PEN K Problems

Part of PEN Problems

Subcontests

(33)

1

K 34

Show that there exists a bijective function

f: \mathbb{N}_{0}\to \mathbb{N}_{0}

such that for all

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

f(3mn+m+n)=4f(m)f(n)+f(m)+f(n).

1

K 33

Find all functions

f: \mathbb{Q}\to \mathbb{Q}

such that for all

x,y,z \in \mathbb{Q}

f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).

1

K 32

Find all functions

f: \mathbb{Z}^{2}\to \mathbb{R}^{+}

such that for all

i, j \in \mathbb{Z}

f(i,j)=\frac{f(i+1, j)+f(i,j+1)+f(i-1,j)+f(i,j-1)}{4}.

1

K 31

Find all strictly increasing functions

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

f(f(n))=3n.

1

K 30

Find all functions

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

such that for all

n\in \mathbb{N}

f(f(f(n)))+f(f(n))+f(n)=3n.

1

K 29

Find all functions

f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}

such that for all

x,y \in \mathbb{Z}\setminus\{0\}

f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}

1

K 28

Find all surjective functions

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

such that for all

n\in \mathbb{N}

f(n) \ge n+(-1)^{n}.

1

K 27

Find all functions

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

such that for all

m,n\in \mathbb{N}

f(f(m)+f(n))=m+n.

1

K 26

f: \mathbb{N}\to\mathbb{N}_{0}

satisfies for all

m,n\in\mathbb{N}

f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.

f(1982)

1

K 25

Consider all functions

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

f(t^2 f(s)) = s(f(t))^2

s

t

N

. Determine the least possible value of

f(1998)

1

K 24

f

is defined on the positive integers by

\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.

for all positive integers

n

. Determine the number of positive integers

n

, less than or equal to 1988, for which

f(n) = n

1

K 23

{\mathbb Q}^{+}

be the set of positive rational numbers. Construct a function

f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}

f(xf(y)) = \frac{f(x)}{y}

x, y \in{\mathbb Q}^{+}

1

K 22

Find all functions

f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}

such that for all

x\in \mathbb{Q}^+

f(x+1)=f(x)+1

f(x^2)=f(x)^2

1

K 20

Find all functions

f: \mathbb{Q}\to \mathbb{Q}

such that for all

x,y \in \mathbb{Q}

f(x+y)+f(x-y)=2(f(x)+f(y)).

1

K 19

Find all functions

f: \mathbb{Q}^{+}\to \mathbb{Q}^{+}

such that for all

x,y \in \mathbb{Q}

f \left( x+\frac{y}{x}\right) =f(x)+\frac{f(y)}{f(x)}+2y, \; x,y \in \mathbb{Q}^{+}.

1

K 18

Find all functions

f: \mathbb{Q}\to \mathbb{R}

such that for all

x,y\in \mathbb{Q}

f(xy)=f(x)f(y)-f(x+y)+1.

1

K 17

Find all functions

h: \mathbb{Z}\to \mathbb{Z}

such that for all

x,y\in \mathbb{Z}

h(x+y)+h(xy)=h(x)h(y)+1.

1

K 16

Find all functions

f: \mathbb{Z}\to \mathbb{Z}

such that for all

m,n\in \mathbb{Z}

f(m+f(n)) = f(m)+n.

1

K 15

Find all functions

f: \mathbb{Z}\to \mathbb{Z}

such that for all

m,n\in \mathbb{Z}

f(m+f(n))=f(m)-n.

1

K 14

Find all functions

f:\mathbb{Z} \to \mathbb{Z}

such that for all

m\in\mathbb{Z}

f(m+8) \le f(m)+8

f(m+11) \ge f(m)+11

1

K 13

Find all functions

f: \mathbb{Z}\to \mathbb{Z}

such that for all

m\in \mathbb{Z}

f(f(m))=m+1.

1

K 12

Find all functions

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

such that for all

m,n\in \mathbb{N}

f(2)=2

f(mn)=f(m)f(n)

f(n+1)>f(n)

1

K 11

Find all functions

f: \mathbb{N}_{0}\to \mathbb{N}_{0}

such that for all

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

mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).

1

K 10

Find all functions

f: \mathbb{N}_{0}\to \mathbb{N}_{0}

such that for all

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

f(m+f(n))=f(f(m))+f(n).

1

K 9

Find all functions

f: \mathbb{N}_{0}\rightarrow \mathbb{N}_{0}

such that for all

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

f(f(n))+f(n)=2n+6.

1

K 8

Find all functions

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

such that for all

n\in \mathbb{N}

f(f(f(n)))+6f(n)=3f(f(n))+4n+2001.

1

K 7

Find all functions

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

such that for all

n\in \mathbb{N}

f(f(n))+f(n)=2n+2001 \text{ or }2n+2002.

1

K 6

Find all functions

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

such that for all

n\in \mathbb{N}

f^{(19)}(n)+97f(n)=98n+232.

1

K 5

Find all functions

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

such that for all

n\in \mathbb{N}

f(f(m)+f(n))=m+n.

1

K 4

Find all functions

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

such that for all

n\in \mathbb{N}

f(f(f(n)))+f(f(n))+f(n)=3n.

1

K 3

Find all functions

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

such that for all

n\in \mathbb{N}

f(n+1) > f(f(n)).

1

K 2

Find all surjective functions

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

such that for all

m,n\in \mathbb{N}

m \vert n \Longleftrightarrow f(m) \vert f(n).

1

K 1

Prove that there is a function

f

from the set of all natural numbers into itself such that

f(f(n))=n^2

n \in \mathbb{N}