MathDB

P7

Part of 2023 Bangladesh Mathematical Olympiad

Problems(2)

3^m*2^n representation

Source: BdMO 2023 Secondary National P7

2/12/2023
Prove that every positive integer can be represented in the form 3m1⋅2n1+3m2⋅2n2+⋯+3mk⋅2nk3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k} where m1>m2>⋯>mk≥0m_1 > m_2 > \dots > m_k \geq 0 and 0≤n1<n2<⋯<nk0 \leq n_1 < n_2 < \dots < n_k are integers.
number theorystrong induction
Reflection of Perpendicular Bisector

Source: BdMO 2023 Higher Secondary National P7

2/12/2023
Let ΔABC\Delta ABC be an acute triangle and ω\omega be its circumcircle. Perpendicular from AA to BCBC intersects BCBC at DD and ω\omega at KK. Circle through AA, DD and tangent to BCBC at DD intersect ω\omega at EE. AEAE intersects BCBC at TT. TKTK intersects ω\omega at SS. Assume, SDSD intersects ω\omega at XX. Prove that XX is the reflection of AA with respect to the perpendicular bisector of BCBC.
geometryreflectionpower of a pointperpendicular bisector