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Algebra Pre-Olympiad Marathon Page 7 of 11 [ 212 posts ]
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Message Apr 27, 2012, 6:24 pm • # 121
admin25 solution
new problem
Introduction to Algebr a Introduction A complete Algebra I course, with many additional topics from Algebra II. Over 1000 problems! Great for MATHCOUNTS and AMC preparation.
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I am not an a ctual admin. admin. There's been some confusion about this, so don't ask me to unban unba n you. Please. "Laziness is nothing more more than the act of resting before you get tired." ~Anonymous ~Anonymous
Apr 27, 2012, 9:18 pm • # 122
pr0likethis pr0liketh is
Beast Academy The first books in our new elementary school math curriculum are now available!
solution From squaring (1):
Posts: 78 Posts: 785 5
Call this this (4)
From squaring (2):
Call this (5)
From squaring (4):
Call this (6)
Using (1), (4), and (6), we can construct a cubic with roots
Using rational roots theo rem and synthetic division division (Or another method that p leases you), we see
The solutions to So we find find that
is a solution
are are all permutations of
New problem
______ ______ _____ "Math is not about following directions, it's about making new directions" -Lockhart
Apr 28, 2012, 5:34 am • # 123
GillesE
Posts: 2 Posts: 2 Location: Near Location: Near 1+1i+1 1+1i+1 +1l
solution Hidden Text
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AoPS Forum - Algebra Pre-Olympiad Marathon • Art of Problem Solving We have:
Furthermore:
(1) (2) (3)
And so:
(4) (5)
Where
and
Adding (4) and (5) yields:
Substituting back yields:
Therefor the answ er is
.
new problem: new
______ ______ _____ Noli turbare circulos meos (Archimedes of Syracus)
Apr 29, 2012, 12:18 pm • # 124
GillesE
Posts: 2 Location: Near 1+1i+1j+1l
Sorry, I just res olved the problem and I d on't think it's an entirely algebraic problem. So I propose a different problem:
new problem
______ ______ _____ Noli turbare circulos meos (Archimedes of Syracus)
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AoPS Forum - Algebra Pre-Olympiad Marathon • Art of Problem Solving Apr 29, 2012, 8:33 pm • # 125
Carolstar9 Let
be a divisor of
. So,
is a divisor too.
Let the number of divisors be Posts: 1008 Location: Why don't you guess?
and G.M be
So
So
So
May 02, 2012, 11:44 am • # 126
gstream
Posts: 48
@Carolstar9: A new p roblem perhaps?
May 03, 2012, 3:00 am • # 127
Carolstar9
I don't have any suitable problem right now. So others, feel free to p ost one .
Posts: 1008 Location: Why don't you guess?
May 03, 2012, 4:45 am • # 128
Faustus New Problem
______ ______ _____ Posts: 1219 Location: north carolina
What is mind? Doesn't matter What is matter? Never mind
May 03, 2012, 6:22 am • # 129
gstream
Posts: 48
solution
new problem
May 04, 2012, 11:54 am • # 130
Virgil Nicula
Quote: PP.
If
are the roots of
, then compute
.
Posts: 6956 Location: Bradenton, FL Blog: View Blog
Proof 1. Eliminate
betw een the relations
and obtain
,
i.e. the equation Observe that
Proof 2. Observe that for an
with the roots
.
.
we have
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and
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AoPS Forum - Algebra Pre-Olympiad Marathon • Art of Problem Solving .
Equation with roots
is
, where
. In
conclusion,
.
2 thanks
May 07, 2012, 11:17 am • # 131
gstream
Posts: 48
Is there nobody w ho is w illing to solve my problem? Should I post the s olution?
May 07, 2012, 1:18 pm • # 132
swkratis
Posts: 37
I think that it is very ea sy for this marathon.
May 07, 2012, 2:23 pm • # 133
gstream
Posts: 48
You are free to post a new one, after having solved this s imple problem. But remember, it's only pre-olympiad stuff, so not so difficult. But I have to say that my problem is indeed not the most difficult I could imagine
May 07, 2012, 5:05 pm • # 134
Virgil Nicula PP. Find all the real values of
Proof. Observe that
in terms of
and
Posts: 6956 Location: Bradenton, FL
s uch that
.
and
. Appear two case :
Blog: View Blog
in this case
.
in this case .
In conclusion,
.
May 16, 2012, 10:20 pm • # 135
narto928
Posts: 132 Location: Dead
Solve the equation
in terms of
.
May 16, 2012, 11:23 pm • # 136
youarebad My Solution
New Problem
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May 17, 2012, 12:18 am • # 137
Faustus Take the smallest no. ,
Thus, Posts: 1219 Location: north carolina
times. Mode is . Now take
's such that
.
New Prob lem Peace.
Faustus.
______ ______ _____ What is mind? Doesn't matter What is matter? Never mind
May 17, 2012, 6:15 am • # 138
Markomak1 solution new problem-Sequence
In case the s econd problem is not suitable for this section, pleas e let me know and i will change it, thanks! Posts: 512 Location: Skopje
______ ______ _____ Marko Karbevski(18), Maths rookie.My facebook: http://www.facebook.com/markokarbevski feel free to add me Please rate my posts(if they are so lutions or tips) and if you wish, please s end constru ctive criticism. Thank you!
Jun 08, 2012, 2:01 am • # 139
Faustus What is
in the problem?
Peace.
Posts: 1219 Location: north carolina
Faustus
______ ______ _____ What is mind? Doesn't matter What is matter? Never mind
Jul 20, 2012, 6:09 am • # 140
abhishek_sinha
Posts: 18 Location: IISc, Bangalore
I think the correct recursion is
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