Individual contest - day 2

youngest group Tuesday, 25 September 2007

21. Find all integer solutions in triples (x, y, z) of equation:

x^{3} + 2y^{3} = 4z^{3}

22. Show that for nonnegative a, b, c an inequality holds:

a^{3} + b^{3} + c^{3} + a^{2}b + b^{2}c + c^{2}a 2(ab^{2} + bc^{2} + ca^{2})

23. A polygon A_{1}A_{2}A_{3}...A_{n} is given. Construct such points B_{1}, B_{2}, ..., B_{n}, for which points A_{1}, A_{2}, ..., A_{n} are midpoints of segments B_{1}B_{2}, B_{2}B_{3}, ..., B_{n−1} B_{n}, B_{n}B_{1} respectively.

24. Quadrilateral ABCD is inscribed in a circle with center O. Diagonals AC and BD intersect at P. Circumcircles of triangles ABP and CDP intersect at points P and Q. Show that if points P, Q and O are pairwise distinct then ∠P QO is right.

25. Skrzypen doesn’t want to grant his 37 Skrzypusies access to his private movie resources. Therefore he has constructed a safe and put a movie �The Pirates” inside. Now Skrzypen would like to add a number of locks to the safe and give every Skrzypus keys to some of them. The distribution of keys should follow conditions:

(a) every 36 Skrzypusies are able to open the safe,

(b) no 35 Skrzypusies are able to open the safe.

What is the minimal number of locks Skrzypen has to assemble?

Individual contest - day 2

younger group Tuesday, 25 September 2007

22. Show that for nonnegative a, b, c an inequality holds:

a^{3} + b^{3} + c^{3} + a^{2}b + b^{2}c + c^{2}a 2(ab^{2} + bc^{2} + ca^{2})

23. A polygon A_{1}A_{2}A_{3}...A_{n} is given. Construct such points B_{1}, B_{2}, ..., B_{n}, for which points A_{1}, A_{2}, ..., A_{n} are midpoints of segments B_{1}B_{2}, B_{2}B_{3}, ..., B_{n−1} B_{n}, B_{n}B_{1} respectively.

24. Quadrilateral ABCD is inscribed in a circle with center O. Diagonals AC and BD intersect at P. Circumcircles of triangles ABP and CDP intersect at points P and Q. Show that if points P, Q and O are pairwise distinct then ∠P QO is right.

25. Skrzypen doesn’t want to grant his 37 Skrzypusies access to his private movie resources. Therefore he has constructed a safe and put a movie �The Pirates” inside. Now Skrzypen would like to add a number of locks to the safe and give every Skrzypus keys to some of them. The distribution of keys should follow conditions:

(a) every 36 Skrzypusies are able to open the safe,

(b) no 35 Skrzypusies are able to open the safe.

What is the minimal number of locks Skrzypen has to assemble?

26. Is it possible to assign one of 2007 colors to every point of a plane in such a way that in every closed disk there are points of all the colors?

Individual contest - day 2

older group Tuesday, 25 September 2007

22. Show that for nonnegative a, b, c an inequality holds:

a^{3} + b^{3} + c^{3} + a^{2}b + b^{2}c + c^{2}a 2(ab^{2} + bc^{2} + ca^{2})

23. A polygon A_{1}A_{2}A_{3}...A_{n} is given. Construct such points B_{1}, B_{2}, ..., B_{n}, for which points A_{1}, A_{2}, ..., A_{n} are midpoints of segments B_{1}B_{2}, B_{2}B_{3}, ..., B_{n−1} B_{n}, B_{n}B_{1} respectively.

24. Quadrilateral ABCD is inscribed in a circle with center O. Diagonals AC and BD intersect at P. Circumcircles of triangles ABP and CDP intersect at points P and Q. Show that if points P, Q and O are pairwise distinct then ∠P QO is right.

27. A sequence {a_{n}} of positive integers satisfies following conditions:

a_{1} = 2, a_{n} = ⌊\frac{3a_{n−1}}{2} ⌋

Show that in this sequence there is infinitely many even numbers and infinitely many odd numbers.

28. Does there exists a sequence a_{1}, a_{2}, a_{3}... of real numbers and non-constant polynomial P (x) such that equality a_{m} + a_{n} = P (mn) holds for all positive integers m, n?

29. Is it possible to assign one of 2007 colors to every point of a plane in such a way that on every circle there are points of all the colors?

Individual contest - day 2

oldest group Tuesday, 25 September 2007

24. Quadrilateral ABCD is inscribed in a circle with center O. Diagonals AC and BD intersect at P. Circumcircles of triangles ABP and CDP intersect at points P and Q. Show that if points P, Q and O are pairwise distinct then ∠P QO is right.

28. Does there exists a sequence a_{1}, a_{2}, a_{3}... of real numbers and non-constant polynomial P (x) such that equality a_{m} + a_{n} = P (mn) holds for all positive integers m, n?

29. Is it possible to assign one of 2007 colors to every point of a plane in such a way that on every circle there are points of all the colors?

210. Determine the smallest odd integer n 1 satisfying following condition: there exist infinitely many integers, which squares are equal to sum of squares of n consecutive integers.

211. Show that for every positive integer n:

∑_{i+j=n} \binom{2i}{i} \binom{2j}{j} = 4^{n}