MathDB

1

Part of 2021 Durer Math Competition Finals

Problems(3)

prime = a^3-b^3 => p=6k+1

Source: 2021 Dürer Math Competition Finals Day 1 E1

1/2/2022
Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder 11 after division by 66.
number theoryprime
in the k-th month the weeks consist of 6^{k-1} days

Source: 2021 Dürer Math Competition Finals Day2 E1 https://artofproblemsolving.com/community/c2749870_2021_

1/8/2022
In Sixcountry there are 12 12 months, but each month consists of 66 weeks. The month are named the same way we do, from January to December, but in each month the weeks have different lengths. In the kk-th month the weeks consist of 6k−16^{k-1} days. What is the number of days of the spring (March, April, May together)?
algebra
computational geo, right triangle, another right angle

Source: 2021 Dürer Math Competition Finals Day2 E+1 https://artofproblemsolving.com/community/c2749870_

1/9/2022
Given a right angled triangle ABCABC in which ∠ACB=90o\angle ACB = 90^o. Let DD be an inner point of ABAB, and let EE be an inner point of ACAC. It is known that ∠ADE=90o\angle ADE = 90^o, and that the length of the segment ADAD is 88, the length of the segment DEDE is 1515, and the length of segment CECE is 33. What is the area of triangle ABCABC?
geometryright triangle