MathDB

1

min of x_1 +x_2^2/ 2 + x_3^3/ 3 +...+ x_n^n /n if sum 1/x_i=n

n

be a nonzero natural number, and

x_1, x_2,..., x_n

positive real numbers that

\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n

. Find the minimum value of the expression

x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}

.

8

1

h (x) = \lambda f_m (x) + (1- \lambda) g_m (x) if g_m (x) <=h (x)<= f_m (x)

The families of second degree functions

f_m, g_m: R\to R,

are considered ,

f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1

,

g_m (x) = m^2x^2 + mx - 1

, where

m

is a real nonzero parameter. Show that, for any function

h

of the second degree with the property that

g_m (x) \le h (x) \le f_m (x)

for any real

x

, there exists

\lambda \in [0, 1]

which verifies the condition

h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)

, whatever real

x

is.

2

1

Number Theory

Prove that: a) there are infinitely many natural numbers of the form 3p + 1, p is positive integer , which can be represented as the difference of 2 cubes of positive integers; b) there are infinitely many natural numbers of the form 5q + 1, q is positive integer , which can be represented as the difference of two cubes of positive integers.

3

1

Inequality

Let

a_1,a_2,...a_n

be positive numbers. And let

s=a_1+a_2+...+a_n

,and

p=a_1*a_2*...*a_n

.Prove that

2^{n}*\sqrt{p} \leq 1+\frac{s}{1!}+\frac{s^2}{2!}+...+\frac{s^n}{n!}

7

1

Prime exponennt

Let

p

be a prime number and

a

and

n

positive nonzero integers. Prove that if

2^p + 3^p = a^n

then

n=1

4

1

Find all solution of function

Let the

A

be the set of all nonenagative integers. It is given function such that

f:\mathbb{A}\rightarrow\mathbb{A}

with

f(1) = 1

and for every element

n

od set

A

following holds: 1)

3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))

; 2)

f(2n) < 6f(n)

, Find all solutions of

f(k)+f(l) = 293

,

k<l

.

5

1

angle MBC is equal to angle FC iff ABC is equilateral

Let

ABC

be an acute-angled triangle, and let

F

be the foot of its altitude from the vertex

C

. Let

M

be the midpoint of the segment

CA

. Assume that

CF=BM

. Then the angle

MBC

is equal to angle

FCA

if and only if the triangle

ABC

is equilateral.

1

Moldova TST 2005 for JBMO, day 1, problem 1

Let the triangle

ABC

with

BC

the smallest side. Let

P

on (

AB

) such that angle

PCB

equals angle

BAC

. and

Q

on side (

AC

) such that angle

QBC

equals angle

BAC

. Show that the line passing through the circumenters of triangles

ABC

and

APQ

is perpendicular on

BC

.

2005 Junior Balkan Team Selection Tests - Moldova

Subcontests

min of x_1 +x_2^2/ 2 + x_3^3/ 3 +...+ x_n^n /n if sum 1/x_i=n

h (x) = \lambda f_m (x) + (1- \lambda) g_m (x) if g_m (x) <=h (x)<= f_m (x)

Number Theory

Inequality

Prime exponennt

Find all solution of function

angle MBC is equal to angle FC iff ABC is equilateral

Moldova TST 2005 for JBMO, day 1, problem 1

2005 Junior Balkan Team Selection Tests - Moldova

Subcontests

min of x_1 +x_2^2/ 2 + x_3^3/ 3 +...+ x_n^n /n if sum 1/x_i=n

h (x) = \lambda f_m (x) + (1- \lambda) g_m (x) if g_m (x) &lt;=h (x)&lt;= f_m (x)

Number Theory

Inequality

Prime exponennt

Find all solution of function

angle MBC is equal to angle FC iff ABC is equilateral

Moldova TST 2005 for JBMO, day 1, problem 1

h (x) = \lambda f_m (x) + (1- \lambda) g_m (x) if g_m (x) <=h (x)<= f_m (x)