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**tony123****Member**- Registered: 2007-08-03
- Posts: 229

1.Positive integers m and n satisfy

Find

2.Find the positive integer n such that

3.A trapezoid has side lengths

and in some order. Find its area4.A rectangle with integer side lengths has the property that its area minus 5 times its perimeter equals

Find the minimum possible perimeter of this rectangle.5.For real numbers a, b, and c, the roots of the polynomial

arithmetic progression. Find

*Last edited by tony123 (2024-03-14 21:20:40)*

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,610

Thanks Tony,

More (hopefully) later.

Bob

ps. Still stuck with http://www.mathisfunforum.com/viewtopic.php?pid=432520#p432520

Maybe I need a little hint please.

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You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,610

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Keep_Relentless****Member**- From: Queensland, Australia
- Registered: 2024-05-05
- Posts: 61

For Q4, is it the area minus (5 times the perimeter),

As in A - 5P = 2023

Or is it (the area minus 5) times the perimeter,

As in (A - 5) * P = 2023

I'm guessing it's the first one??

*Last edited by Keep_Relentless (2024-05-08 23:03:09)*

"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

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**Keep_Relentless****Member**- From: Queensland, Australia
- Registered: 2024-05-05
- Posts: 61

*Last edited by Keep_Relentless (2024-05-08 23:02:32)*

"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

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**Keep_Relentless****Member**- From: Queensland, Australia
- Registered: 2024-05-05
- Posts: 61

Oh I forgot the sides are integers.

*Last edited by Keep_Relentless (2024-05-08 23:32:24)*

"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

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**Keep_Relentless****Member**- From: Queensland, Australia
- Registered: 2024-05-05
- Posts: 61

I think for Q3 you use Brahmagupta's formula

"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

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**Keep_Relentless****Member**- From: Queensland, Australia
- Registered: 2024-05-05
- Posts: 61

Regarding Q5, help is needed on finding roots of quintic equations and what's needed to see the answer, thanks

*Last edited by Keep_Relentless (2024-05-09 00:33:29)*

"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

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**Keep_Relentless****Member**- From: Queensland, Australia
- Registered: 2024-05-05
- Posts: 61

Can anyone show how to do Q5??

"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

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**Bob****Administrator**- Registered: 2010-06-20
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I called the roots a, a+d, a+2d, a+3d and a+4d and expanded that quintic. There were things about the resulting expression that suggested a + b + c = 9, but I'm not confident that's the correct answer so this one is still on my 'to-do list'.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**KerimF****Member**- From: Aleppo-Syria
- Registered: 2018-08-10
- Posts: 238

tony123 wrote:

5.For real numbers a, b, and c, the roots of the polynomial

form an

arithmetic progression. Find

This exercise has a solution if the right side of the equation is 32 instead of 320.

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).

But only a human may have the freedom and ability to oppose his natural robotic nature.

But, by opposing it, such a human becomes no more of this world.

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**KerimF****Member**- From: Aleppo-Syria
- Registered: 2018-08-10
- Posts: 238

Well, I also found the solution in case the right side is 320.

a+b+c=49

Kerim

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).

But only a human may have the freedom and ability to oppose his natural robotic nature.

But, by opposing it, such a human becomes no more of this world.

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**Keep_Relentless****Member**- From: Queensland, Australia
- Registered: 2024-05-05
- Posts: 61

Interesting, any chance you can explain how you worked it out? We all had a go and didn't get there.

K_R

"The most incomprehensible thing about the world is that it is comprehensible." -Albert Einstein.

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**KerimF****Member**- From: Aleppo-Syria
- Registered: 2018-08-10
- Posts: 238

Keep_Relentless wrote:

Can anyone show how to do Q5??

(x-A)*(x-B)*(x-C)*(x-D)*(x-E) = 0

x^5 - (x^4)*(A + B + C + D + E) + (x^3)*(A*B + A*C + A*D + B*C + B*D + C*D + A*E + B*E + C*E + D*E) - (x^2)*(A*B*C + A*B*D + A*C*D + B*C*D + A*B*E + A*C*E + A*D*E + B*C*E + B*D*E + C*D*E) + x*(A*B*C*D + A*B*C*E + A*B*D*E + A*C*D*E + B*C*D*E] - A*B*C*D*E = 0

or

x^5 - (x^4)*(A + B + C + D + E) + (x^3)*(A*B + A*C + A*D + B*C + B*D + C*D + A*E + B*E + C*E + D*E) - (x^2)*(A*B*C + A*B*D + A*C*D + B*C*D + A*B*E + A*C*E + A*D*E + B*C*E + B*D*E + C*D*E) + x*(A*B*C*D + A*B*C*E + A*B*D*E + A*C*D*E + B*C*D*E] = A*B*C*D*E

We have:

x^5 - 10x^4 + ax^3 + bx^2 + cx = 320

By comparing, we get

A + B + C + D + E = 10

A*B + A*C + A*D + B*C + B*D + C*D + A*E + B*E + C*E + D*E = a

A*B*C + A*B*D + A*C*D + B*C*D + A*B*E + A*C*E + A*D*E + B*C*E + B*D*E + C*D*E = b

A*B*C*D + A*B*C*E + A*B*D*E + A*C*D*E + B*C*D*E = c

A*B*C*D*E = 320

A + B + C + D + E = 10

In general, Sum = [2*m + (n-1)*r]*n/2

where

m is the first term

r is the common difference

n the number of terms

[2*A + (5-1)*r]*5/2 = 10

A + 2*r = 2

r = (2 - A)/2

For A = -4 , r = 3

Therefore

A = -4

B = -1

C = 2

D = 5

E = 8

Their sum is 10

And their product is 320

a = A*B + A*C + A*D + B*C + B*D + C*D + A*E + B*E + C*E + D*E = -5

b = A*B*C + A*B*D + A*C*D + B*C*D + A*B*E + A*C*E + A*D*E + B*C*E + B*D*E + C*D*E = 190

c = A*B*C*D + A*B*C*E + A*B*D*E + A*C*D*E + B*C*D*E = -136

Therefore:

a + b + c = 49

Kerim

I did a mistake (sorry, a mistyping, not about the asnswer); will someone correct it?

*Last edited by KerimF (2024-05-14 01:41:52)*

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).

But only a human may have the freedom and ability to oppose his natural robotic nature.

But, by opposing it, such a human becomes no more of this world.

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**Europe2048****Member**- Registered: 2024-01-03
- Posts: 38

KerimF, use the hide tag if you don't want to straight-up reveal the answer.

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