MathDB

1

a_{n+1} = [\sqrt[3]{a_n +n}]^3

(a_n)

is defined by

a_0 = 0

and

a_{n+1} = [\sqrt[3]{a_n +n}]^3

for

n \ge 0

. (a) Find

a_n

in terms of

n

. (b) Find all

n

for which

a_n = n

.

8

1

x + Q(y + P(x))= y + Q(x + P(y)) when Q(0)= 0

Determine all real polynomials

P(z)

for which there exists a unique real polynomial

Q(x)

satisfying the conditions

Q(0)= 0

,

x + Q(y + P(x))= y + Q(x + P(y))

for all

x,y \in R

.

3

1

sequence with powers of primes, a_{j+1} = f (a_j)

Define

f (n) = n + 1

if

n = p^k > 1

is a power of a prime number, and

f (n) =p_1^{k_1}+... + p_r^{k_r}

for natural numbers

n = p_1^{k_1}... p_r^{k_r}

(

r > 1, k_i > 0

). Given

m > 1

, we construct the sequence

a_0 = m, a_{j+1} = f (a_j)

for

j \ge 0

and denote by

g(m)

the smallest term in this sequence. For each

m > 1

, determine

g(m)

.

5

1

x^3 + y = 3x + 4 ,2y^3 + z = 6y + 6, 3z^3 + x = 9z + 8.

Solve in real numbers the system

\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}

1

2^x - 3^y = 7

Solve in positive integers

x,y

the equation

2^x - 3^y = 7

.

6

1

(\sqrt{a}+\sqrt{b})/2)^2 <=(a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b)/4 <=

If

a,b \ge 0

are real numbers, prove the inequality

\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}

For each of the inequalities, find the cases of equality.

2

1

max volume of tetrahedra with sum of 3 faces < = 1

Consider all tetrahedra

ABCD

in which the sum of the areas of the faces

ABD, ACD, BCD

does not exceed

1

. Among such tetrahedra, find those with the maximum volume.

9

1

sum of inradii in terms of side of an equilateral

Point

P

is taken on the extension of side

AB

of an equilateral triangle

ABC

so that

A

is between

B

and

P

. Denote by

a

the side length of triangle

ABC

, by

r_1

the inradius of triangle

PAC

, and by

r_2

the exradius of triangle

PBC

opposite

P

. Find the sum

r_1+r_2

as a function in

a

.

4

1

Fibonacci numbers

The Fibonacci numbers are defined by F_0 \equal{} 1, F_1 \equal{} 1, F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n. The positive integers

A, B

are such that

A^{19}

divides

B^{93}

and

B^{19}

divides

A^{93}

. Show that if

h < k

are consecutive Fibonacci numbers then

(AB)^h

divides (A^4 \plus{} B^8)^k

1993 Austrian-Polish Competition

Subcontests

a_{n+1} = [\sqrt[3]{a_n +n}]^3

x + Q(y + P(x))= y + Q(x + P(y)) when Q(0)= 0

sequence with powers of primes, a_{j+1} = f (a_j)

x^3 + y = 3x + 4 ,2y^3 + z = 6y + 6, 3z^3 + x = 9z + 8.

2^x - 3^y = 7

(\sqrt{a}+\sqrt{b})/2)^2 <=(a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b)/4 <=

max volume of tetrahedra with sum of 3 faces < = 1

sum of inradii in terms of side of an equilateral

Fibonacci numbers

1993 Austrian-Polish Competition

Subcontests

a_{n+1} = [\sqrt[3]{a_n +n}]^3

x + Q(y + P(x))= y + Q(x + P(y)) when Q(0)= 0

sequence with powers of primes, a_{j+1} = f (a_j)

x^3 + y = 3x + 4 ,2y^3 + z = 6y + 6, 3z^3 + x = 9z + 8.

2^x - 3^y = 7

(\sqrt{a}+\sqrt{b})/2)^2 &lt;=(a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b)/4 &lt;=

max volume of tetrahedra with sum of 3 faces &lt; = 1

sum of inradii in terms of side of an equilateral

Fibonacci numbers

(\sqrt{a}+\sqrt{b})/2)^2 <=(a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b)/4 <=

max volume of tetrahedra with sum of 3 faces < = 1