Shortlisted Problems for the 24th IMO

Paris (France), 1983

Original HTML-version by Enrique Valeriano Cuba

Problem 1. (AUS 1)
The localities P1, ..., P1983 are served by ten international airlines A1, ..., A10. 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}:

[s (m)]/m > [s (k)]/k

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.


Problem 4. (BEL 5)
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 = (aj) and B = (bj) be any two of such sequences.
We say that A precedes B if ak < bk and ai = bi for i < k. Find the terms of the first sequence of the set under this order.

Problem 6. (CAN 2)
Suppose that {x1, x2, ..., xn} are positive integers for which x1 + x2 + ... + xn = 2(n + 1). Show that there exists an integer r with 0 £ r £ n - 1 for which the following n - 1 inequalities hold:

xr+1 £ 3
xr+1 + xr+2 £ 5
....
xr+1 + xr+2 + ... + xn £ 2(n-r) + 1
....
xr+1 + xr+2 + ... + xn + x1 + ... + xi £ 2(n + i - r) + 1; (1 £ i < r - 1)
....
xr+1 + xr+2 + ... + xn + x1 + ... + xr-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 {an} be defined by

a0 = 0
an+1 = (an + 1)a + (a + 1)an + 2.Sqrt [a(a + 1)an(an + 1)]; (n = 1, 2, ...)

Show that for each positive integer n, an 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 N1(t), N2(t), N3(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

|Ni(t) - Nj(t)| < 2, i ¹ j, i, j = 1, 2, ...

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:

|P(x) - p/q|<1/(q2)
for all x in I.

Problem 10. (FIN 2')
Let f : [0,1] ® R be continuous and satisfy:

f (2x) = b.f (x); 0 £ x £ 1/2,
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:

Problem 12. (LUX 2)
Let E be the set of the 19833 points of the space R3 whose 3 coordinates are integers between 0 and 1982 (including 0 and 1982). A colouring of E is a map from E to the set {red, blue}. How many colourings of E are there, satisfying the following property: The number of red vertices among the 8 vertices of any parallelepiped (whose edges are 4 by 4 parallele to the axis) is a multiple of 4.

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:

Problem 15. (DDR 1)
Let F(n) be the set of polynomials:

P(x) = a0 + a1x + ... + anxn

with a0, a1, ... ,an Î R and 0 £ a0 = an £ a1 = an-1 £ ... £ a[n/2] = a[(n + 1)/2].

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

Problem 16. (DDR 3)
Let P1, P2, ..., Pn be distinct points of the plane, n ³ 2. Prove that

where PiPj is the euclidean distance between Pi and Pj.

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

xbc + yca + zab

with nonnegative integers x, y, z.

Problem 18. (ROM 1)
Let (Fn)n ³ 1 be the Fibonacci sequence: F1 = F2 = 1; Fn+2 = Fn+1 + Fn, n ³ 1 and P(x) the polynomial of degree 990, verifying:

P (k) = Fk for k = 992, ..., 1982.

Prove that P (1983) = F1983 - 1.

Problem 19. (ROM 3)
SOlve the system of equations:

x1.|x1| = x2.|x2| + (x1 - a).|x1 - a|
x2.|x2| = x3.|x3| + (x2 - a).|x2 - a|
....................................
xn.|xn| = x1.|x1| + (xn - a).|xn - a|

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 = 21983

Problem 21. (SWE 4)
Let n be a positive integer having at least two different prime factors. Show that there exist a permutation a1, a2, ..., an of the integeres 1, 2, ..., n such that:

S k.cos( (2.p.ak)/n) = 0

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:

a2b(a - b) + b2c(b - c) + c2a(c - a) ³ 0

and determine when there is equality.

Problem 23. (USR 1)
Let be K one of the two intersection points of the circles W1 and W2. O1 and O2 are the centers of W1 and W2. The two common tangents to the circles meet W1 and W2 in P1 and P2 the first, and Q1 and Q2 the second, respectively. Let be M1 and M2 the midpoints of P1Q1 and P2Q2, respectively. Prove that Ð O1KO2 = Ð M1KM2

Problem 24. (USR 2)
Let be dn the last no nule digit of the decimal representation of n!. Prove that dn it is non-periodical. In other words, prove that there is no a positive integer T such that:

" n ³ n0: dn+T = dn