MathDB

1

Part of 2023 India Regional Mathematical Olympiad

Problems(2)

RMO 2023 P1

Source:

10/29/2023
Let N\mathbb{N} be the set of all positive integers and S={(a,b,c,d)∈N4:a2+b2+c2=d2}S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}. Find the largest positive integer mm such that mm divides abcd for all (a,b,c,d)∈S(a, b, c, d) \in S.
number theoryRMO 2023SquaresDivisibility
RMO KV 2023 P1

Source: RMO 2023 P1

10/29/2023
Given a triangle ABCABC with ∠ACB=120∘.\angle ACB = 120^{\circ}. A point LL is marked in the side ABAB such that CLCL bisects ∠ACB.\angle ACB. Points NN and KK are chosen in the sides ACAC and BCBC such that CK+CN=CL.CK+CN=CL. Prove that the triangle KLNKLN is equilateral.
geometry