45-th Mathematical Olympiad in Poland First Round, September–December, 1993

1. Prove that the system of equations

\{ \begin{matrix}a^{2} − b = c^{2} \\ b^{2} − a = d^{2} \end{matrix}

has no integer solutions a, b, c, d.

2. The sequence of functions f_{0}, f_{1}, f_{2}, … is given by the conditions:

f_{0}(x) = |x|  for all x ∈ R
f_{n+1}(x) = |f_{n}(x) − 2|  for n = 0, 1, 2, … and all x ∈ R .

For each positive integer n, solve the equation f_{n}(x) = 1.

3. Prove that if a, b, c are the lengths of the sides of a triangle, then

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} ≤ \frac{1}{a + b − c} + \frac{1}{c + a − b} + \frac{1}{b + c − a} .

4. Given is a circle with center O, point A inside the circle and a chord P Q which is not a diameter and passing through A. The lines p and q are tangent to the given circle at P and Q respectively. The line ℓ passing through A and perpendicular to OA intersects the lines p and q at K and L respectively. Prove that |AK| = |AL|.

5. Prove that if the polynomial x^{3} + ax^{2} + bx + c has three distinct real roots, so does the polynomial

x^{3} + ax^{2} + \frac{1}{4}(a^{2} + b)x + \frac{1}{8}(ab − c) .

6. The function f: R → R is continuous. Prove that if for every real number x, there exists a positive integer n, such

that

(\underbrace{f ∘ f ∘ … ∘ f}_{n})(x) = 1 ,

then f(1) = 1.

7. Given convex quadrilateral ABCD. We construct the similar triangles AP B, BQC, CRD, DSA outside ABCD

so that

P̸ AB = ^{}Q̸BC = ^{}R̸CD = ^{}S̸DA , ^{}P̸ BA = ^{}Q̸CB = ^{}R̸DC = ^{}S̸AD .

Prove that if P QRS is a parallelogram, so is ABCD.

8. Given positive integers a, b, c such that a^{3} is divisible by b, b^{3} is divisible by c, c^{3} is divisible by a. Prove that (a + b + c)^{13} is divisible by abc.

9. In a conference 2n personalities take part. Each person has at least n acquaintaces among the others. Prove that it is possible to quarter the participants into two-person rooms, so that each participant would share the room with his/her acquaintace.

10. Given positive real numbers p, q with p + q = 1. Prove that for all positive integers m, n the following inequality

holds

(1 − p^{m})^{n} + (1 − q^{n})^{m} ≥ 1 .

11. A triangle with perimeter 2p is inscribed in a circle of radius R and also circumscribed on a circle of radius r. Prove that p < 2(R + r).

12. Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the the opposite edges of this tetrahedron are equal.