MathDB

2

Part of 1992 Vietnam National Olympiad

Problems(2)

last digit of divisor

Source: 30-th Vietnamese Mathematical Olympiad 1992

2/17/2007
For any positive integer aa, denote f(a)=∣{b∈N∣b∣af(a)=|\{b\in\mathbb{N}| b|a and\text{and} bmod  10∈{1,9}}∣b\mod{10}\in\{1,9\}\}| and g(a)=∣{b∈N∣b∣ag(a)=|\{b\in\mathbb{N}| b|a and\text{and} bmod  10∈{3,7}}∣b\mod{10}\in\{3,7\}\}|. Prove that f(a)≥g(a)∀a∈Nf(a)\geq g(a)\forall a\in\mathbb{N}.
number theory unsolvednumber theory
rotation rectangle

Source: 30-th Vietnamese Mathematical Olympiad 1992

2/17/2007
Let HH be a rectangle with angle between two diagonal ≤450\leq 45^{0}. Rotation HH around the its center with angle 00≤x≤36000^{0}\leq x\leq 360^{0} we have rectangle HxH_{x}. Find xx such that [H∩Hx][H\cap H_{x}] minimum, where [S][S] is area of SS.
geometrygeometric transformationrotationrectangleinequalities unsolvedinequalities