A Problem (finally!)

by v_Enhance, Jun 26, 2010, 11:06 PM

I decided I need to actually put some problems on this blog to make it less trivial, so I'll start off with a semi-easy one :P

Problem 1. For two $x,y \in \mathbb C$, we have $x+y = 2N$ and $xy = 2N^2$. Compute $x^2+y^2$, and prove that $x,y \in \mathbb R$ if and only if $N=0$.
This post has been edited 2 times. Last edited by v_Enhance, Jun 27, 2010, 3:02 AM

Comment

1 Comment

The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
First part: $x^2+y^2=(x+y)^2-2xy=(2N)^2-2 \cdot 2N^2=4N^2-4N^2=\boxed{0}$.

Second part: First, we assume $N=0$ and try to prove that $x,y \in \mathbb{R}$. We know $xy=2N^2=0$, so either $x=0$ or $y=0$. If $x=0$, then $0+y=0$, so $y=0$. Similarly, if $y=0$, $x=0$. Thus, the only ordered pair that works is $(x,y)=(0,0)$, which is real, so we have proven the "if" part.

Next, we assume $x,y \in \mathbb{R}$. From above, $x^2+y^2=0$ and, by the Trivial Inequality, $x^2+y^2\ge 0$. Since we have equality, we know $x=y=0$. Thus, $2N=x+y=0$, so $N=0$. This completes the "only if" part and the proof. $\blacksquare$
This post has been edited 1 time. Last edited by PowerOfPi, Jun 27, 2010, 4:31 AM

by PowerOfPi, Jun 27, 2010, 4:19 AM

Archives
- April 2023
Announcing BOATIS
+ February 2023
If I make a post here does it show up in the blogroll?
+ January 2017
Instructions for Creating a Contest Instance
+ March 2015
New Olympiad Handouts Up
+ February 2015
Anyone want to help me proofread some new olympiad handouts?
+ December 2014
New Article on Chinese Remainder Theorem
Geo Book ETA is very roughly December 2015
+ November 2014
Isogonal Conjugates!
Publishing is hard
+ September 2014
Update on Olympiad Geometry Book
+ August 2014
Oly Combinatorics
+ July 2014
Combinatorics or vodka
Probabilistic Handout Done
MIT Website (including IMO discussion)
+ April 2014
Olympiad Geometry Book
+ March 2014
v_E vs. AIME #14
+ February 2014
2500 Posts!
+ May 2013
This Blog
+ April 2013
Modular Arithmetic
+ March 2013
New ELMO Selection Procedure
MOP vs RSI
Life
NIMO Winter Contest
+ February 2013
2^(n-1) + 3^(n-1) != 0 mod n
darkmafiabot
Dear google
Tank
+ January 2013
Side Projects
hmm NIMO
Complex determinants
+ December 2012
USA Contests Section
Congrats to the adMITted!
+ November 2012
Inequalities Forum
November NIMO in 5 hours!
The only decent CHMMC problem I saw this year
PUMaC
Number theory
SOS
Barying Pascal
+ October 2012
Why can I not find USAMO 2011 solutions
SOS article so far
Motivating a certain identity
SOS and Chinese Dumbass Notation
+ September 2012
Pretty Pictures
Permutations Problem
Sept NIMO Tomorrow at 5PM PDT
blargh
+ August 2012
It's Been A While
Irreducible Polynomial
Midpoints of Altitudes
Deja Vu
CGMO Ramblings
Amusing German Pre-TST Problem
Chinese Dumbass Notation
Powers of primitive roots
Algebra
+ July 2012
Cross Ratio and Bary
Titu's Lemma OP
Random post about life
Brokard
Yay I can still synthetic
MOP Post #5: Count the One-Liners
IMO 2012, Problem 1
MOP Post #4: Romance
MOP Post #3: An Excellent Solution
An amazingly strong inequality :O
A *Very* Strong Inequality
MOP Post #2: Singing Troupe
+ June 2012
MOP Post #1: Touring the AMC Office
Hello
Spam
Recursion
What an EXCELLENT problem
De-lighted to SIR!
Moderators
+ May 2012
I need better style
PROjective COrollaries
Almost Done
MathCounts
Featured!
ARML Homework
Green
Another book that you should read
Problems from the Book
+ April 2012
USAMO Asthetics
USAMO tomorrow
And there goes another one
Today marks an EXCELLENT day for geometry.
Bary Article Almost Done
The Calculus Manifesto
E Chen
USAMO Qualifiers
Mock SAT1 Math
+ March 2012
+1 Bary || AIME1 Stats are out
SHao + JHuang
10
BAMO Awards
A Poem
darn no apmo
CHMMC
+ February 2012
What I do in PE
SMT
Barybash
HMM
lol
gee
queue
q
g
A Troll Inequality
Group Theory
+ January 2012
WordPress
My new name
OMO 42: Ptolemy + Coordinates!
From the AMC FAQ
barybash
FRIKKIN
Guess what I learned today
+ December 2011
2011 lastpost
A One-Liner Inequality
Russia 2004
Vectors
My first competition problem
All you need to do is construct a parallelogram!
WOOT: Inversion
2007AIME = 1979 IMO Longlist
+ November 2011
Geometry is no longer my worst subject!
(p-1)! + 1
USAMO 2010.5
Flowchart: How to do geo
:sigh:
A Proj Geo Problem
TSTST Complaint Thread Got Revived
Fakesolving
Santa Cruz
Band
Diophantine Equation
+ October 2011
Ok I give up on this one
Things I need to learn
calc class optimization problems
Is it just me
PO1 QQ
Win Quote
10/01/11
+ September 2011
A nice (and harder) geo
Probability two integers are relatively prime
Floors
An easy geo
Honors Pre-Calc
An easy geo
Geometry
ISL
Self-troll
+ August 2011
Gabriel NT
Another ramble on teaching...
A nice problem
Random Inequality
FTW
starcraft
AMSP: NT3 Test
AMSP: Worst mafia ever
+ July 2011
UCSC AMSP tomorrow
Titu Andreescu
NT Exercise
Calc Exercise
hello
+ June 2011
JMO Certificate
ARML
Happy birthday
+ May 2011
Mafia
Darjn
A2
Handbooks
JMO Winner
congratulations
gratz~
MathPath
Yay!
rrusczyk
+ April 2011
n=91
My hopes for USAJMO
Python
Spring Break
Adobe
AMCDirector
Carnot
Darjn
Hoping for a hard JMO
SCVMA
Jensen
Purple Comet
WOOT Shirt
What I do when I'm sick
Nice LaTeX Tool
6 days...
Mock USA(J)MO
+ March 2011
Haxx Time
Compendium
If I make an April Fool's joke a few days later
Happiness
Mimicry
Jensen's Inequality
Dodgeball
FML
:(
From my Health Book
Finally
Education
OMG THE C(j)ONTRIBS
Wars
xkcd forums
Nightmare
59+ on USAMTS
usamo.org
385, 386
Quizzes
+ February 2011
The Wall
R(j)AWR
Inversion
Why is it that I
PE Class
LeechBlock
Just noticed
smt fml
February
High School
Doneosour
HMMT!
errr
Sarcasm
3-person mafia
+ January 2011
Finals
Proposal 42
Attempting to mod my first game...
Sarcasm
HMMT Practice
Old Posts
Warm-up Problems
MafiaBot
+ December 2010
Wow I suck at math
Traveling
R0meo + Juli3t - 0th Edition
KevinChen_Wang is a very interesting person
riemannHypothesis2.71828
Romeo and Juliet- 1337
|-|0\|/ /|\4|\|j |*30|*|_3 |=/-\||_3|) +|-|15 |*|^08|_3/|\?
How to grade finals
If anyone still uses Python...
Advance Notice
I found the next 3 years!!!
Cold weather
Mathleague + USAMTS + WOOT
Boo
WOOT PO3
This may not be welcome, but......
Weather
Wason Selection Task
Ohwow
+ November 2010
Why
Weather
Reaper
USAMTS Round 1
Um....
Mandelbrot
Repetition
SantaClara Math Contest
Facebook
I find this interesting
what...
Google and darjn
.
MathLeague
Math grade
Lol post
1984
+ October 2010
That problem is too trivial.
Trivial Problem
Problem
Just wow.
School Geo
Logarithm Worksheet
(Really) Random Problem
Lagrange FAIL
FTW
Lagrange Multipliers
Research papers
AoPS Books
Why
NCCQM
Today I learned
Is it just me
darjn FTW
Ever seen a thread where 100% of the posts in it are fail?
USAMTS is out!
Homework
Artistic Inablity
I
+ September 2010
Homework
Speedreaper
Blasphemy, Part 2
Advisory
Blasphemy
Boo...
Asymptote
Transitive Property of Location + Extension
Today at school something incredible happened.
Coaching
USAMTS Forum
The biggest obstacle to the high school mathematician...
Atheism
USAMTS
Dreaming
Really annoying blog descriptions
An Actual Problem
A List
+ August 2010
Whoa
Avatar
2499 Blog Visits!
Randomosity
WOOT
Mafia
whee
MATH
P$\neq$NP
AWESOME WEBSITE
1337
Meh
+ July 2010
whee
fail
lolz
LOLL levans
I think
a transcript which already has been posted at polka's blog...
oooh
Randomosity
Finally read the IMO results
Yay MathPath!
FTW Chat
Whee
Mafia!
Conway
I lost the game
+ June 2010
Yay
levans
MathPath Compies
A Problem...
Another random problem
Yay MathPath!
hm
A Problem (finally!)
Thanks Phire
This is going to be one interesting year....
Yay!
I like how...
Remodel!
darjn
Eugenex Chenx
Yay Solitaire
Now that you know the Euclidean Algorithm...
W00T!!
Wheee summer and miscellany
Almost Done
Yawns
Eh fail?
I want a refund
Yay for CSS Pichus!
Hi World!
ARML 2010 and other miscellany
+ May 2010
SFBA ARML
Holy Cow
Percentages
Improvement!
+ April 2010
This is trivial. QED
So why exactly is this spam...
USAMTS Scores
USAJMO Qualification!!
+ March 2010
Do you have Trubey?
Failing School
State MC Fail
CA MathCounts!!!
2010 AIME I Fail
Oh Noes AIME
First Post
Shouts
Submit

  • no more school for a while :O

    by SamuelWang, Apr 16, 2025, 7:49 PM

  • first shout of 2026

    by Yiyj1, Apr 11, 2025, 2:10 AM

  • first shout in 2025

    by Captainscrubz, Jan 28, 2025, 5:32 PM

  • 289322 is crazy

    by Tem8, Nov 20, 2024, 3:13 AM

  • orz evan :orz: thx for diamond

    by Likeminded2017, Nov 17, 2024, 5:44 AM

  • a relic of the past

    by Clew28, Jul 25, 2024, 8:55 PM

  • why'd you look at this? You leave me no choice...

    by aidending, Jul 14, 2024, 11:49 PM

  • hi evan thanks for the diamond

    by Scilyse, Feb 8, 2024, 10:16 PM

  • woa hi evan

    by crazyeyemoody907, Aug 20, 2023, 9:14 PM

  • just google "nontrivial progress" for this blog! :D

    by gracemoon124, Jul 29, 2023, 10:40 PM

  • HI GUYS!

    by unknown12, Apr 16, 2023, 3:04 AM

  • isn't the shoutbox a bit too big?

    by cinnamon_e, Apr 1, 2023, 7:56 PM

  • Omg market moment

    by GrantStar, Mar 19, 2023, 12:50 AM

  • no $ $ $ $

    by v4913, Mar 18, 2023, 9:56 PM

  • do shouts bring this up the blogroll?

    by vsamc, Mar 4, 2023, 4:46 PM

181 shouts
Contributors
AIME15 • AwesomeToad • dragon96 • hrithikguy • phiReKaLk6781 • PowerOfPi • pythag011 • turak • v_Enhance
About Owner
  • Posts: 6876
  • Joined: Dec 30, 2008
Blog Stats
  • Blog created: Mar 13, 2010
  • Total entries: 383
  • Total visits: 293299
  • Total comments: 1317
Search Blog
a