MathDB

1

Chess tournament

In a chess tournament , each of two players have only one game played. After 2 rounds 5 players left the tournament. At the final of tournament was found that the number of total games played is 100. How many players were at the start of the tournament?

8.4

1

equation 4

Solve in equation:

x^2+y^2+z^2+w^2=3(x+y+z+w)

where

x,y,z,w

are positive integers.

9.4

1

rational terms

Find all rational terms of sequence defined by formula

a_n=\sqrt{\frac{9n-2}{n+1}}, n \in N

10.3

1

Nice

Determine strictly positive real numbers

a_{1},a_{2},...,a_{n}

if for any

n\in N^*

takes place equality: a_{1}^2\plus{}a_{2}^2\plus{}...\plus{}a_{n}^2\equal{}a_{1}\plus{}a_{2}\plus{}...\plus{}a_{n}\plus{}\frac{n(n^2\plus{}6n\plus{}11)}{3}

11.8

1

Moldova NMO, 11th grade, Day 2, Problem 8

The continuous function and twice differentiable function

f: \mathbb{R}\rightarrow\mathbb{R}

satisfies

2007^{2}\cdot f(x)+f''(x)=0

. Prove that there exist two such real numbers

k

and

l

such that

f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)

.

11.7

1

Moldova NMO, 11th grade, Day 2, Problem 7

Given a tetrahedron

VABC

with edges

VA

,

VB

and

VC

perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is

3p

. Find the maximal volume of

VABC

.

11.6

1

Moldova NMO, 11th grade, Day 2, Problem 6

Define

(b_{n})

to be:

b_{0}=12

,

b_{1}=\frac{\sqrt{3}}{2}

adn

b_{n+1}+b_{n-1}=b_{n}\cdot\sqrt{3}

. Find

b_{0}+b_{1}+\dots+b_{2007}

. Note. Maybe this seems too easy, but I want to post all the problems...

11.5

1

Moldova NMO, 11th grade, Day 2, Problem 5

Real numbers

a_{1},a_{2},\dots,a_{n}

satisfy

a_{i}\geq\frac{1}{i}

, for all

i=\overline{1,n}

. Prove the inequality:

\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).

12.8

1

Integral identity-inequality

Find all continuous functions

f\colon [0;1] \to R

such that

\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}

12.7

1

Weak Stirling limit

Find the limit

\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}

12.6

1

Distance between two curves

Show that the distance between a point on the hyperbola

xy=5

and a point on the ellipse

x^{2}+6y^{2}=6

is at least

\frac{9}{7}

.

12.5

1

Polynomial equation

Find all polynomials

P\in \mathbb C[X]

such that

P(X^{2})=P(X)^{2}+2P(X)

11.4

1

Moldova NMO, 2007, XI Grade, Problem 4

The function

f: \mathbb{R}\rightarrow\mathbb{R}

satisfies

f(\textrm{cot}x)=\sin2x+\cos2x

, for any

x\in(0,\pi)

. Find the minimum and maximum value of

g: [-1;1]\rightarrow\mathbb{R}

,

g(x)=f(x)\cdot f(1-x)

.

11.3

1

Moldova NMO, 2007, XI Grade, Problem 3

ABCDA_{1}B_{1}C_{1}D_{1}

is a cube with side length

4a

. Points

E

and

F

are taken on

(AA_{1})

and

(BB_{1})

such that

AE=B_{1}F=a

.

G

and

H

are midpoints of

(A_{1}B_{1})

and

(C_{1}D_{1})

, respectively. Find the minimum value of the

CP+PQ

, where

P\in[GH]

and

Q\in[EF]

.

11.1

1

Moldova NMO, 2007, XI Grade, Problem 1

Define the sequence

(x_{n})

:

x_{1}=\frac{1}{3}

and

x_{n+1}=x_{n}^{2}+x_{n}

. Find

\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]

, wehere

[

]

denotes the integer part.

11.2

1

Find Limit

Define

a_{n}

as satisfying:

\left(1+\frac{1}{n}\right)^{n+a_{n}}=e

. Find

\lim_{n\rightarrow\infty}a_{n}

.

12.4

1

Basically integral inequality

If the function

f\colon [1,2]\to R

is such that

\int_{1}^{2}f(x) dx=\frac{73}{24}

, then show that there exists a

x_{0}\in (1;2)

such that

x_{0}^{2}<f(x_{0})<x_{0}^{3}

[Edit:

f

is continuous]

12.3

1

Inequality with expenential functions and logarithms

For

a,b \in [1;\infty)

show that

ab\leq e^{a-1}+b\ln b

12.2

1

Area of region bounded by two curves

For

p\in (0;\infty)

find the area of the region bounded by the curves

y^{2}=4px

and

16py^{2}=5(x-p)^{3}

12.1

1

Polynomial divisibility

For

a\in C^{*}

find all

n\in N

such that

X^{2}(X^{2}-aX+a^{2})^{2}

divides

(X^{2}+a^{2})^{n}-X^{2n}-a^{2n}

2007 Moldova National Olympiad

Subcontests

Chess tournament

equation 4

rational terms

Nice

Moldova NMO, 11th grade, Day 2, Problem 8

Moldova NMO, 11th grade, Day 2, Problem 7

Moldova NMO, 11th grade, Day 2, Problem 6

Moldova NMO, 11th grade, Day 2, Problem 5

Integral identity-inequality

Weak Stirling limit

Distance between two curves

Polynomial equation

Moldova NMO, 2007, XI Grade, Problem 4

Moldova NMO, 2007, XI Grade, Problem 3

Moldova NMO, 2007, XI Grade, Problem 1

Find Limit

Basically integral inequality

Inequality with expenential functions and logarithms

Area of region bounded by two curves

Polynomial divisibility