2 Equalities in Different Veins

by agbdmrbirdyface, Aug 25, 2016, 12:37 AM

Both of these problems were problems I solved way back in spring for a math camp during spring break (we don't question why).
Areteem Institute, Algebraic Transformations wrote:
Given that $abcd = 1$, show that
$\frac{a}{abc + ab + a + 1} + \frac{b}{bcd + bc + b + 1} + \frac{c}{cda + cd + c + 1} + \frac{d}{dab + da + d + 1} = 1$

Solution:
Areteem Institute, Algebraic Transformations wrote:
Given that $1991x^3 = 1992y^3 = 1993z^3 $ and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$, prove that $\sqrt[3]{1991} + \sqrt[3]{1992} + \sqrt[3]{1993} = \sqrt[3]{1991x^2 + 1992y^2 + 1993z^2}$

Solution:

Comment

5 Comments

The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Wait that was pr0

Just like you
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:11 AM

by skipiano, Aug 25, 2016, 1:08 AM

The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Oh wait these problems

I think I saw them among others when I went to MathZoom back in 2012!

Good to see camps are recycling problems!
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:13 AM

by djmathman, Aug 25, 2016, 1:20 AM

The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wait MathZoom/Areteem Institute had a location in my area this year for spring break
so I took advantage of it and I went for a week
it was pretty decent

Nice
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:14 AM

by agbdmrbirdyface, Aug 25, 2016, 2:37 AM

The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sorry for double posting but
Motivation

Nice
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:15 AM

by agbdmrbirdyface, Aug 25, 2016, 2:40 AM

The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
these are pretty problems

3pretty5you
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 5:03 PM

by cjquines0, Aug 25, 2016, 3:08 PM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

avatar

shiningsunnyday
Archives
- May 2017
Revival?
+ April 2017
Revisiting a September 2016 Problem, Part II
Revisiting a September 2016 Problem
Older ISL Combo (collab with Champion999)
Birthday power
Yay for inequality
Whoever wrote this... doesn't know what concise means :\
Bary Bash (oops)
Slick combinatorial-ish argument
Silly geo
Complex Bash (oops)
Nice radical axis argument
Bounding perfect squares
Chasing cute angles
Power and with respect
Pascal's... triangle?
Easy NT Easy Life
MOPing some floors
+ March 2017
Easiest USAMO 2/5?
shiningsunnyday wants me to post
A cute quadratic
Power of the WOOT shirt?
G_U's set bounding?
N2 by T2?
Redemption
+ February 2017
Where did shiningsunnyday go?
+ January 2017
Functional equation, dysfunctional mind?
The Legend of the IMO Problem 6
AMC 12 Lockdown Training [Day 3]
AMC 12 Lockdown Training [Day 2]
AMC 12 Lockdown Training [Day 1]
Bored during English class
making this blog great again, 111 problem at a time
Combin$ $ $ $ $ $
Some Combinatorics
Conjugate Pairs in Polynomials
Why do people hate Trig?
If dj likes it, it's gotta be good (and it was!)
Divisors at the round table
It's 2017!
Two different flavored algebra problems
+ December 2016
Ineq!⠀⠀⠀
Geo!⠀⠀⠀⠀
LOOK WHAT GEO GEM I FOUND
Inequalities from 109 (ft. nukes and bashing)
An challenge for anyone willing to try
Deck the Halls
Irrational guys trying to be rational
Sesame Street Christmas
Cute algebra manipulation
Back to NT - fun!
EVEN though this problem may seem ODD at first
Easier (but nice) geo
easy geo
First USAMO Combo P4
Geometry When I was Bored
Nice "Combinatorics"
Dealing with a wacky condition
Right Angle on Intouch Chord
Radical Axis and Three Tangents Lemma Applied
More combinatorics of sets fun
Slick problem that can be solved with posets
Second favorite config of all time
3 Tangents Lemma Today
Pre-Winter Geometry Problem
How to fe?
Midpoints and the Incircle Meet in EGMO
Configuration Recognition
Midnight fun(ctional equations)
Geometry is too hard
Here's to a prolific winter-break USAMO marathon
Lifting me HIGHER!
Simson says... go do geo!
I like the NIMO Summer Contest
MODerately easy problem to cool down an insane 10 days
AIME Combinatorics
Telescopes, again
I Always Get Away with Trial and Error
Finite Differences
Easy as... 3.14?
+ November 2016
Two Problems from the 2015 USA TST + CODE GEASS SEASON THREE
Dear SSD, please work on graph theory
Thanksgiving PIE?
The Return of Combinatorial ... something idk
Oh sweet 16
Vlad is back
Geo for girls
Combinatorial whuuu...?
Djmathman's 100 Geometry Problems
Directed graphs
Redemption
Michael likes cycles
Nobody took Modulo 8 :(
Thirteenth
Useful graph theory lemma
End of killer week cool down problem
SOS! ssd has kidnapped me
Traditional Filipino cuisine
Nice and Difficult FE
Mein gott what have you done America
Mein gott what have you done to Malay
Graduation to Chinese TST Problems
Simson Lines Proved One Way
The Infamous Problem 6, No Vieta Jumping
Mein gott what have you done to my prime
Mein gott what have you done to my diagram
Affirmative Action
Let's get the party started
Wooohoooo
omg i haven't done geo in a month
Mein gott what have you done to my dominoes
Mein gott what have you done to my coat
+ October 2016
Geometry before Halloween
Let me study those curves of yours
A sequence of 0s and 1s
“Angle Chasing”
very nice "geo" for ssd
NT's and FE's: my living nightmares
Easy identity
Three weird equalities
Kicking the IMO P1 while it's down
Water polo is kind of great
Cute little NT trick
Lovely summation
Entering APMOde
A medley of lovely NT ideas!
Bullying an innocent IMO P1
+ September 2016
Bary earned First Blood! +2014 XP!
Very 'pell' liang de problem
I did it
Whoever guesses the pun gets a prize
Equalities > Inequalities
Since I got tired of sucking at non-geo
My very first
Exponential inequality
trig-gered
The problem I solved in bed
Water polo is great
[MOM] What I do home alone on a Sunday night...
[MOM] 2 cute problems to add to your AIME arsenal of tricks
geo-geo-geoooo
geo back2back
easy geo
How does 11:06 A.M. => 9:35 P.M. imply "2 A.M." both times
Darn 2 AM Number Theory
+ August 2016
Two-value polynomial interpolation!?
A beautifully disgusting problem
Phi of Stuff is Power of Two
The Fat Problem
A(n) (Easy?) USAMO Problem
1994 IMO P1
2 Equalities in Different Veins
2 inequalities of the same vein
Really nice concurrency trick
That China TST Problem
You're the CSS to my HTML
Combo loves me I love combo
My Favorite Problem from AMSP
Inverting into a Trash Can
Filler post
I like algebra because it's the only bra I can ever unlock..
The Great Aviary Chandelier
T(NT) is da bomb
Sets of four reals
More Harmonies
Geographic Graph Theory
ISL 2007 N3
Let's all harmonize to math!
+ July 2016
Contrib NT
Contributor's Geo~ Go Big or Go Home
Contrib Alg
Contrib Combo [LAMPS]
Baby pic!
Yay first TSTST problem
Doing Euler with my ruler!
Set up please
+ June 2016
Two really "cyc" problems
Wait barn
Are you a rational thinker?
Get riddle the boring stuff!
Gee? Woah... much intrigue
Back-to-back USAMO and IMO problems in one day!
I'm out of puns
Oh we meet again sweet Aime
What lingerie does a mermaid wear?
Wait farm I'm dumb
Life is complex...
I can't suck at NT forever...
A "Triggy" (get it?) Problem
Think outside the square!
Mods on Monday
2010 is a good year
Welcome!
Shouts
Submit

  • this guy is an absolute legend. much love wherever you are Michael

    by LeonidasTheConquerer, Aug 5, 2024, 9:37 PM

  • amazing blog

    by anurag27826, Jun 17, 2023, 7:20 AM

  • hi i randomly found this

    by purplepenguin2, Mar 1, 2023, 8:43 AM

  • can i be a contributor please?

    by cinnamon_e, Mar 10, 2022, 6:58 PM

  • orzorzorzorzorzorozo

    by samrocksnature, Jul 16, 2021, 8:25 PM

  • 2021 post

    by the_mathmagician, May 5, 2021, 3:28 PM

  • Let $ ABC$ be an equilateral triangle of side length $ 1$. Let $ D$ be the point such that $ C$ is the midpoint of $ BD$, and let $ I$ be the incenter of triangle $ ACD$. Let $ E$ be the point on line $ AB$ such that $ DE$ and $ BI$ are perpendicular. $ \

    by ARay10, Aug 25, 2020, 5:55 PM

  • Nice blog! :)

    by User526797, Jan 12, 2020, 4:48 PM

  • oh my gosh it's been so longggggg.... contrib? what does that mean?

    by adiarasel, Dec 1, 2019, 8:31 PM

  • 2019 post

    by piphi, Aug 10, 2019, 6:32 AM

  • hi contrib please

    by Emathmaster, Dec 27, 2018, 5:38 PM

  • hihihihihi contrib plzzzzz

    by haha0201, Aug 20, 2018, 3:58 PM

  • contrib please

    by Max0815, Aug 1, 2018, 12:35 AM

  • contrib /charmander

    by mathmaster2000, Apr 16, 2017, 4:59 PM

  • for contrib

    by SomethingNeutral, Mar 30, 2017, 7:57 PM

270 shouts
Tags
geometry
algebra
combinatorics
number theory
USAMO
induction
Solved
Angle Chasing
IMO
Inequality
AMSP
Cyclic Quadrilaterals
Manipulation
modular arithmetic
similar triangles
EGMO
Projective
Combinatorial Number Theory
functional equation
105
AMC 12
APMO
enumerative
graph theory
homothety
Log
Orders and Primitive Roots
radical axis
Romania TST
Sequence
Sets
trigonometry
104
110
area
common configurations
concurrency
Contradiction
diophantine
Divisibility
ISL
not geometry
Polynomials
Tstst
USAJMO
102
108
ACoPS
AIME
algorithms
bash
basic tools
BDSM
bijection
Bounding
collinearity
combinatorial geometry
combo breaker
complex numbers
configurations
Crt
Elmo
Euler's
expected value
Exponents
geo 3
geometric inequality
Lemmas
logic
mandelbrot
nt
nullstellensatz
pie
poll
PSFTB
rational and irrational
unsolved
USA TST
weird quadrilateral names
WOOT
About Owner
  • Posts: 1350
  • Joined: Dec 19, 2014
Blog Stats
  • Blog created: Jun 11, 2016
  • Total entries: 193
  • Total visits: 30959
  • Total comments: 579
Search Blog