1

Part of 2021 Czech-Polish-Slovak Junior Match

Problems(2)

equal areas fro quadrilaterals in a trapezoid, AD=PC, BC=PD, <CMD =90^o

Source: 2021 Czech-Polish-Slovak Match Junior, team p1 CPSJ

Consider a trapezoid

ABCD

AB

CD

| AB | > | CD |

M

be the midpoint of

AB

. Let the point

P

ABCD

| AD | = | PC |

| BC | = | PD |

. Prove that if

| \angle CMD | = 90^o

, then the quadrilaterals

AMPD

BMPC

have the same area.

geometrytrapezoidequal areas

sum of products of numbers in rows and columns in 2x2 table is 2021

Source: Czech-Polish-Slovak Match Junior 2021, individual p1 CPSJ

You are given a

2 \times 2

array with a positive integer in each field. If we add the product of the numbers in the first column, the product of the numbers in the second column, the product of the numbers in the first row and the product of the numbers in the second row, we get

2021

. a) Find possible values for the sum of the four numbers in the table. b) Find the number of distinct arrays that satisfy the given conditions that contain four pairwise distinct numbers in arrays.