11.4

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

king in 8x8 chessboard (IV Soros Olympiad 1997-98 Correspondence 11.4)

Source:

In the lower left corner of the

8 \times 8

chessboard there is a king. He can move one cell either to the right, or up, or diagonally - to the right and up. How many ways can the king go to the upper right corner of the board if his route does not contain cells located on opposite sides of the diagonal going from the lower left to the upper right corner of the board?

combinatoricscombinatorial geometry

max area of projection of cylinder on a plane

Source: IV Soros Olympiad 1997-98 R2 11.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

Find the largest value of the area of the projection of the cylinder onto the plane if its radius is

r

and its height is

h

(orthogonal projection).

geometry3D geometrygeometric inequalitycylinderprojection

set of 1998 different naturals (IV Soros Olympiad 1997-98 R3 11.4)

Source:

There is a set of

1998

different natural numbers. It is known that none of them can be represented as the sum of several other numbers in this set. What is the smallest value that the largest of these numbers can take?