MathDB

8

Part of 1997 Ukraine National Mathematical Olympiad

Problems(3)

easy inequality

Source: Ukraine 1997 grade 9

7/21/2009
For every natural number n n prove the inequality: \sqrt{2\plus{}\sqrt{2\plus{}\sqrt{2\plus{}...\plus{}\sqrt{2}}}}\plus{}\sqrt{6\plus{}\sqrt{6\plus{}\sqrt{6\plus{}...\plus{}\sqrt{6}}}}<5 where there are n n twos and n n sixs on the left hand side.
inequalitiesinductioninequalities unsolved
find k

Source: Ukraine 1997 grade 10

7/21/2009
Let d(n) d(n) denote the largest odd divisor of a positive integer n n. The function f:N→N f: \mathbb{N} \rightarrow \mathbb{N} is defined by f(2n\minus{}1)\equal{}2^n and f(2n)\equal{}n\plus{}\frac{2n}{d(n)} for all n∈N n \in \mathbb{N}. Find all natural numbers k k such that: f(f(...f(1)...))\equal{}1997. (where the paranthesis appear k k times)
functionalgebra unsolvedalgebra
solid geometry

Source: Ukraine 1997 grade 11

7/22/2009
On the edges AB,BC,CD,DA AB,BC,CD,DA of a parallelepiped ABCDA1B1C1D1 ABCDA_1 B_1 C_1 D_1 points K,L,M,N K,L,M,N are selected, respectively. Prove that the circumcenters of the tetrahedra A1AKN,B1BKL,C1CLM,D1DMN A_1 AKN, B_1 BKL, C_1 CLM, D_1 DMN are vertices of a parallelogram.
geometrycircumcircleparallelogrammodular arithmetic3D geometryspheregeometry proposed