivalently, the kings and rooks haven’t moved).
for each of the following cases : (a) without moving any pawns. (b) without moving any queen.

A3 B1

1

FEs on Naturals are really nice sometimes!

Find all functions

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

such that

f(m + f(n)f(m)) = nf(m) + m

holds for all

m,n \in \mathbb{N}

.

A5 B5

1

Line perpendicular to the Euler line(Again!) with a beautiful concurrency

Let

\triangle ABC

be an acute angled triangle. Let

G

be the centroid and let

D

be the foot of the altitude from

A

onto

BC

. Let ray

GD

intersect

(ABC)

at

X

and let

AG

intersect nine point circle at

Y

not on

BC

. Let

Z

be the intersection of the

\text{A-tangent}

to

(ABC)

and

\text{A-midline}

. Prove that perpendicular from

Z

to the Euler line,

AX

and

DY

concur.
The line joining the midpoints of

AB

and

AC

is called the

\text{A-midline}

.

(ABC)

denotes the circumcircle of

\triangle ABC

STEMS 2021-22 Math Cat A-B

Subcontests

Good functions from integers to reals

Another strange representation of a prime

Bound on the complex roots of an irreducible monic polynomial.

Finally!A chess game in a mathematical Olympiad

FEs on Naturals are really nice sometimes!

Line perpendicular to the Euler line(Again!) with a beautiful concurrency