MathDB

1

S_m(n) \le n + m (\sqrt[4]{2^m}-1), S_m(n)=\sum \sqrt[k^2]{k^m}]

m, n

, denote

S_m(n)=\sum_{1\le k \le n} \left[ \sqrt[k^2]{k^m}\right]

Prove that

S_m(n) \le n + m (\sqrt[4]{2^m}-1)

8

1

all numbers written in the unit squares of the grid are equal

In each unit square of an infinite square grid a natural number is written. The polygons of area

n

with sides going along the gridlines are called admissible, where

n > 2

is a given natural number. The value of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon

P

is congruent to

P

.)

7

1

a^ab^b has exactly 98 zeros

Consider all pairs

(a, b)

of natural numbers such that the product

a^ab^b

written in decimal system ends with exactly

98

zeros. Find the pair

(a, b)

for which the product

ab

is the smallest.

1

(x_1y_1 + x_2y_2 - 1)^2 >= (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)

Let

x_1, x_2,y _1,y_2

be real numbers such that

x_1^2 + x_2^2 \le 1

. Prove the inequality

(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)

5

1

x^3 - 17x^2 + ax - b^2 = 0 , 3 integer roots

Determine all pairs

(a, b)

of positive integers for which the equation

x^3 - 17x^2 + ax - b^2 = 0

has three integer roots (not necessarily different).

9

1

r+KX+LY+MZ=2R, with inradius, circumradius and midpoints of sides and arcs

Given a triangle

ABC

, points

K,L,M

are the midpoints of the sides

BC,CA,AB

, and points

X,Y,Z

are the midpoints of the arcs

BC,CA,AB

of the circumcircle not containing

A,B,C

respectively. If

R

denotes the circumradius and

r

the inradius of the triangle, show that

r+KX+LY+MZ=2R

.

6

1

5 points on a circle, concurrent or parallel wanted - Austrian-Polish 1998

Different points

A,B,C,D,E,F

lie on circle

k

in this order. The tangents to

k

in the points

A

and

D

and the lines

BF

and

CE

have a common point

P

. Prove that the lines

AD,BC

and

EF

also have a common point or are parallel.

2

1

APMC 1998

For n points

P_1;P_2;...;P_n

in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points

P_i;P_{i+1} (i=1;2;...n-1)

is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color?

3

1

system

Find all pairs of real numbers

(x, y)

satisfying the following system of equations

2-x^{3}=y, 2-y^{3}=x

.

1998 Austrian-Polish Competition

Subcontests

S_m(n) \le n + m (\sqrt[4]{2^m}-1), S_m(n)=\sum \sqrt[k^2]{k^m}]

all numbers written in the unit squares of the grid are equal

a^ab^b has exactly 98 zeros

(x_1y_1 + x_2y_2 - 1)^2 >= (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)

x^3 - 17x^2 + ax - b^2 = 0 , 3 integer roots

r+KX+LY+MZ=2R, with inradius, circumradius and midpoints of sides and arcs

5 points on a circle, concurrent or parallel wanted - Austrian-Polish 1998

APMC 1998

system

1998 Austrian-Polish Competition

Subcontests

S_m(n) \le n + m (\sqrt[4]{2^m}-1), S_m(n)=\sum \sqrt[k^2]{k^m}]

all numbers written in the unit squares of the grid are equal

a^ab^b has exactly 98 zeros

(x_1y_1 + x_2y_2 - 1)^2 &gt;= (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)

x^3 - 17x^2 + ax - b^2 = 0 , 3 integer roots

r+KX+LY+MZ=2R, with inradius, circumradius and midpoints of sides and arcs

5 points on a circle, concurrent or parallel wanted - Austrian-Polish 1998

APMC 1998

system

(x_1y_1 + x_2y_2 - 1)^2 >= (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)