1.
Each field of 20 x 20 table is filled with at most one star. To every field we write the number of stars in the square 3 x 3 with our field in the middle. Instead of 9 we write -10. What is maximal sum over whole table?

2a.
Given some 10 x 10, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that you can obtain a table, with nonnegative sums over each row and over each column.

2b.
What is the least possible number of the checkers being required for the 10 x 10 chess-board to provide the property: Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker?

3a.
Two equal chess-boards (20 x 20) have the same centre, but one is rotated by 45 degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides. \endexample

3b.
Given positive numbers a₁, a₂, ..., a_m and b₁, b_2,..., b_n and is known that a₁ + a₂ + ... + a_m = b₁ + b₂ + ... + b_n. Prove that you can fill an empty table with m rows and n columns with no more than (m+n-1) positive number in such a way, that for all i, j the sum of the numbers in the i-th row will equal to a_i, and the sum of the numbers in the j-th column equal to b_j.

4a.
You are trying to find the 4-field ship (a rectangle 1 x 4), situated on the 7 x 7 playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely?

4b.
The white fields of 1999 x 2000 chess-board are filled with either +1 or -1. For every black field, the product of neighbouring numbers is +1. Prove that all numbers are +1.

5a.
The lock on a safe consist of three wheels A, B and C each of which may be set in eight different positions. Due to defect in the mechanism, the door will open when any two of the wheels are in the correct position. What is minimum number of tries that can guaranteed to open the safe?

5b.
The squares of an infinite chessboard are numbered successively as folows : in the corner we put 0, and then in every other square we put the smallest nonnegative integer that does not appear to its left in the same row or below it in the same column. What number will appear at intersection of the 1820th row and the 135th column?

1.
The orthocenter of triangle ABC is H. The points H₁, H₂,H₃ we obtain by reflecting point H by the sides of triangle. Prove that circumcenter of triangle H₁H₂H₃ is also circumcenter of triangle ABC.

2a.
Let segments be drawn from the incenter I to the vertices of triangle ABC to partition the triangle into three smaller triangles. If O₁, O₂, O₃ are circumcenters of these little triangles, prove that the circumcenter O of triangle ABC is also the circumcenter of triangle O₁O₂O₃.

2b.
Let the sides of convex quadrilateral ABCD be extended and the bisectors of the exterior angles meet at points K₁, K₂, K₃ and $K₄$. Prove that K₁K₂K₃K₄ is always a cyclic quadrilateral.

3a.
In the cyclic quadrilateral ABCD the diagonal BD bisects diagonal AC. Prove that

2 . |BD|² = |AB|² + |BC|² + |CD|² + |DA|²

3b.
The circumcenter of triangle ABC is O and CD is median to AB. The centroid of triangle ACD is E. Prove that OE is perpendicular to CD if and only if triangle ABC is isosceles with |AB| = |AC|.

4a.
Given cyclic quadrilateral ABCD. It is known that |CD| = |AD|+|BC|. Prove that intersection of angle bisectors of angles DAB and ABC lies on segment CD.

4b.
Let the circumcenter and orthocenter of a triangle be called O and H, respectively. Our problem is to construct a triangle ABC from the following scanty information about AB and OH : all you are given is their lenghts and the fact that they are parallel.

5a.
In any triangle, prove that lines from the vertices of the medial triangle, which are perpendicular to the opposite sides of the Gergonne triangle (three point where the incircle touches sides), are always concurrent.

5b.
A circle is outscribed around the triangle ABC. Chords, from the middle of the arc AC to the middles of the arcs AB and BC, intersect sides AB and BC in the points D and E. Prove that the line DE is parallel to AC.

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BKMS
RNDr. Jaroslav Gurian, CSc.
KATČ, MFF UK
Mlynsk dolina
842 15 BRATISLAVA
SLOVAKIA