A few problems a bit harder than the others

1. Show that for all a, n positive integers:

n| ∑_{i=0}^{n−1} a^{(i,n)}

where (k, l) denotes the greatest common divisor of k and l.

2. Let n be a positive integer. For k = 1, 2, …, n we define a_{k} = \frac{1}{\binom{n}{k}} and b_{k} = 2^{k−n} . Show that equality holds:

\frac{a_{1} − b_{1}}{1} + \frac{a_{2} − b_{2}}{2} + … + \frac{a_{n} − b_{n}}{n} = 0

3. Show that if a, b, c are positive, then an inequality holds:

\frac{a}{\sqrt{ab + b^{2}}} + \frac{b}{\sqrt{bc + c^{2}}} + \frac{c}{\sqrt{ca + a^{2}}} \frac{3}{\sqrt{2}}

4. Given is square chessboard 19 × 19. Strictly inside this chessboard we specify a smaller square chessboard 17 × 17. Determine, if the smaller chessboard can be covered using disjoint horizontal rectangles 5 × 1 and vertical 1 × 4 placed inside the bigger chessboard.

5. Find all fours of integers x, y, z, t satisfying a system of equations:

\{ \begin{matrix}xz − 2yt = 3 \\ xt + yz = 1 \end{matrix}

6. Show, that there exists such an integer k, that k · 2^{n} + 1 is composite for every n 1.

7. Let P (x), Q(x), R(x), S(x) be such real polynomials, that for every real x equation holds:

P (x^{5}) + xQ(x^{5}) + x^{2}R(x^{5}) = (x^{4} + x^{3} + x^{2} + x + 1)S(x)

Show, that P (1) = 0.

8. Point P lies inside a triangle ABC. Let D, E, F be perpendicular projections of P onto lines BC, CA, AB. Let O be the circumcentre of triangle DEF, and r be radius of circumcircle of this triangle. Show that:

area(ABC) 3r\sqrt{3r^{2} − 3|OP |^{2}}

9. A circle o is given and two triangles inscribed into it. The common part of them is a hexagon. Show that main diagonals of this hexagon intersect in one point.

10. Median AM of triangle ABC, in which AB ≠ AC, intersects inscribed circle ω in points K and L. Lines parallel to BC through K and L respectively intersect ω once more in X and Y. Lines AX and AY intersect side BC in points P and Q respectively. Show, that BP = CQ.

11. On the plane there is given a disk with radius 1. It has been covered with a square with a side of length 2. The square has been divided into 2007 parallel stripes and one has been thrown away. Show that the disk cannot be covered using remaining stripes.