**Problem 1. (AUS 1)**

The localities P_{1}, ..., P_{1983} are served
by ten international airlines A_{1}, ..., A_{10}.
It's noted that there is a direct service
(without stops) between any two of these localities and all
airline schedules are both ways.
Prove that at least one of the airlines can offer a round trip
with an odd number of landings.

**Problem 2. (BEL 1)**

Let n be a positive integer. Let
s (n) be the sum of the natural
divisors d of n (including 1 and n). We say that an integer
m ³ 1 is "superabondant"
if " k Î
{1, 2, ..., m-1}:

Prove that there exists an infinite number of superabondant numbers.

**Problem 3. (BEL 3)**

We say that a set E of points of the euclidian plane is
"pythagorician" if for any partition of E in two sets A
and B, at least one of the sets contains the vertices of
a rectangle triangle. Decide whether the following sets
are or not pythagorician.

- (a) A circle.
- (b) An equilateral triangle (that is the set of the 3 vertices and the points of the 3 edges).

On the sides of the triangle ABC, are constructed the similar isosceles triangles ABP (AP = PB), AQC (AQ = QC) and BRC (BR = RC). The first two are constructed externally to the triangle ABC, but the third is placed in the same halfplane determined by the line BC as the triangle ABC. Prove that APRQ is a parallelogram.

**Problem 5. (BRA 1)**

Consider the set of all strictly decreasing sequences of n
natural numbers having the property that in each sequence
no term divides any other term of the sequence. Let
A = (a_{j}) and B = (b_{j}) be any two of
such sequences.

We say that A precedes B if a_{k} < b_{k}
and a_{i} = b_{i} for i < k. Find the terms of
the first sequence of the set under this order.

**Problem 6. (CAN 2)**

Suppose that {x_{1}, x_{2}, ..., x_{n}}
are positive integers for which
x_{1} + x_{2} + ... + x_{n} = 2(n + 1).
Show that there exists an integer r with 0
£ r £
n - 1 for which the following n - 1 inequalities hold:

x_{r+1} £ 3

x_{r+1} + x_{r+2} £ 5

....

x_{r+1} + x_{r+2} + ... +
x_{n} £ 2(n-r) + 1

....

x_{r+1} + x_{r+2} + ... +
x_{n} + x_{1} + ... + x_{i}
£ 2(n + i - r) + 1; (1
£ i < r - 1)

....

x_{r+1} + x_{r+2} + ... +
x_{n} + x_{1} + ... + x_{r-1}
£ 2(n) - 1;

Prove that if all the inequalities are strict, then r is unique, and that otherwise there are exactly two such r.

**Problem 7. (CAN 5)**

Let a be a positive integer and let {a_{n}} be defined
by

a_{0} = 0

a_{n+1} = (a_{n} + 1)a + (a + 1)a_{n}
+ 2.Sqrt [a(a + 1)a_{n}(a_{n} + 1)];
(n = 1, 2, ...)

Show that for each positive integer n, a_{n} is a
positive integer.

**Problem 8. (ESP)**

In a test participate 3n students who are located in three
rows of n students each. The students leave the test room
one by one. If N_{1}(t), N_{2}(t),
N_{3}(t) denote the numbers of students in the first,
second and third row respectively at the time t, find the
probability that for each t during the test

**Problem 9. (FIN 1)**

Let p and q > 0 be integers. Show that there exists an interval
I of length 1/q and a polynomial P with integral coefficients
such that:

**Problem 10. (FIN 2')**

Let *f* : [0,1] ® **R**
be continuous and satisfy:

f (x) = b - (1 - b).f (2x - 1); 1/2 £ x £ 1,

where b = (1 + c)/(2 + c), c > 0. Show that 0 < f(x) - x < c for every x, 0 < x < 1.

**Problem 11. (UKG 4)**

Find all functions f defined on the positive real numbers
and taking positive real values, which satisfy the conditions:

- (i) f( xf(y)) = y. f(x) for all positive real x, y.
- (ii) f(x) ® 0 as x ® +¥.

Let E be the set of the 1983

**Problem 13. (POL 2)**

Prove or disprove: From the interval [1, 30000] one can
select a set of 1000 integers containing no arithmetic triple
(three numbers in arithmetic progression).

**Problem 14. (POL 3)**

Decide whether does there exist a set M of natural
numbers satisfying the following conditions:

- (a) For any natural number m > 1 there are a, b Î M such that a + b = m.
- (b) If a, b, c, d Î M; a, b, c, d < 10 and a + b = c + d, then a = c or a = d.

**Problem 15. (DDR 1)**

Let F(n) be the set of polynomials:

with a

Prove that if f Î F(m) and g Î F(n), then f.g Î F(m + n).

**Problem 16. (DDR 3)**

Let P_{1}, P_{2}, ..., P_{n} be
distinct points of the plane, n ³
2. Prove that

where P_{i}P_{j} is the euclidean distance
between P_{i} and P_{j}.

**Problem 17. (RFA 3)**

Let a, b, c be positive integers satisfying (a,b) = (b,c) =
(c,a) = 1. Show that 2abc - ab - bc - ca is the
largest integer not representable as

with nonnegative integers x, y, z.

**Problem 18. (ROM 1)**

Let (F_{n})_{n ³ 1
} be the Fibonacci sequence: F_{1} =
F_{2} = 1; F_{n+2} = F_{n+1} +
F_{n}, n ³ 1 and P(x)
the polynomial of degree 990, verifying:

Prove that P (1983) = F

**Problem 19. (ROM 3)**

SOlve the system of equations:

x

....................................

x

where a > 0, in the set of real numbers.

**Problem 20. (SWE 3)**

Find the greatest integer less than or equal to
S k^{(1/1983 - 1)},
where the sum is taken from k = 1 to k = 2^{1983}

**Problem 21. (SWE 4)**

Let n be a positive integer having at least two different
prime factors. Show that there exist a permutation
a_{1}, a_{2}, ..., a_{n} of the
integeres 1, 2, ..., n such that:

where the sum is taken from k = 1 to k = n.

**Problem 22. (USA 3)**

If a, b and c are sides of a triangle, prove that:

and determine when there is equality.

**Problem 23. (USR 1)**

Let be K one of the two intersection points of the circles
W_{1} and W_{2}. O_{1} and O_{2}
are the centers of W_{1} and W_{2}.
The two common tangents to the circles meet
W_{1} and W_{2} in
P_{1} and P_{2} the first, and
Q_{1} and Q_{2} the second, respectively.
Let be M_{1} and M_{2} the midpoints of
P_{1}Q_{1} and P_{2}Q_{2},
respectively. Prove that
Ð O_{1}KO_{2} =
Ð M_{1}KM_{2}

**Problem 24. (USR 2)**

Let be d_{n} the last no nule digit of the decimal
representation of n!. Prove that d_{n} it is
non-periodical. In other words, prove that there is no a positive
integer T such that: