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1.Positive integers m and n satisfy
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
Last edited by tony123 (2024-03-14 21:20:40)
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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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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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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)
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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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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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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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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)
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Can anyone show how to do Q5??
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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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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.
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But only a human may have the freedom and ability to oppose his natural robotic nature.
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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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Interesting, any chance you can explain how you worked it out? We all had a go and didn't get there.
K_R
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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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KerimF, use the hide tag if you don't want to straight-up reveal the answer.
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