STEMS 2021 Math Cat C

Part of 2021 India STEMS

Subcontests

(5)

1

primes in a sequence

p \in \mathbb{N} \setminus \{0, 1\}

be a fixed positive integer. Prove that for every

K > 0

, there exist infinitely many

n

N

such that there are atleast

\dfrac{KN}{\log(N)}

primes among the following

N

numbers given by

n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.

Proposed by Bimit Mandal

1

Infinitely many functions which win and lose

M>1

be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function

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

f(M) \ne M

f(k)<2k

k \in \mathbb{N}

f^{f(n)}(n)=n

n \in \mathbb{N}

\ell>0

f^{\ell}(n)=f\left(f^{\ell-1}(n)\right)

f^0(n)=n

[/*]
Tom wins otherwise. Prove that for infinitely many

M

, Tom wins, and for infinitely many

M

, Jerry wins.
Proposed by Anant Mudgal

1

Minimal number of polynomials to write as SOS

n

be a fixed positive integer. - Show that there exist real polynomials

p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]

(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)

- Find the least natural number

k

n

, such that the above polynomials

p_1, p_2, \cdots, p_k

1

Common points in discs

Find the largest constant

c

, such that if there are

N

discs in the plane such that every two of them intersect, then there must exist a point which lies in the common intersection of

cN + O(1)

1

Number field having infinitely many roots

Does there exist a nonzero algebraic number

\alpha

|\alpha| \neq 1

such that there exists infinitely many positive integers

n

for which there's

\beta_n \in \mathbb{C}

\beta_n \in \mathbb{Q}(\alpha)

\beta_n^n = \alpha