MathDB

2

Part of 2010 All-Russian Olympiad

Problems(5)

All-Russian Olympiad 2010 grade 9 P-2

Source:

9/9/2010
There are 100100 random, distinct real numbers corresponding to 100100 points on a circle. Prove that you can always choose 44 consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.
inequalitiescombinatorics proposedcombinatorics
All-Russian Olympiad 2010 grade 9 P-6

Source:

9/9/2010
Each of 10001000 elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.
pigeonhole principleinductioncombinatorics proposedcombinatorics
All-Russian Olympiad 2010 grade 10 P-6

Source:

9/9/2010
Into triangle ABCABC gives point KK lies on bisector of ∠BAC \angle BAC. Line CKCK intersect circumcircle ω \omega of triangle ABCABC at M≠CM \neq C. Circle Ω \Omega passes through AA, touch CMCM at KK and intersect segment ABAB at P≠AP \neq A and ω\omega at Q≠AQ \neq A. Prove, that PP, QQ, MM lies at one line.
geometrycircumcircleangle bisectorgeometric transformationgeometry proposed
All-Russian Olympiad 2010 11 grade P-2

Source:

9/10/2010
On an n×nn\times n chart, where n≥4n \geq 4, stand "++" signs in the cells of the main diagonal and "−-" signs in all the other cells. You can change all the signs in one row or in one column, from −- to ++ or from ++ to −-. Prove that you will always have nn or more ++ signs after finitely many operations.
combinatorics proposedcombinatorics
All-Russian Olympiad 2010 11 grade P-6

Source:

9/10/2010
Could the four centers of the circles inscribed into the faces of a tetrahedron be coplanar?
(vertexes of tetrahedron not coplanar)
geometry3D geometrytetrahedronincenteranalytic geometrygeometry proposed