2

Part of 2019 Durer Math Competition Finals

Problems(3)

fill in each cell of an 8x8 table with a 0 or 1

Source: 2019 Dürer Math Competition Finals E1.2

Albrecht fills in each cell of an

8 \times 8

0

1

. Then at the end of each row and column he writes down the sum of the

8

digits in that row or column, and then he erases the original digits in the table. Afterwards, he claims to Berthold that given only the sums, it is possible to restore the

64

digits in the table uniquely. Show that the

8 \times 8

table contained either a row full of

0

’s or a column full of

1

combinatoricstable

perfect squares after multiplying 2-digit numbers with 588

Source: 2019 Dürer Math Competition Finals Day2 E2 https://artofproblemsolving.com/community/c1621835_2019_

Anne multiplies each two-digit number by

588

in turn, and writes down the so-obtained products. How many perfect squares does she write down?

Perfect Squarenumber theory

right triangle when sidelengths are integers and circumradius prime

Source: 2019 Dürer Math Competition Finals E+1.2

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

geometrysidelengthsIntegersprimenumber theoryright triangle