MathDB

2

Part of 2005 India IMO Training Camp

Problems(3)

Prove that one can find...

Source: Indian IMOTC 2005 Day 1 Problem 2

9/17/2005
Prove that one can find a n0∈Nn_{0} \in \mathbb{N} such that ∀m≥n0\forall m \geq n_{0}, there exist three positive integers aa, bb , cc such that (i) m3<a<b<c<(m+1)3m^3 < a < b < c < (m+1)^3; (ii) abcabc is the cube of an integer.
geometry3D geometrynumber theory solvednumber theory
Prove that there exist integers st....

Source: Indian IMOTC 2005 Day 4 Problem 2

9/23/2005
Given real numbers a,α,β,σ and ϱa,\alpha,\beta, \sigma \ and \ \varrho s.t. σ,ϱ>0\sigma, \varrho > 0 and σϱ=116\sigma \varrho = \frac{1}{16}, prove that there exist integers xx and yy s.t. −σ≤(x+α(ax+y+β)≤ϱ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho
number theory unsolvednumber theory
Phi(n)

Source: Indian IMOTC 2005 Day 5 Problem 2

9/23/2005
Determine all positive integers n>2n > 2 , such that 12φ(n)≡1( mod 6) \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6)
number theory solvednumber theory