“Baltic Way – 92” Mathematical Team Contest Vilnius, November 5–8, 1992

1. Let p, q be two consecutive odd prime numbers. Prove that p+q is a product of at least 3 natural numbers > 1 (not necessarily different).

2. Denote by d(n) the number of all positive divisors of a natural number n (including 1 and n). Prove that there are infinitely many n, such that n/d(n) is an integer.

3. Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

4. Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers ?

5. It is given that a^{2}+b^{2}+(a+b)^{2} = c^{2}+d^{2}+(c+d)^{2}. Prove that a^{4}+b^{4}+(a+b)^{4} = c^{4}+d^{4}+(c+d)^{4}.

6. Prove that the product of the 99 numbers \frac{k^{3} − 1}{k^{3} + 1}, k = 2, 3, …, 100 is greater than 2/3.

7. Let a = ^{1992} \sqrt{1992}. Which number is greater

a^{a^{a^{^{···}^{a}}}} } 1992 or 1992 ?

8. Find all integers satisfying the equation 2^{x} · (4 − x) = 2x + 4.

9. A polynomial f(x) = x^{3} + ax^{2} + bx + c is such that b < 0 and ab = 9c. Prove that the polynomial f has three different real roots.

10. Find all fourth degree polynomials p(x) such that the following four conditions are satisfied:

(i) p(x) = p(−x) for all x,

(ii) p(x) ≥ 0 for all x,

(iii) p(0) = 1,

(iv) p(x) has exactly two local minimum points x_{1} and x_{2} such that |x_{1} − x_{2}| = 2.

11. Let Q^{+} denote the set of positive rational numbers. Show that there exists one and only one function f: Q^{+} → Q^{+} satisfying the following conditions:

(i) If 0 < q < 1/2 then f(q) = 1 + f(q/(1 − 2q)),

(ii) If 1 < q ≤ 2 then f(q) = 1 + f(q − 1),

(iii) f(q) · f(1/q) = 1 for all q ∈ Q^{+}.

12. Let N denote the set of natural numbers. Let ϕ: N → N be a bijective function and assume

that there exists a finite limit

lim_{n→∞} \frac{ϕ(n)}{n} = L .

What are the possible values of L?

13. Prove that for any positive x_{1}, x_{2}, …, x_{n}, y_{1}, y_{2}, …, y_{n} the inequality

∑_{i=1}^{n} \frac{1}{x_{i}y_{i}} ≥ \frac{4n^{2}}{∑_{i=1}^{n}(x_{i} + y_{i})^{2}}

holds.

14. There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from another one. Prove that there is a town such that all remaining towns can be reached from it.

15. Noah has 8 species of animals to fit into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accomodation. Prove that there is a way to assign the animals to their cages so that each species shares with compatible species.

16. All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?

17. Quadrangle ABCD is inscribed in a circle with radius 1 in such a way that the diagonal AC is a diameter of the circle, while the other diagonal BD is as long as AB. The diagonals intesect at P. It is known that the length of P C is 2/5. How long is the side CD?

18. Show that in a non-obtuse triangle the perimenter of the triangle is always greater than two times the diameter of the circumcircle.

19. Let C be a circle in plane. Let C_{1} and C_{2} be nonintersecting circles touching C internally at points A and B respectively. Let t be a common tangent of C_{1} and C_{2} touching them at points D and E respectively, such that both C_{1} and C_{2} are on the same side of t. Let F be the point of intersection of AD and BE. Show that F lies on C.

20. Let a ≤ b ≤ c be the sides of a right triangle, and let 2p be its perimeter. Show that

p(p − c) = (p − a)(p − b) = S ,

where S is the area of the triangle.