IMO预选题1983(shortlist)

SHORLISTED PROBLEMS FOR THE 24th IMO

France, 1983

1. The towns 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.

2. Let n be a positive integer. Let σ(n ) be the sum of the natural divisors of n (including 1 and n ). We say that an integer m ≥ 1 is "superabundant" if ? k ∈ {1, 2, ..., m -1}we have k

k m m )()(σσ>. Prove that there exists an infinite number of superabundant numbers.

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

? (a) A circle.

? (b) An equilateral triangle (that is the set of the 3 vertices and the points of the 3 edges).

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

5. 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 minimal sequence of the set under this order.

6. 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 ≤ 2n - 1;

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

7. Let a be a positive integer and let (a n ) be defined by a 0 = 0, )1()1(2)1()1(1++++++=+n n n n n a a a a a a a a a . Show that for each positive integer n , a n is a positive integer.

8. 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 we have |N i (t ) - N j (t )| < 2, i ≠ j , i, j = 1, 2, ... .

9. Let p and q > 0 be two 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/q 2 for all x in I.

10. Let f : [0,1] → R be a continuous function that satisfies: f (2x ) = bf (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.

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

? (i ) f(xf(y)) = yf(x) for all positive real x, y .

? (ii ) f (x ) → 0 as x → +∞.

12. Let E be the set of the 19833 points of the space R 3 whose three 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 parallel to the axes) is a multiple of 4.

13. 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).

14. Decide whether there exists a set M of positive integer numbers satisfying the following conditions:

? (a) For any natural number m > 1 there are a, b ∈ M such that a + b = m .

? (b) If a, b, c, d ∈ M, a, b, c, d < 10 and a + b = c + d , then a = c or a = d .

15. Let F(n ) be the set of all polynomials: P(x ) = a 0 + a 1x + ... + a n x n with a 0, a 1, ... ,a n ∈ R and 0 ≤ a 0 = a n ≤ a 1 = a n -1 ≤ ... ≤ a [n /2] = a [(n + 1)/2]. Prove that if f ∈ F(m ) and g ∈ F(n ), then fg ∈ F(m + n ).

16. Let P 1, P 2, ..., P n be distinct points of the plane, n ≥ 2. Prove that

j i n j i j i n j i P P n P P ≤<≤≤<≤?≥11min )1(2

3max .

17. Let a, b, c be positive integers satisfying gcd (a, b ) = gcd (b, c ) = gcd (c, a ) = 1. Show that 2abc - ab - bc - ca is the largest integer that can not be represented as xbc + yca + zab with nonnegative integers x, y, z .

18. Let (F n )n ≥ 1 be the Fibonacci sequence: F 1 = F 2 = 1; F n +2 = F n +1 + F n , n ≥ 1 and P(x ) a polynomial of degree 990 verifying P(k ) = F k for k = 992, ..., 1982. Prove that P(1983) = F 1983 - 1.

19. Solve the system of equations:???íì??+=??+=a x a x x x x x a x a x x x x x n n n

n )()(11112211M where a > 0.

20. Find the greatest integer less than or equal to ?=?1983

21

1)1983/1(k k .

21. 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 02cos 1

=?=n k k n a k π.

22. If a, b and c are sides of a triangle, prove that a 2b(a - b) + b 2c(b - c) + c 2a(c - a) ≥ 0 and

determine when equality holds.

23. 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 1Q 1 and P 2Q 2, respectively. Prove that ∠ O 1KO 2 = ∠ M 1KM 2.

24. Let be d n the last non-zero digit of the decimal representation of n !. Prove that d n is not

periodic from a certain point on.