MathDB

10.6

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(3)

numbers 0-2 in nxn board (IV Soros Olympiad 1997-98 Correspondence 10.6)

Source:

6/1/2024
Is it possible to arrange n×nn \times n in the cells of a square table the numbers 00,1 1 or 22 so that the sums of the numbers in rows and columns took on all different values from 11 to 2n2n? Consider two cases: a) nn is an odd number; b) nn is an even number.
combinatorics
man gets lost in a large forest,

Source: IV Soros Olympiad 1997-98 R2 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
A man gets lost in a large forest, the boundary of which is a straight line. (We can assume that the forest fills the half-plane.) It is known that the distance from a person to Granina forest does not exceed 22 km.
a) Suggest a path along which he will certainly be able to get out of the forest after walking no more than 1414 km. (Of course, a person does not know in which direction the border of the forest is, BUT he has the opportunity to move along any pre-selected curve. It is believed that a person left the forest as soon as he reached its border, while the border of the forest is invisible to him, no matter how close he would have approached it.)
b) Find a path with the same property and length no more than 1313 km.
geometrycombinatoricscombinatorial geometry
a fire at a speed of 1 km per hour

Source: IV Soros Olympiad 1997-98 R3 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/2/2024
A fire that starts in the steppe spreads in all directions at a speed of 11 km per hour. A grader with a plow arrived on the fire line at the moment when the fire engulfed a circle with a radius of 11 km. The grader moves at a speed of 1414 km per hour and cuts a strip with a plow that cuts off the fire. Indicate the path along which the grader should move so that the total area of the burnt steppe does not exceed: a) 4π4 \pi km2^2; b) 3π3 \pi km2^2. (We can assume that the grader’s path consists of straight segments and circular arcs.)
combinatoricscombinatorial geometry