MathDB

P7

Part of 2024 Bangladesh Mathematical Olympiad

Problems(2)

Bash? Anyone??

Source: BdMO 2024 Secondary National P7 Higher Secondary National P2

3/18/2024
Let ABCDABCD be a square. EE and FF lie on sides ABAB and BCBC, respectively, such that BE=BFBE = BF. The line perpendicular to CECE, which passes through BB, intersects CECE and ADAD at points GG and HH, respectively. The lines FHFH and CECE intersect at point PP and the lines GFGF and CDCD intersect at point QQ. Prove that the line DPDP is perpendicular to the line BQBQ.
geometryperpendicular linescartesian coordinates
Ceiling in FE

Source: BdMO 2024 Higher Secondary National P7

3/18/2024
Find all functions f:N→Nf:\mathbb{N} \to \mathbb{N} such thatf(⌈f(m)n⌉)=⌈mf(n)⌉f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceilfor all m,n∈Nm,n \in \mathbb{N}.
Proposed by Md. Ashraful Islam Fahim
functional equationceiling functionnumber theoryalgebra