MathDB

1

computational of angles on media in triangle

P

be a point on the median

CM

of a triangle

ABC

with

AC\ne BC

and the acute angle

\gamma

at

C

, such that the bisectors of

\angle PAC

and

\angle PBC

intersect at a point

Q

on the median

CM

. Determine

\angle APB

and

\angle AQB

.

Problem 4

1

two monochromatic triangles, given 7 points

Seven points are given in space, no four of which are on a plane. Each of the segments with the endpoints in these points is painted black or red. Prove that there are two monochromatic triangles (not necessarily both of the same color) with no common edge. Does the statement hold for six points?

Problem 3

1

insphere of pyramid, planar cross-sections with equal area

A pyramid

MABCD

with the top-vertex

M

is circumscribed about a sphere with center

O

so that

O

lies on the altitude of the pyramid. Each of the planes

ACM,BDM,ABO

divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes

ACM

and

ABO

inside the pyramid are in ratio

(\sqrt2+2):4

. Determine the angle

\delta

between the planes

ACM

and

ABO

, and the dihedral angle of the pyramid at the edge

AB

.

Problem 2

1

sextic has only real roots

Find all real parameters

a

for which all the roots of the equation

x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1

are real.

Problem 1

1

prime factors of f(a),f(a+1),...,f(a+n-1), ≥k in count

Let

f(x)

be a non-constant polynomial with integer coefficients and

n,k

be natural numbers. Show that there exist

n

consecutive natural numbers

a,a+1,\ldots,a+n-1

such that the numbers

f(a),f(a+1),\ldots,f(a+n-1)

all have at least

k

prime factors. (We say that the number

p_1^{\alpha_1}\cdots p_s^{\alpha_s}

has

\alpha_1+\ldots+\alpha_s

prime factors.)

Problem 6

1

greatest odd divisor, limit of sequence

Let

\alpha_a

denote the greatest odd divisor of a natural number

a

, and let

S_b=\sum_{a=1}^b\frac{\alpha_a}a

Prove that the sequence

S_b/b

has a finite limit when

b\to\infty

, and find this limit.

1985 Bulgaria National Olympiad

Subcontests

computational of angles on media in triangle

two monochromatic triangles, given 7 points

insphere of pyramid, planar cross-sections with equal area

sextic has only real roots

prime factors of f(a),f(a+1),...,f(a+n-1), ≥k in count

greatest odd divisor, limit of sequence

1985 Bulgaria National Olympiad

Subcontests

computational of angles on media in triangle

two monochromatic triangles, given 7 points

insphere of pyramid, planar cross-sections with equal area

sextic has only real roots

prime factors of f(a),f(a+1),...,f(a+n-1), &ge;k in count

greatest odd divisor, limit of sequence

prime factors of f(a),f(a+1),...,f(a+n-1), ≥k in count