Problems from the Bus
by djmathman, May 23, 2013, 10:56 PM
And no, I'm not making that up. This probably needs a bit of explanation.
Today, all of the Physics students were invited to a "Physics Day" trip to Six Flags (since Physics is not a required science and is taught in 12th grade normally if one does not double up). Being a student in AP Calculus, the invitation was extended to me, and so I went.
And there was a chance of rain.
So to prepare for any long indoor sessions from rain I "created" a "Practice Olympiad" of three questions to work on. I solved two of the problems on the bus, and it turns out that no rain affected the amusement park until we were about to leave.
Here are the two questions and their solutions:
Today, all of the Physics students were invited to a "Physics Day" trip to Six Flags (since Physics is not a required science and is taught in 12th grade normally if one does not double up). Being a student in AP Calculus, the invitation was extended to me, and so I went.
And there was a chance of rain.
So to prepare for any long indoor sessions from rain I "created" a "Practice Olympiad" of three questions to work on. I solved two of the problems on the bus, and it turns out that no rain affected the amusement park until we were about to leave.
Here are the two questions and their solutions:
All-Russian MO 2003 Grade 9 #6 wrote:
Let 
be positive numbers with the sum 
. Prove the inequality
![\[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.\]](//latex.artofproblemsolving.com/c/6/f/c6fce19452f41bc112788355e1679c5b277e8e07.png)
be positive numbers with the sum 
. Prove the inequality![\[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.\]](http://latex.artofproblemsolving.com/c/6/f/c6fce19452f41bc112788355e1679c5b277e8e07.png)
![\[\sum\left(\dfrac1{1-a}-\dfrac2{1+a}\right)=\sum \dfrac{3a-1}{1-a^2}\geq 0.\]](http://latex.artofproblemsolving.com/d/a/6/da66303f6824670117724bc407c809f6da74731f.png)
By the Engel form of Cauchy-Schwartz, we have
Sharygin 2009 #14 wrote:
Given triangle 
of area 1. Let 
be the perpendicular from 
to the bisector of angle 
. Determine the area of triangle 
.
of area 1. Let 
be the perpendicular from 
to the bisector of angle 
. Determine the area of triangle 
.
and 
.![$[ABC]=1$](http://latex.artofproblemsolving.com/4/a/0/4a08c4c1431ba612a7f5f04de67b275518a068b2.png)
we have 
so 
.
,
.This post has been edited 2 times. Last edited by djmathman, Sep 5, 2013, 1:33 AM
example does not hold since it was given that 
are all positive.
gives:
,
April 2025
November 2024