MathDB

5

Part of 2023 India Regional Mathematical Olympiad

Problems(2)

RMO 2023 P-5

Source:

10/29/2023
Let n>k>1n>k>1 be positive integers. Determine all positive real numbers a1,a2,…,ana_1, a_2, \ldots, a_n which satisfy ∑i=1nkaik(k−1)aik+1=∑i=1nai=n. \sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n .
inequalities
RMO KV P-5

Source:

10/29/2023
The side lengths a,b,ca,b,c of a triangle ABCABC are positive integers. Let:\\
Tn=(a+b+c)2n−(a−b+c)2n−(a+b−c)2n+(a−b−c)2nT_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}
for any positive integer nn. If T22T1=2023\frac{T_{2}}{2T_{1}}=2023 and a>b>ca>b>c , determine all possible perimeters of the triangle ABCABC.
algebraravi s substitutiongeometryperimeter