MathDB

2

172 two-way direct airways between 20 cities

172

two-way direct airways between

20

cities, at most one between any two cities. Prove that one can reach any city from any other city with at most one transfer.

five nonnegative real numbers with sum 1 around a circle

Given any five nonnegative real numbers with the sum

1

, show that they can be arranged around a circle in such a way that the five products of two consecutive numbers sum up to at most

1/5

.

2

perfect squares sequence, a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)

The sequence

a_0,a_1,a_2,...

is defined by

a_0 = 0, a_1 = a_2 = 1

and

a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)

for all

n \in N

. Show that all

a_n

are perfect squares .

A(n) =[(n+1)/2] [(n+2)/2} , distinct points in the plane

Let

A(n)

be the least possible number of distinct points in the plane with the following property: For every

k = 1,2,...,n

there is a line containing precisely

k

of these points. Show that

A(n) =\left[\frac{n+1}{2}\right] \left[\frac{n+2}{2}\right]

1

2

f(x) =x^2 + 2bx + c >= for all integers n, then f(x) >= 0

Consider the trinomial

f(x) = x^2 + 2bx + c

with integer coefficients

b

and

c

. Prove that if

f(n) \ge 0

for all integers

n

, then

f(x) \ge 0

even for all rational numbers

x

.

Product of two integers is congruent 1 mod the third

Find all triples

(a,b,c)

of positive integers such that the product of any two of them when divided by the third leaves the remainder

1

.

4

2

midpoints of the six edges of a tetrahedron lie on a sphere

Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.

worm in the plane

In the plane there is a worm of length 1. Prove that it can be always covered by means of half of a circular disk of diameter 1. Note. Under a "worm", we understand a continuous curve. The "half of a circular disk" is a semicircle including its boundary.

1990 Bundeswettbewerb Mathematik

Subcontests

172 two-way direct airways between 20 cities

five nonnegative real numbers with sum 1 around a circle

perfect squares sequence, a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)

A(n) =[(n+1)/2] [(n+2)/2} , distinct points in the plane

f(x) =x^2 + 2bx + c >= for all integers n, then f(x) >= 0

Product of two integers is congruent 1 mod the third

midpoints of the six edges of a tetrahedron lie on a sphere

worm in the plane

1990 Bundeswettbewerb Mathematik

Subcontests

172 two-way direct airways between 20 cities

five nonnegative real numbers with sum 1 around a circle

perfect squares sequence, a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)

A(n) =[(n+1)/2] [(n+2)/2} , distinct points in the plane

f(x) =x^2 + 2bx + c &gt;= for all integers n, then f(x) &gt;= 0

Product of two integers is congruent 1 mod the third

midpoints of the six edges of a tetrahedron lie on a sphere

worm in the plane

f(x) =x^2 + 2bx + c >= for all integers n, then f(x) >= 0