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Calculating combinatorial numbers
lgx57   0
11 minutes ago
Try to simplify this expression:

$$\sum_{i=1}^n \sum_{j=1}^i C_{n}^i C_{n}^j$$
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lgx57
11 minutes ago
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Find (a+1/a)(b+1/b)(c+1/c)
speck   6
N Jan 19, 2016 by Virgil Nicula
Let ${a,b,c}$ be roots of the polynomial $P(x) = x^3 - 2x^2 + 3x - 4$. Determine $\left( a + \dfrac{1}{a} \right) \left(b + \dfrac{1}{b} \right) \left( c + \dfrac{1}{c} \right)$.

Bonus: Find a polynomial with integer coefficients with roots ${ \left(a + \dfrac{1}{a} \right), \left( b + \dfrac{1}{b} \right), \left( c + \dfrac{1}{c} \right) }$.

I have a nice solution in mind, but I'm not sure if there's an easier way.
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speck
Jan 17, 2016
Virgil Nicula
Jan 19, 2016
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Find (a+1/a)(b+1/b)(c+1/c)
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speck
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speck
#1 h  Jan 17, 2016, 4:06 PM • 2 Y
Y by Adventure10, Mango247
Let ${a,b,c}$ be roots of the polynomial $P(x) = x^3 - 2x^2 + 3x - 4$. Determine $\left( a + \dfrac{1}{a} \right) \left(b + \dfrac{1}{b} \right) \left( c + \dfrac{1}{c} \right)$.

Bonus: Find a polynomial with integer coefficients with roots ${ \left(a + \dfrac{1}{a} \right), \left( b + \dfrac{1}{b} \right), \left( c + \dfrac{1}{c} \right) }$.

I have a nice solution in mind, but I'm not sure if there's an easier way.
This post has been edited 1 time. Last edited by speck, Jan 17, 2016, 4:16 PM
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AopsIsCool
143 posts
AopsIsCool
#2 h  Jan 17, 2016, 4:32 PM • 1 Y
Y by Adventure10
This can be rewritten as Click to reveal hidden text.
Does this help? Now use vieta's.
Edit: 42nd post!
This post has been edited 2 times. Last edited by AopsIsCool, Jan 17, 2016, 4:33 PM
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trumpeter
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trumpeter
#3 h  Jan 17, 2016, 8:03 PM • 9 Y
Y by Generic_Username, ythomashu, champion999, HVishy, heidimiha, spartan168, dhusb45, Adventure10, Mango247
Solution
This post has been edited 1 time. Last edited by trumpeter, Jan 17, 2016, 10:40 PM
Reason: formatted better
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speck
1727 posts
speck
#4 h  Jan 17, 2016, 10:28 PM • 3 Y
Y by champion999, Adventure10, Mango247
trumpeter wrote:
Solution

Hmm, that's actually a lot nicer than my solution.
Thanks!
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Virgil Nicula
7054 posts
Virgil Nicula
#6 h  Jan 19, 2016, 8:41 AM • 2 Y
Y by Adventure10, Mango247
See PP6 from here
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AopsIsCool
143 posts
AopsIsCool
#7 h  Jan 19, 2016, 1:17 PM • 2 Y
Y by Adventure10, Mango247
alternatively
This post has been edited 1 time. Last edited by AopsIsCool, Jan 19, 2016, 1:19 PM
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Virgil Nicula
7054 posts
Virgil Nicula
#8 h  Jan 19, 2016, 3:53 PM • 1 Y
Y by Adventure10
See the first proof of PP6 from here
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