MathDB

3

Part of 2022 New Zealand MO

Problems(2)

x^2 + y^2 = 2, \frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2

Source: 2022 NZMO - New Zealand Maths Olympiad Round 1 p3

10/9/2022
Find all real numbersx x and yy such that x2+y2=2x^2 + y^2 = 2 x22−y+y22−x=2.\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.
algebrasystem of equations
resiprocal sums differ by less than 0.01

Source: 2022 NZMO - New Zealand Maths Olympiad Round 2 p3

10/8/2022
Let SS be a set of 1010 positive integers. Prove that one can find two disjoint subsets A={a1,...,ak}A =\{a_1, ..., a_k\} and B={b1,...,bk}B = \{b_1, ... , b_k\} of SS with ∣A∣=∣B∣|A| = |B| such that the sums x=1a1+...+1akx =\frac{1}{a_1}+ ... +\frac{1}{a_k} and y=1b1+...+1bky =\frac{1}{b_1}+ ... +\frac{1}{b_k} differ by less than 0.010.01, i.e., ∣x−y∣<1/100|x - y| < 1/100.
combinatoricsSubsets