PEN E Problems

Part of PEN Problems

Subcontests

(40)

1

E 39

c

be a nonzero real number. Suppose that

g(x)=c_0x^r+c_1x^{r-1}+\cdots+c_{r-1}x+c_r

is a polynomial with integer coefficients. Suppose that the roots of

g(x)

b_1,\cdots,b_r

k

be a given positive integer. Show that there is a prime

p

p>\max(k,|c|,|c_r|)

, and moreover if

t

is a real number between

0

1

j

1,\cdots,r

|(\text{ }c^r\text{ }b_j\text{}g(tb_j)\text{ })^pe^{(1-t)b}|<\dfrac{(p-1)!}{2r}.

Furthermore, if

f(x)=\dfrac{e^{rp-1}x^{p-1}(g(x))^p}{(p-1)!}

\left|\sum_{j=1}^r\int_0^1 e^{(1-t)b_j}f(tb_j)dt\right|\leq \dfrac{1}{2}.

1

E 38

c > \dfrac{8}{3}

, then there exists a real number

\theta

\lfloor\theta^{c^n}\rfloor

is prime for every positive integer

n

1

E 31

n

r

are nonnegative integers such that no number of the form

n^2+r-k(k+1) \text{ }(k\in\mathbb{N})

-1

or a positive composite number. Show that

4n^2+4r+1

1

9

1

E 26

Find the smallest prime which is not the difference (in some order) of a power of

2

3

1

E 25

\ln n \geq k\ln 2

n

is a natural number and

k

is the number of distinct primes that divide

n

1

E 41

n

\lim_{r \rightarrow\infty}\,\lim_{s \rightarrow\infty}\,\lim_{t \rightarrow \infty}\,\sum_{u=0}^{s}\left(1-\left(\cos\,\frac{(u!)^{r} \pi}{n} \right)^{2t} \right)=n

PS : I posted it because it's in the PDF file but not here ...

1

E 40

Prove that there do not exist eleven primes, all less than

20000

, which form an arithmetic progression.

1

E 37

It's known that there is always a prime between

n

2n-7

n \ge 10

. Prove that, with the exception of

1

4

6

, every natural number can be written as the sum of distinct primes.

1

E 36

Prove that there are infinitely many twin primes if and only if there are infinitely many integers that cannot be written in any of the following forms:

6uv+u+v, \;\; 6uv+u-v, \;\; 6uv-u+v, \;\; 6uv-u-v,

for some positive integers

u

v

1

E 35

There exists a block of

1000

consecutive positive integers containing no prime numbers, namely,

1001!+2

1001!+3

\cdots

1001!+1001

. Does there exist a block of

1000

consecutive positive integers containing exactly five prime numbers?

1

E 34

p_{n}

n

th prime number. For all

n \ge 6

\pi \left( \sqrt{p_{1}p_{2}\cdots p_{n}}\right) > 2n.

1

E 33

Prove that there are no positive integers

a

b

such that for all different primes

p

q

1000

ap+bq

1

E 32

n \ge 5

be an integer. Show that

n

is prime if and only if

n_{i} n_{j} \neq n_{p} n_{q}

for every partition of

n

4

n=n_{1}+n_{2}+n_{3}+n_{4}

, and for each permutation

(i, j, p, q)

(1, 2, 3, 4)

1

E 30

Given an odd integer

n>3

k

t

be the smallest positive integers such that both

kn+1

tn

are squares. Prove that

n

is prime if and only if both

k

t

are greater than

\frac{n}{4}

1

E 29

s_n

denote the sum of the first

n

primes. Prove that for each

n

there exists an integer whose square lies between

s_n

s_{n+1}

1

E 28

n^{\pi(2n)-\pi(n)}<4^{n}

for all positive integer

n

1

E 27

Prove that for each positive integer

n

n

consecutive positive integers none of which is an integral power of a prime number.

1

E 24

p_{n}

again denote the

n

th prime number. Show that the infinite series

\sum^{\infty}_{n=1}\frac{1}{p_{n}}

1

E 23

p_{1}=2, p_{2}={3}, p_{3}=5, \cdots, p_{n}

n

prime numbers, where

n \ge 3

\frac{1}{{p_{1}}^{2}}+\frac{1}{{p_{2}}^{2}}+\cdots+\frac{1}{{p_{n}}^{2}}+\frac{1}{p_{1}p_{2}\cdots p_{n}}< \frac{1}{2}.

1

E 22

p

be a prime number. Prove that there exists a prime number

q

such that for every integer

n

n^p -p

is not divisible by

q

1

E 21

p

is a prime, then

p^{p}-1

has a prime factor that is congruent to

1

p

1

E 20

Verify that, for each

r \ge 1

, there are infinitely many primes

p

p \equiv 1 \; \pmod{2^r}

1

E 19

p

be an odd prime. Without using Dirichlet's theorem, show that there are infinitely many primes of the form

2pk+1

1

E 18

Without using Dirichlet's theorem, show that there are infinitely many primes ending in the digit

9

1

E 17

a

b

n

be positive integers with

\gcd (a, b)=1

. Without using Dirichlet's theorem, show that there are infinitely many

k \in \mathbb{N}

\gcd(ak+b, n)=1

1

E 16

Prove that for any prime

p

in the interval

\left]n, \frac{4n}{3}\right]

p

\sum^{n}_{j=0}{{n}\choose{j}}^{4}.

1

E 15

Show that there exist two consecutive squares such that there are at least

1000

primes between them.

1

E 14

Prove that there do not exist polynomials

P

Q

such that \pi(x)\equal{}\frac{P(x)}{Q(x)} for all

x\in\mathbb{N}

1

E 13

Find all natural numbers

n

for which every natural number whose decimal representation has

n-1

1

7

1

E 12

Show that there are infinitely many primes.

1

E 11

In 1772 Euler discovered the curious fact that

n^2 +n+41

n

0,1,2, \cdots, 39

. Show that there exist

40

consecutive integer values of

n

for which this polynomial is not prime.

1

E 10

Represent the number

989 \cdot 1001 \cdot 1007 +320

as a product of primes.

1

E 9

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle in a given direction (that is, the numbers

a

b

c

d

are replaced by

a-b

b-c

c-d

d-a

). Is it possible after

1996

such steps to have numbers

a

b

c

d

such that the numbers

|bc-ad|

|ac-bd|

|ab-cd|

1

E 8

Show that for all integer

k>1

, there are infinitely many natural numbers

n

k \cdot 2^{2^n} + 1

1

E 7

Show that there exists a positive integer

k

such that k \cdot 2^{n} \plus{} 1 is composite for all

n \in \mathbb{N}_{0}

1

E 6

Find a factor of

2^{33}-2^{19}-2^{17}-1

that lies between

1000

5000

1

E 4

1280000401

1

E 3

Find the sum of all distinct positive divisors of the number

104060401

1

E 2

a, b, c, d

be integers with

a>b>c>d>0

ac+bd=(b+d+a-c)(b+d-a+c)

ab+cd

1

E 1

Prove that the number

512^{3} +675^{3}+ 720^{3}