MathDB

1

Elements which appear in an odd number of sets chosen

A_1

,

A_2

,

\ldots

,

A_n

are given and we denote by

d(n)

the number of elements which appear exactly in an odd number of sets chosen from

A_1

,

A_2

,

\ldots

,

A_n

. Prove that for any

k

,

1\leq k\leq n

the number

{ d(n) - \sum\limits^n_{i=1} |A_i| + 2\sum\limits_{ i<j} |A_i \cap A_j | - \cdots + (-1)^k2^{k-1} \sum\limits_{i_1 <i_2 <\cdots < i_k} | A_{i_1} \cap A_{i_2} \cap \cdots \cap A_{i_k}}|

is divisible by

2^k

. Ioan Tomescu, Dragos Popescu

15

1

Interval without integer numbers

Let

[a,b]

be a given interval of real numbers not containing integers. Prove that there exists

N>0

such that

[Na,Nb]

does not contain integer numbers and the length of the interval

[Na,Nb]

exceedes

\dfrac 16

.

13

1

Sequence and solutions of a quadratic equation

Let

a

be a positive integer. The sequence

\{x_n\}_{n\geq 1}

is defined by

x_1=1

,

x_2=a

and

x_{n+2} = ax_{n+1} + x_n

for all

n\geq 1

. Prove that

(y,x)

is a solution of the equation

|y^2 - axy - x^2 | = 1

if and only if there exists a rank

k

such that

(y,x)=(x_{k+1},x_k)

. Serban Buzeteanu

11

1

x+y+z=0 then |cos x| + |cos y| + |cos z| > 1

Let

x,y,z

be real numbers with

x+y+z=0

. Prove that

|\cos x |+ |\cos y| +| \cos z | \geq 1 .

Viorel Vajaitu, Bogdan Enescu

10

1

Combination of integer polynomials

Let

p > 2

be a prime number. Find the least positive number

a

which can be represented as a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X), where

f(X)

and

g(X)

are integer polynomials. Mircea Becheanu.

9

1

Prove that \prod^n_{k=1} k^{2k-n-1} is an integer

Prove that for all positive integers

n\geq 1

the number

\prod^n_{k=1} k^{2k-n-1}

is also an integer number. Laurentiu Panaitopol.

7

1

Two lines and two circles with centers on the lines

In the plane there are given the lines

\ell_1

,

\ell_2

, the circle

\mathcal{C}

with its center on the line

\ell_1

and a second circle

\mathcal{C}_1

which is tangent to

\ell_1

,

\ell_2

and

\mathcal{C}

. Find the locus of the tangent point between

\mathcal{C}

and

\mathcal{C}_1

while the center of

\mathcal{C}

is variable on

\ell_1

. Mircea Becheanu

8

1

Compute the sum of all ordered sequences

The positive integer

n

is given and for all positive integers

k

,

1\leq k\leq n

, denote by

a_{kn}

the number of all ordered sequences

(i_1,i_2,\ldots,i_k)

of positive integers which verify the following two conditions: a)

1\leq i_1<i_2< \cdots i_k \leq n

; b)

i_{r+1}-i_r \equiv 1 \pmod 2

, for all

r \in\{1,2,\ldots,k-1\}

. Compute the number

a(n) = \sum\limits_{k=1}^n a_{kn}

. Ioan Tomescu

5

1

The cells of a 11x11 chess board are colored in 3 colors

The cells of a

11\times 11

chess-board are colored in 3 colors. Prove that there exists on the board a

m\times n

rectangle such that the four cells interior to the rectangle and containing the four vertices of the rectangle have the same color. Ioan Tomescu

4

1

Another inequality with distinct positive integers

Prove that for all positive integers

0<a_1<a_2<\cdots <a_n

the following inequality holds:

(a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 .

Viorel Vajaitu

3

1

Count the number of regular and star polygons in a circle

Consider all regular convex and star polygons inscribed in a given circle and having

n

sides. We call two such polygons to be equivalent if it is possible to obtain one from the other using a rotation about the center of the circle. How many classes of such polygons exist? Mircea Becheanu

2

1

Another solid inequality with a trihedral angle

Let

OABC

be a trihedral angle such that \angle BOC = \alpha, \angle COA = \beta, \angle AOB = \gamma , \alpha + \beta + \gamma = \pi . For any interior point

P

of the trihedral angle let

P_1

,

P_2

and

P_3

be the projections of

P

on the three faces. Prove that

OP \geq PP_1+PP_2+PP_3

. Constantin Cocea

14

1

Triangle function - find the set of values

Let

\Delta

denote the set of all triangles in a plane. Consider the function

f: \Delta\to(0,\infty)

defined by

f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)

, for any triangle

ABC

with

BC=a\leq CA=b\leq AB = c

. Find the set of values of

f

.

6

1

Sistem of real numbers with sum of reciprocals

Find all vectors of

n

real numbers

(x_1,x_2,\ldots,x_n)

such that

\left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } \\ x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \\ \ & \cdots & \ \\ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right.

Mircea Becheanu

1

A sphere, a plane and tangent circles with a cone

Consider a sphere and a plane

\pi

. For a variable point

M \in \pi

, exterior to the sphere, one considers the circular cone with vertex in

M

and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. Octavian Stanasila

12

1

convex quad

The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral. Laurentiu Panaitopol

1988 Romania Team Selection Test

Subcontests

Elements which appear in an odd number of sets chosen

Interval without integer numbers

Sequence and solutions of a quadratic equation

x+y+z=0 then |cos x| + |cos y| + |cos z| > 1

Combination of integer polynomials

Prove that \prod^n_{k=1} k^{2k-n-1} is an integer

Two lines and two circles with centers on the lines

Compute the sum of all ordered sequences

The cells of a 11x11 chess board are colored in 3 colors

Another inequality with distinct positive integers

Count the number of regular and star polygons in a circle

Another solid inequality with a trihedral angle

Triangle function - find the set of values

Sistem of real numbers with sum of reciprocals

A sphere, a plane and tangent circles with a cone

convex quad

1988 Romania Team Selection Test

Subcontests

Elements which appear in an odd number of sets chosen

Interval without integer numbers

Sequence and solutions of a quadratic equation

x+y+z=0 then |cos x| + |cos y| + |cos z| &gt; 1

Combination of integer polynomials

Prove that \prod^n_{k=1} k^{2k-n-1} is an integer

Two lines and two circles with centers on the lines

Compute the sum of all ordered sequences

The cells of a 11x11 chess board are colored in 3 colors

Another inequality with distinct positive integers

Count the number of regular and star polygons in a circle

Another solid inequality with a trihedral angle

Triangle function - find the set of values

Sistem of real numbers with sum of reciprocals

A sphere, a plane and tangent circles with a cone

convex quad

x+y+z=0 then |cos x| + |cos y| + |cos z| > 1