MathDB

1

Part of 2005 Iran MO (3rd Round)

Problems(5)

a,b,c

Source: Iran 2005

8/27/2005
Suppose a,b,c∈R+a,b,c\in \mathbb R^+. Prove that :(ab+bc+ca)2≥(a+b+c)(1a+1b+1c)\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)
inequalities3-variable inequalitycyclic inequalitycauchy schwarz
Locus

Source: Iran 2005

8/27/2005
From each vertex of triangle ABCABC we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them ℓ1\ell_1, ℓ2\ell_2 and ℓ3\ell_3. a) Prove that ℓ1\ell_1, ℓ2\ell_2 and ℓ3\ell_3 are concurrent at a point PP. b) Find the locus of PP as we move the 3 arbitary lines.
geometryrectanglegeometry proposed
n,p,q

Source: Iran 2005

8/29/2005
Find all n,p,q∈Nn,p,q\in \mathbb N that:2n+n2=3p7q2^n+n^2=3^p7^q
quadraticsnumber theory proposednumber theory
CN functions

Source: Iran 2005

9/1/2005
We call the set A∈RnA\in \mathbb R^n CN if and only if for every continuous f:A→Af:A\to A there exists some x∈Ax\in A such that f(x)=xf(x)=x. a) Example: We know that A={x∈Rn∣∣x∣≤1}A = \{ x\in\mathbb R^n | |x|\leq 1 \} is CN. b) The circle is not CN.
Which one of these sets are CN?
1) A={x∈R3∣∣x∣=1}A=\{x\in\mathbb R^3| |x|=1\}
2) The cross {(x,y)∈R2∣xy=0, ∣x∣+∣y∣≤1}\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}
3) Graph of the function f:[0,1]→Rf:[0,1]\to \mathbb R defined by f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0
functiontrigonometrygeometry3D geometrysphereanalytic geometrytopology
northeast

Source: Iran 2005

9/21/2005
An airplane wants to go from a point on the equator, and at each moment it will go to the northeast with speed vv. Suppose the radius of earth is RR. a) Will the airplane reach to the north pole? If yes how long it will take to reach the north pole? b) Will the airplne rotate finitely many times around the north pole? If yes how many times?
rotationlimitgeometry proposedgeometry