MathDB

1

a_1x_1 + a_2x2 + ... + a_nx_n not zero when sum a_1=0,

a_1, a_2, ... , a_n

is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if

x_i

all belong to the same set, then

a_1x_1 + a_2x_2 + ... + a_nx_n

is non-zero.

8

1

tiling a 28 x48 rectangle with axb tiles

We are given a supply of

a \times b

tiles with

a

and

b

distinct positive integers. The tiles are to be used to tile a

28 \times 48

rectangle. Find

a, b

such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find

a, b

such that the tile has the largest possible area and there is more than one possible tiling.

7

1

sequences of dominoes

D_n

is a set of domino pieces. For each pair of non-negative integers

(a, b)

with

a \le b \le n

, there is one domino, denoted

[a, b]

or

[b, a]

in

D_n

. A ring is a sequence of dominoes

[a_1, b_1], [a_2, b_2], ... , [a_k, b_k]

such that

b_1 = a_2, b_2 = a_3, ... , b_{k-1} = a_k

and

b_k = a_1

. Show that if

n

is even there is a ring which uses all the pieces. Show that for n odd, at least

(n+1)/2

pieces are not used in any ring. For

n

odd, how many different sets of

(n+1)/2

are there, such that the pieces not in the set can form a ring?

6

1

a_1 = a_3 when a_i+1=p(a_1), integer polynomila p(x)

p(x)

is a polynomial with integer coefficients. The sequence of integers

a_1, a_2, ... , a_n

(where

n > 2

) satisfies

a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)

. Show that

a_1 = a_3

.

4

1

x_i4 + 14x_ix_{i+1} + 1 = y_i^4, diophantine system

Find all solutions in positive integers to:

\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}

3

1

x + y^2 + z^4 = 0 , y + z^2 + x^4 = 0 , z + x^2 + y^5 = 0

Show that there are two real solutions to:

\begin{cases} x + y^2 + z^4 = 0 \\ y + z^2 + x^4 = 0 \\ z + x^2 + y^5 = 0\end {cases}

2

1

a^A = b^B = c^C = 1990^{1990}abc where A = b^c, B = c^a, C = a^b

Find all solutions in positive integers to

a^A = b^B = c^C = 1990^{1990}abc

, where

A = b^c, B = c^a, C = a^b

.

5

1

Permutation of a Set

Let

n>1

be an integer and let

f_1

,

f_2

, ...,

f_{n!}

be the

n!

permutations of

1

,

2

, ...,

n

. (Each

f_i

is a bijective function from

\{1,2,...,n\}

to itself.) For each permutation

f_i

, let us define

S(f_i)=\sum^n_{k=1} |f_i(k)-k|

. Find

\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)

.

1

hard [Neuberg circle]

The distinct points

X_1, X_2, X_3, X_4, X_5, X_6

all lie on the same side of the line

AB

. The six triangles

ABX_i

are all similar. Show that the

X_i

lie on a circle.

1990 Austrian-Polish Competition

Subcontests

a_1x_1 + a_2x2 + ... + a_nx_n not zero when sum a_1=0,

tiling a 28 x48 rectangle with axb tiles

sequences of dominoes

a_1 = a_3 when a_i+1=p(a_1), integer polynomila p(x)

x_i4 + 14x_ix_{i+1} + 1 = y_i^4, diophantine system

x + y^2 + z^4 = 0 , y + z^2 + x^4 = 0 , z + x^2 + y^5 = 0

a^A = b^B = c^C = 1990^{1990}abc where A = b^c, B = c^a, C = a^b

Permutation of a Set

hard [Neuberg circle]