MathDB

1.

Part of 2020 Saint Petersburg Mathematical Olympiad

Problems(3)

Pythagorean triple thing

Source: Saint Petersburg MO 2020 Grade 11 Problem 1

5/7/2020
A positive integer is called hypotenuse if it can be represented as a sum of two squares of non-negative integers. Prove that any natural number greater than 1010 is the difference of two hypotenuse numbers.
number theory
weights and stones

Source: Saint Petersburg MO 2020 Grade 10 Problem 1

5/7/2020
Andryusha has 100100 stones of different weight and he can distinguish the stones by appearance, but does not know their weight. Every evening, Andryusha can put exactly 1010 stones on the table and at night the brownie will order them in increasing weight. But, if the drum also lives in the house then surely he will in the morning change the places of some 22 stones.Andryusha knows all about this but does not know if there is a drum in his house. Can he find out?
combinatorics
Maximal number of solutions

Source: Saint Petersburg MO 2020 Grade 9 Problem 1

5/7/2020
What is the maximal number of solutions can the equation have max⁡{a1x+b1,a2x+b2,…,a10x+b10}=0\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0 where a1,b1,a2,b2,…,a10,b10a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10} are real numbers, all aia_i not equal to 00.
algebra