4

Part of 2016 May Olympiad

Problems(2)

triangle and quadrilateral have same areas, when midline bisects a segment

Source: May Olympiad (Olimpiada de Mayo) 2016 L2

ABC

D

E

be points of the sides

BC

AC

respectively. Segments

AD

BE

O

. Suppose that the line connecting midpoints of the triangle and parallel to

AB

, bisects the segment

DE

. Prove that the triangle

ABO

and the quadrilateral

ODCE

have equal areas.

geometryareasmidline

Given a board of $3 \times 3$

Source: May Olympiad 2016 L1 P4

Given a board of

3 \times 3

you want to write the numbers

1, 2, 3, 4, 5, 6, 7, 8

and a number in their boxes positive integer

M

, not necessarily different from the above. The goal is that the sum of the three numbers in each row be the same

a)

Find all the values of

M

for which this is possible.

b)

For which of the values of

M

a)

is it possible to arrange the numbers so that no only the three rows add the same but also the three columns add the same?