MathDB

2

Part of 2022 New Zealand MO

Problems(2)

product of the 31 numbers is 5th power, from pairs of 0-61

Source: 2022 NZMO - New Zealand Maths Olympiad Round 1 p2

10/9/2022
Is it possible to pair up the numbers 0,1,2,3,...,610, 1, 2, 3,... , 61 in such a way that when we sum each pair, the product of the 3131 numbers we get is a perfect f ifth power?
number theorypower of numberPerfect power
a^2 + b^2 + c^2 = 1 , a(2b - 2a - c) >= 1/2

Source: 2022 NZMO - New Zealand Maths Olympiad Round 2 p2

10/8/2022
Find all triples (a,b,c)(a, b, c) of real numbers such that a2+b2+c2=1a^2 + b^2 + c^2 = 1 and a(2b−2a−c)≥12a(2b - 2a - c) \ge \frac12.
algebrainequalities