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## #1 2016-08-22 15:17:52

Mathegocart
Member
Registered: 2012-04-29
Posts: 1,884

### Number Theory Challenges, #1

1. Find the sum of all the integers between 50 and 350 whose final digit is 1.
1a. How much greater is this than the sum of all the integers from 1 to 50?
2. Find the complete set of real solutions for the equation $4y^2+4xy+2x+8$ where y is real.
3. How many primes can be represented as ? Say NONE if no primes can be represented, say ALL if every prime can be represented.
4. Find all solutions (x,y) where x and y are integers, to the equation$x^2+xy+y^2+x+y=1$
5. For what values of a does $x^2+ax+1 = 0$
and $ax^2+x+1 = 0$ have at least one root in common?
6. Find the last digit of (2^1232)*(2^5+3^7)?
7.Find the coefficient of p if the difference between the roots in this equation is 9.

8. Bobbym and his neighbor were solving equations late in the night and they were solving two quadratics,
(1)$bx^2+ax+1 = 0$ and
(2)$ax^2+bx+1 = 0$.
His neighbor switched up bobbym's equation and got (2). For what values of a and b are at least one of the roots the same?

Last edited by Mathegocart (2016-08-22 15:19:58)

The integral of hope is reality.

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## #2 2016-08-22 15:21:56

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Challenges, #1

Hi;

What is that after #8?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2016-08-22 16:15:05

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

### Re: Number Theory Challenges, #1

Last edited by ElainaVW (2016-08-22 16:18:52)

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## #4 2016-08-23 02:12:49

Mathegocart
Member
Registered: 2012-04-29
Posts: 1,884

### Re: Number Theory Challenges, #1

ElainaVW wrote:

You know who he is in real life??

The integral of hope is reality.

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## #5 2016-08-23 02:18:39

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,123
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## #6 2016-08-23 03:55:37

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Challenges, #1

Hi Mathegocart;

Any chance we can get some clarification on 2,3,8 and

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2016-08-23 04:39:56

Mathegocart
Member
Registered: 2012-04-29
Posts: 1,884

### Re: Number Theory Challenges, #1

bobbym wrote:

Hi Mathegocart;

Any chance we can get some clarification on 2,3,8 and

http://latex.artofproblemsolving.com/9/6/9/969d1960185e877ca37aa5ae592fb3ab24c3614f.png

The equation described in 2 equals 0,
In 8, you got equation (1) and your neighbor got (2). For what values of a and b(a and b are distinct), for at least 1 of the solutions are the same?

For (9), what is the range of

Last edited by Mathegocart (2016-08-23 04:40:30)

The integral of hope is reality.

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## #8 2016-08-23 05:22:49

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Challenges, #1

Hi;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #9 2016-08-23 17:50:37

Member
Registered: 2016-04-16
Posts: 1,086

### Re: Number Theory Challenges, #1

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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## #10 2016-08-24 03:54:05

Member
Registered: 2016-04-16
Posts: 1,086

### Re: Number Theory Challenges, #1

Mathegocart wrote:

2. Find the complete set of real solutions for the equation $4y^2+4xy+2x+8$

where y is real.

Something fishy. Why the common factor 2 exists?

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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## #11 2016-08-27 04:12:57

ElainaVW
Member
Registered: 2013-04-29
Posts: 580

### Re: Number Theory Challenges, #1

Mathegocart wrote:

You know who he is in real life??

Yes

Last edited by ElainaVW (2016-08-27 04:14:32)

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## #12 2016-08-27 04:20:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Number Theory Challenges, #1

I have three quotes in case you are thinking of revealing anything you think you know about me:

Cobbie Bean, Cobbie Bean, never tell what you have seen.

Omerta - The code of silence.

He who is deaf, blind, and silent will live a hundred years in peace.