Question

We are all lazy in this time of the day, lets try this one:

Find the number of quadratic polynomials \( ax^2 + bx + c \) such that:
a) \( a, b, c \) are distinct.
b) \(a, b, c \in \{1, 2, 3, ...2008\} \)
c) \( (x + 1)\) divides \( ax^2 + bx + c \)

PS:Just another problem uploaded, on the request of MR.Math in chat.

Answer 1

random guess. 2015028

Answer 2

Nopes :-(

Answer 3

It's the set of all solutions of \(c=b-a\) such that \(b\ne c \ne a\), \(a,b,c \in \{1,2,3,\dots, 2008\}\).

Answer 4

2014024? basically trying to count what Mr.Math posted.

Answer 5

Yes, joe you got that right, but your approach was very close too, you just needed to subtract few cases for b is even.

Answer 6

i was thinking of it like, a+c = b, so the question is really, have many pairs of distinct numbers can you pick from 1,...,2008, where their sum is also within 2008.

Answer 7

The answer you gave joe is the sum of 1 + 2 + 3 + 4 + ... + 2007

Answer 8

earlier*

Answer 9

yeah, i over counted in my first answer. I was counting pairs like (1004,1004), (1005,1005), etc.

Answer 10

yes, when b is even, there will be case of \(a = c\). We need to remove such cases as \( a \neq b \neq c\) Such cases = 2008/2 = 1004

so the answer becomes:2015028 - 1004 = 2014024

Answer 11

cool problem, now i can get back to cleaning lol. I saw the problem and was like, "its going to bug me if I dont try to asnwer it <.<" haha

Answer 12

lol, yeah just for fun :D

Answer 13

\[2(2)+2(4)+2(6)+\cdots+2(2006)=2\sum_{n=1}^{1003}2n=2014024.\]

Answer 14

I did have the answer before you posted it, but the page was freezing.

Answer 15

Let me explain Mr.Math's strategy, a + c = b now b = 3 gives (2, 1), (1, 2).
b = 4 gives (3, 1), (1, 3) so again 2 solution like this ..

so summing we get 2+2+4+4+· · ·+2006+2006 = 2·1003·1004 = 2014024

Answer 16

*n is odd*

Answer 17

Basically, what I did was this:

b a c
1 - -
2 - -
3 1,2 2,1
4 1,3 3,1
5 1,2,3,4 4,3,2,1
6 1,2,4,5 5,4,2,1
We can see that for b=3 we have 2 solutions, same as b=4. For b=5 and b=6, we had 4 solutions. For b=7 and b=8, there will be 6 solutions (under the given constrains). We can use this pattern to formulate the formula I wrote above that gave us the number of quadratic polynomials that satisfy the given conditions.

Answer 18

Cases go from 3 to 2008.

3 = 1 case
4 = 1 + 3, 3 + 1 = 2 cases
5 = 2 + 3, 4 + 1, 3 + 2, 4 + 1 = 4 cases
6 = 4 + 2, 2 + 4, 5 + 1, 1 + 5, = 4 cases
7 = 4 + 3, 3 + 4, 5 + 2, 2 + 5, 1 + 6, 6 + 1 = 6 cases

So odd numbers have n-1, even numbers have n-2

Series to use:

3 - 1 + 4-2 + 5 - 1 + 6 - 2 + ... + 2007 - 1 + 2008 - 2

2 + 2 + 4 + 4 + ... + 2006 + 2006

There are 2006 terms

now there are 1003

2(2) + 2(4) + ... + 2006(2)

Sum of this = \[\sum_{i=1}^{1003} 4i = 2014024\]

Answer 19

and I see I am terribly late. Damn.

Answer 20

lol

Answer 21

Sorry, made a typo, 3 has 2 cases.

Answer 22

We used the same approach @slaaibak!

Answer 23

Now, where is the problem you promised?

Answer 24

I posted it in chat, but I think the buggy chat lost it somehow. I will post it in a sec!

Answer 25

Not in the chat please! thanks :)

Answer 26

That's cheating across :P

Answer 27

Looks like I will ask across to help me with the programming I have to do. :P

Answer 28

My computer is my loyal slave *pets computer. computer barks.*

Answer 29

First problem to warm up: (Not that difficult)

_, T, T, F, F,
S, S, E, N, T,
E, T, T, F, F,
S, S, E, N, T

Missing letter?

Answer 30

E

Answer 31

Nope

Answer 32

I can't find it. but if I have to make a guess, other than E, I would say N.

Answer 33

Nope. Think VERY basic.

Answer 34

it's F actually.

Answer 35

Not F either

Answer 36

That just leaves us with T...

Answer 37

wait don't answer now

Answer 38

kk

Answer 39

I asked this to someone I know, he said T. and when I asked why he replied "Why not? It's a finite thing, we can always create enough superficial demands for it to be anything."
lol :D

Answer 40

hahaha. True, but that does not fit my answer :)

Answer 41

I'll give you a clue. English.

Answer 42

I'm not good at English! :(

Answer 43

yeah, same... I didn't get it either.

Answer 44

I want a problem in Zimbabwean language.

Answer 45

I don't think it gets any more basic than E, honestly.

Answer 46

Verstaan jy wat ek nou sê?

Answer 47

Hierdie is 'n taal wat in Zimbabwe gebruik word, so jy behoort dit te verstaan.

Answer 48

Verstaan \(\approx\) Verstehen = Understand ?\[\]

Answer 49

Yeah, I got that. So it's T! I knew it :D

Answer 50

Alright, what comes next in this sequence: 7, 4, 5, 4, 2, 4, [ ]

Answer 51

3

Answer 52

no, no

Answer 53

7,5,2,-2

Answer 54

Gosh I can't do simple addition and subtraction.

-2 or 2

Answer 55

No across.

Answer 56

It's 4 !

Answer 57

No :-(

Answer 58

( Across, yes, verstaan = understand. Some of the german words are a bit similar to my language)

Answer 59

5 ?

Answer 60

Here is the answer :

Alright|=7, |what|=4, |comes|=5, |next|=4, |in|=2, |this|=4

I am sure that this will bring a smile in your face :-)

Answer 61

Haha,unfortunately, the pattern isn't even mathematical... :D

Answer 62

4

Answer 63

you are so messed up

Answer 64

lol :D

Answer 65

lol,

Answer 66

lol. Another clue. My pattern isn't mathematical either..

Answer 67

8! :P

Answer 68

haha. It's a letter tho. Is everyone fine with me giving the answer?

Answer 69

You answer is O

Answer 70

yep! It's O. did you google, or figure it out?

Answer 71

One,two, three, four,...

Answer 72

Yep! Lame, I know... Well done!

Answer 73

You both suck!

Answer 74

Time for MrMath or across to give a puzzle.

Answer 75

http://www.youtube.com/watch?v=RCn7Xk_6sb8

Answer 76

lol :D

Answer 77

I have this link once to my German friend in Germany and he told me it was blocked! :0

Answer 78

Germany blocks most YouTube videos, especially music videos.

Answer 79

Really?!! Why?

Answer 80

Oh I got, copyrights.

Answer 81

Yes, why ?

Answer 82

My life without youtube would be incomplete.

Answer 83

I thought it had to do with the sensitivity about Nazism or something like that.

Answer 84

http://www.afterdawn.com/news/article.cfm/2009/04/02/youtube_blocks_music_videos_in_germany

Answer 85

No, it has nothing to do with the Hitler reference.

Answer 86

Hey my friend reaction to the answer of slaaibak's question:

"Maybe we should rename "one" to begin with an "e" so as to preserve symmetry. I'm sure mathematicians will go for it with that rationale."

Answer 87

In my language, one begins with an E :D But yes, we should start a petition!

Answer 88

haha, :D

Answer 89

One also begins with an E auf Deutsch.

Answer 90

Poor German!

Answer 91

Germans*

Answer 92

Alright, time to sleep for me.

Answer 93

Gute Nacht, FFM. Träume schön und bis später!

Answer 94

haha ,across, the only German I know is Auf wiedersehen :P

Answer 95

Good night! Hoop jy slaap lekker.

Answer 96

Horra Borra Torra Fool!! (Good night in Zimbabwean)

Answer 97

lol :D Subho ratri (Good night in my mother tongue)