10.6

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

2 parallel lines and a rhombus

Source: III Soros Olympiad 1996-97 R1 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

There is a rhombus with acute angle

b

a

. Two parallel lines, the distance between which is equal to the height of the rhombus, intersect all four sides of the rhombus. What can be the sum of the perimeters of two triangles cut off from a rhombus by straight lines? (These two triangles lie outside the strip between parallel lines.)

geometryrhombus

76 numbere cards in a circle

Source: III Soros Olympiad 1996-97 R2 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

76

cards with different numbers written on them. These cards are laid out on the table in a circle, number down. Try to find some three cards in a row such that the number written on the middle of these three cards is greater than on each of the two neighboring ones. You can turn over no more than

10

cards in succession. How should one proceed to be sure to find three cardboard boxes for which the specified condition is met?

|y-2x|=x <=> |mx + ny| = y (III Soros Olympiad 1996-97 R3 10.6)

Source:

m

n

such that the set of points whose coordinates

x

y

satisfy the equation

|y-2x|=x

, coincides with the set of points specified by the equation

|mx + ny| = y

analytic geometryalgebra