10.7

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(2)

perpendicular bisects MN - VI Soros Olympiad 1999-00 Round 1 10.7

Source:

Let a line, perpendicular to side

AD

of parallelogram

ABCD

passing through point

B

, intersect line

CD

M

, and a line, passing through point

B

and perpendicular to side

CD

, intersect line

AD

N

. Prove that the line passing through point

B

perpendicular to the diagonal

AC

, passes through the midpoint of the segment

MN

geometrybisects segment

1-100 in 10x10 grid

Source: VI Soros Olympiad 1990-00 R1 10.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

1, 2, 3, ..., 99, 100

are randomly arranged in the cells of a square table measuring

10\times 10

(each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1

82

. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.

combinatoricscombinatorial geometry