MathDB

1

Part of 2018 European Mathematical Cup

Problems(2)

At least two of a, b, c are equal

Source: 7th European Mathematical Cup , Junior Category, Q1

12/25/2018
Let a,b,ca, b, c be non-zero real numbers such that a2+b+c=1a,b2+c+a=1b,c2+a+b=1c.a^2+b+c=\frac{1}{a}, b^2+c+a=\frac{1}{b}, c^2+a+b=\frac{1}{c}. Prove that at least two of a,b,ca, b, c are equal.
algebraemc
Determine all positive integers n

Source: 7th European Mathematical Cup , Senior Category , Q1

12/25/2018
A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition is odd if all its elements are odd. Determine all positive integers nn such that the number of even partitions of nn is equal to the number of odd partitions of nn. Remark: A partition of a positive integer nn is a non-decreasing sequence of positive integers whose sum of elements equals nn. For example, (2;3;4),(1;2;2;2;2)(2; 3; 4), (1; 2; 2; 2; 2) and (9)(9) are partitions of 9.9.
number theoryalgebra