MathDB

1

x_1+ x_2+ x_3 +x_4=1,a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c, ...

a_1 < a_2 < a_3 < a_4

be given positive numbers. Find all real values of parameter

c

for which the system

\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\ a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\ a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}

has a solution in nonnegative

(x_1,x_2,x_3,x_4)

real numbers.

8

1

c_{n}=(4n-6)/n c_{n-1} is integer, prod (2^{n+1-i}-1)^{2^i} divides(2^{n+1}-1)!

(a) Prove that

(2^{n+1}-1)!

is divisible by

\prod_{i=0}^n (2^{n+1-i}-1)^{2^i }

, for every natural number n (b) Define the sequence (

c_n

) by

c_1=1

and

c_{n}=\frac{4n-6}{n}c_{n-1}

for

n\ge 2

. Show that each

c_n

is an integer.

9

1

equal rectangles on sides of regular p-gon, 3 colours, symmetry axis

To each side of the regular

p

-gon of side length

1

there is attached a

1 \times k

rectangle, partitioned into

k

unit cells, where

k

and

p

are given positive integers and p an odd prime. Let

P

be the resulting nonconvex star-like polygonal figure consisting of

kp + 1

regions (

kp

unit cells and the

p

-gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?

6

1

l_1, l_2, l_3$ are pairwise perpendicular, and so are l_4, l_5, l_6

Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled

\ell_1,...,\ell_6

in such a way that

\ell_1, \ell_2, \ell_3

are pairwise perpendicular, and so are

\ell_4, \ell_5, \ell_6

.

4

1

a, b, a + b either all belong to A or all belong to B where A,B partitions of N

The set

N

has been partitioned into two sets A and

B

. Show that for every

n \in N

there exist distinct integers

a, b > n

such that

a, b, a + b

either all belong to

A

or all belong to

B

.

3

1

F contains a subfamily consisting of pairwise disjoint discs, joint area >=S/9

A bounded planar region of area

S

is covered by a finite family

F

of closed discs. Prove that

F

contains a subfamily consisting of pairwise disjoint discs, of joint area not less than

S/9

.

1

a^5 + b^5 <= 1 and x^5 + y^5 <= 1 then a^2x^3 + b^2y^3 <= 1 for a,b,x,y>=0

Nonnegative real numbers

a, b,x,y

satisfy

a^5 + b^5 \le

1 and

x^5 + y^5 \le 1

. Prove that

a^2x^3 + b^2y^3 \le 1

.

7

1

4 points, 6 lines, prove or disprove that the segments form a hexagon

Let

P_1,P_2,P_3,P_4

be four distinct points in the plane. Suppose

\ell_1,\ell_2, … , \ell_6

are closed segments in that plane with the following property: Every straight line passing through at least one of the points

P_i

meets the union

\ell_1 \cup \ell_2\cup … \cup\ell_6

in exactly two points. Prove or disprove that the segments

\ell_i

necessarily form a hexagon.

2

1

Nice Equation

Find all triples of positive integers

(p, q, n)

with

p

and

q

prime, such that

p(p+1)+q(q+1) = n(n+1)

.

1983 Austrian-Polish Competition

Subcontests

x_1+ x_2+ x_3 +x_4=1,a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c, ...

c_{n}=(4n-6)/n c_{n-1} is integer, prod (2^{n+1-i}-1)^{2^i} divides(2^{n+1}-1)!

equal rectangles on sides of regular p-gon, 3 colours, symmetry axis

l_1, l_2, l_3$ are pairwise perpendicular, and so are l_4, l_5, l_6

a, b, a + b either all belong to A or all belong to B where A,B partitions of N

F contains a subfamily consisting of pairwise disjoint discs, joint area >=S/9

a^5 + b^5 <= 1 and x^5 + y^5 <= 1 then a^2x^3 + b^2y^3 <= 1 for a,b,x,y>=0

4 points, 6 lines, prove or disprove that the segments form a hexagon

Nice Equation

1983 Austrian-Polish Competition

Subcontests

x_1+ x_2+ x_3 +x_4=1,a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c, ...

c_{n}=(4n-6)/n c_{n-1} is integer, prod (2^{n+1-i}-1)^{2^i} divides(2^{n+1}-1)!

equal rectangles on sides of regular p-gon, 3 colours, symmetry axis

l_1, l_2, l_3$ are pairwise perpendicular, and so are l_4, l_5, l_6

a, b, a + b either all belong to A or all belong to B where A,B partitions of N

F contains a subfamily consisting of pairwise disjoint discs, joint area &gt;=S/9

a^5 + b^5 &lt;= 1 and x^5 + y^5 &lt;= 1 then a^2x^3 + b^2y^3 &lt;= 1 for a,b,x,y&gt;=0

4 points, 6 lines, prove or disprove that the segments form a hexagon

Nice Equation

F contains a subfamily consisting of pairwise disjoint discs, joint area >=S/9

a^5 + b^5 <= 1 and x^5 + y^5 <= 1 then a^2x^3 + b^2y^3 <= 1 for a,b,x,y>=0