Individual contest, day 1

youngest group Monday, 24 September 2007

11. Show that every convex 3n-gon can be divided into triangles such that their sides are diagonals of the 3n-gon and every vertex of the 3n-gon is a vertex of an odd number of triangles.

12. Show that in a regular 12-gon A_{1}A_{2} …A_{12} diagonals A_{1}A_{5}, A_{3}A_{8} i A_{4}A_{11} intersect in one point.

13. Let x, y, z ∈ [1, 2]. Show that:

3(x + y + z) ⩾ xy + yz + zx + 6

14. Six little Skrzypens swim in a little, cilculal pond with ladius 1. Show that in evely moment thele ale two Skrzypens with distance at most 1 between them.

15. Let a, b be positive integers of different parity. Show that there exists such c ∈ ℤ, that numbers c + a, c + b, c + ab are all perfect squares.

Individual contest, day 1

younger group Monday, 24 September 2007

12. Show that in a regular 12-gon A_{1}A_{2} …A_{12} diagonals A_{1}A_{5}, A_{3}A_{8} i A_{4}A_{11} intersect in one point.

13. Let x, y, z ∈ [1, 2]. Show that:

3(x + y + z) ⩾ xy + yz + zx + 6

14. Six little Skrzypens swim in a little, cilculal pond with ladius 1. Show that in evely moment thele ale two Skrzypens with distance at most 1 between them.

15. Let a, b be integers of different parity. Show that there exists such c ∈ ℤ, that numbers c + a, c + b, c + ab are all perfect squares.

17. Show that every natural number n can be expressed as a sum of distinct numbers with form 2^{s}3^{r}, such that there are no two elements of this representation such that one of them divides the second.

Individual contest, day 1

older group Monday, 24 September 2007

14. Six little Skrzypens swim in a little, cilculal pond with ladius 1. Show that in evely moment thele ale two Skrzypens with distance at most 1 between them.

15. Let a, b be integers of different parity. Show that there exists such c ∈ ℤ, that numbers c + a, c + b, c + ab are all perfect squares.

16. Show that for positive a, b, c an inequality holds:

\frac{1}{b(a + b)} + \frac{1}{c(b + c)} + \frac{1}{a(c + a)} ⩾ \frac{27}{2(a + b + c)^{2}}

17. Show that every natural number n can be expressed as a sum of distinct numbers with form 2^{s}3^{r}, such that there are no two elements of this representation such that one of them divides the second.

18. Circles S_{1} and S_{2} with centers O_{1} and O_{2} respectively, intersect at points M and N. Line t is tangent to S_{1} in A and to S_{2} in B and lies closer to point M than to N. Line perpendicular to AM passing through point B intersects line O_{1}O_{2} in point C. Show that ∠BMC = 90^{∘}.

Individual contest, day 1

oldest group Monday, 24 September 2007

15. Let a, b be integers of different parity. Show that there exists such c ∈ ℤ, that numbers c + a, c + b, c + ab are all perfect squares.

17. Show that every natural number n can be expressed as a sum of distinct numbers with form 2^{s}3^{r}, such that there are no two elements of this representation such that one of them divides the second.

18. Circles S_{1} and S_{2} with centers O_{1} and O_{2} respectively, intersect at points M and N. Line t is tangent to S_{1} in A and to S_{2} in B and lies closer to point M than to N. Line perpendicular to AM passing through point B intersects line O_{1}O_{2} in point C. Show that ∠BMC = 90^{∘}.

19. Given is a positive integer c. A sequence {p_{n}} is defined according to the following rule: p_{1} is a prime number and for k 1 p_{k+1} is any of the prime divisors of p_{k} + c, not appearing in sequence p_{1}, p_{2}, ..., p_{k}. Show that the sequence {p_{n}} is finite.

110. Find all functions f : ⟨0, +∞) −→ ⟨0, +∞) satisfying conditions:

(a) For every nonnegative x, y with positive sum f(xf(y))f(y) = f(\frac{xy}{x+y} ).

(b) f(1) = 0.

(c) f(x) > 0 for x > 1.