The Fifteenth IMO Moscow, Soviet Union, July 5–16, 1973

3.15 The Fifteenth IMO Moscow, Soviet Union, July 5–16, 1973

3.15.1 Contest Problems

First Day (July 9)

1. Let O be a point on the line l and −−→ OP −−→

1 , OP 2 ,..., OP n unit vectors such that points P 1 ,P 2 ,...,P n and line l lie in the same plane and all points P i lie in the same half-plane determined by l. Prove that if n is odd, then

is the length of vector OM ).

2. Does there exist a finite set M of points in space, not all in the same plane, such that for each two points A , B ∈ M there exist two other points C,D ∈ M such that lines AB and CD are parallel but not equal?

3. Determine the minimum of a 2 +b 2 if a and b are real numbers for which the equation

x 4 + ax 3 + bx 2 + ax + 1 = 0

has at least one real solution. Second Day (July 10)

4. A soldier has to investigate whether there are mines in an area that has the form of equilateral triangle. The radius of his detector’s range is equal to one-half the altitude of the triangle. The soldier starts from one vertex of the triangle. Determine the shortest path through which the soldier has to pass in order to check the entire region.

5. Let G be a set of functions f : R → R of the form f (x) = ax + b, where a and b are real numbers and a 6= 0. Suppose that G satisfies the following conditions:

(1) If f , g ∈ G, then g ◦ f ∈ G, where (g ◦ f )(x) = g[ f (x)]. (2) If f ∈ G and f (x) = ax + b, then the inverse f −1 of f belongs to G

(f −1 (x) = (x − b)/a). (3) For each f ∈ G there exists a number x f ∈ R such that f (x f )=x f . Prove that there exists a number k ∈ R such that f (k) = k for all f ∈ G.

6. Let a 1 ,a 2 ,...,a n

be positive numbers and q a given real number, 0 < q < 1. Find

n real numbers b 1 ,b 2 ,...,b n that satisfy:

(1) a k <b k for all k = 1, 2, . . . , n; (2) q b < k +1

b k < 1 q for all k = 1, 2, . . . , n − 1; (3) b

1 +q

1 +b 2 + ··· + b n < 1 −q (a 1 +a 2 + ··· + a n ).

3.15 IMO 1973 87

3.15.2 Shortlisted Problems

1. (BGR 6) Let a tetrahedron ABCD be inscribed in a sphere S. Find the locus of points P inside the sphere S for which the equality

holds, where A 1 ,B 1 ,C 1 , and D 1 are the intersection points of S with the lines AP , BP,CP, and DP, respectively.

2. (CZS 1) Given a circle K, find the locus of vertices A of parallelograms ABCD with diagonals AC ≤ BD, such that BD is inside K.

3. (CZS 6) IMO 1 Prove that the sum of an odd number of unit vectors passing through the same point O and lying in the same half-plane whose border passes through O has length greater than or equal to 1.

4. (UNK 1) Let P be a set of 7 different prime numbers and C a set of 28 different composite numbers each of which is a product of two (not necessarily different) numbers from P. The set C is divided into 7 disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of C are there?

5. (FRA 2) A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.

6. (POL 2) IMO 2 Does there exist a finite set M of points in space, not all in the same plane, such that for each two points A , B ∈ M there exist two other points

C , D ∈ M such that lines AB and CD are parallel?

7. (POL 3) Given a tetrahedron ABCD, let x = AB · CD, y = AC · BD, and z = AD · BC. Prove that there exists a triangle with edges x,y,z.

8. (ROU 1) Prove that there are exactly k [k/2] arrays a 1 ,a 2 ,...,a k +1 of nonnega- tive integers such that a 1 = 0 and |a i −a i +1 | = 1 for i = 1,2,...,k.

9. (ROU 2) Let Ox , Oy, Oz be three rays, and G a point inside the trihedron Oxyz. Consider all planes passing through G and cutting Ox , Oy, Oz at points A, B,C, respectively. How is the plane to be placed in order to yield a tetrahedron OABC with minimal perimeter?

10. (SWE 3) IMO 6 Let a 1 ,a 2 ,...,a n

be positive numbers and q a given real number,

0 < q < 1. Find n real numbers b 1 ,b 2 ,...,b n that satisfy: (1) a k <b k for all k = 1, 2, . . . , n; (2) q < b k +1 b k < 1 q for all k = 1, 2, . . . , n − 1;

11. (SWE 4) IMO 3 Determine the minimum of a 2 +b 2 if a and b are real numbers for which the equation

x 4 + ax 3 + bx 2 + ax + 1 = 0

88 3 Problems has at least one real solution.

12. (SWE 6) Consider the two square matrices

1 −1 1 −1 1 with entries 1 and −1. The following operations will be called elementary:

(1) Changing signs of all numbers in one row; (2) Changing signs of all numbers in one column; (3) Interchanging two rows (two rows exchange their positions); (4) Interchanging two columns.

Prove that the matrix B cannot be obtained from the matrix A using these opera- tions.

13. (YUG 4) Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to 1.

14. (YUG 5) IMO 4 A soldier has to investigate whether there are mines in an area that has the form of an equilateral triangle. The radius of his detector is equal to one-half of an altitude of the triangle. The soldier starts from one vertex of the triangle. Determine the shortest path that the soldier has to traverse in order to check the whole region.

15. (CUB 1) Prove that for all n ∈ N the following is true:

∏ sin

16. (CUB 2) Given a , θ ∈ R, m ∈ N, and P(x) = x 2m −2|a| m x m cos θ +a 2m , factorize P (x) as a product of m real quadratic polynomials.

17. (POL 1) IMO 5 Let F be a nonempty set of functions f : R → R of the form

f (x) = ax + b, where a and b are real numbers and a 6= 0. Suppose that F satisfies the following conditions: (1) If f , g ∈ F , then g ◦ f ∈ F , where (g ◦ f )(x) = g[ f (x)]. (2) If f ∈ F and f (x) = ax + b, then the inverse f −1 of f belongs to F

(f −1 (x) = (x − b)/a). (3) None of the functions f (x) = x + c, for c 6= 0, belong to F . Prove that there exists x 0 ∈ R such that f (x 0 )=x 0 for all f ∈F.

3.16 IMO 1974 89