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1. ### A taylor series problem....

Let F(t)=f(t)/g(t) be a rational function with integer coefficients, assume g(0)=1, then the Taylor expansion of F(t) at 0 has integer coefficients, and more over, these coefficients satisfy a recursion relation of the form c_n+k=a_{k-1}c_{n+k-1}+ ... + a_0c_n (k and all a_i are all fixed integers) for all but finitely many n?

(for example try computing a MacLauren series for (1+2x)/(1-x^3)

2. ### One last functional equation

(f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y

Find f(x) where f(x) is a polynomial.

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3. ### Emoji based math puzzle - level hard!

I get two possible solutions: 1100 or 76461.  Though if we want only positive values for each emoji then my answer of 1100 is the correct one.

Though I am treating the fact that like how one row has two emojis of alligators and is different than the other rows, then

the eagles being doubled is significant.

4. ### Another Functional Equation

2f(1/x)-f(x)+2f(2/x)-f(x/2) = x, x is defined on the reals except where x =0

find f(x) =

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5. ### Solving a system of functional equations

h(f(x)) + g(h(x)) + f(g(x)) = 2x^2 + 11x + 14

f(h(x)) + h(g(x)) + g(f(x)) = 2x^2 - 15x + 66

f(g(x)) = g(f(x))

h(g(x)) - g(h(x)) = -16x + 72

h(f(x)) + f(h(x)) = 2x^2 + 10x  + 30

f(x) * g(x) = h(x) - 3x - 40

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f(x) =
g(x) =
h(x) =

6. ### Climbing Stairs

How many different paths can I make up a flight of 20 stairs if I can take the steps either one at a time or two at a time (in any order)?

7. ### Easy question about a system (or is it)

1 hour ago, Nightfox__ said:

X comes around 1.207 at r = 1 + (1/ 2) . Are you saying a larger value of x is possible??

On the right track but see if you can find what x is approaching.

8. ### Easy question about a system (or is it)

I have a negative value for x as my min and a different x max

36 minutes ago, Nightfox__ said:

Max of x is 1/2+(1/√2) and min 0??

This is not the answer but as an example of this possibility:

9. ### Easy question about a system (or is it)

Suppose we have the following system

x^2+y^2=r; x+y=r, such that the line crosses the circle at exactly two places.

Obviously with two equations and three variables, we have a solution set of answers that can satisfy the given conditions.  What I want to know is of the given solutions that satisfies this problem, what is the smallest and largest values x can possibly be?

10. ### Find the missing value

I think you are on the right track rocdocmac

11. ### Find the missing value

On 4/24/2019 at 6:59 AM, rocdocmac said:
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For 11 up to 9999 it's a lie! After that, maybe too.

Some figures get close, however, e.g. 25, 37, 449, 4499, one or two off the mark..

you have not proven nor disproven this

12. ### Find the missing value

I have in mind a number which, when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 2. Am I telling the truth?

13. ### Bringing back an old one.... Making Toast

On 4/17/2019 at 6:02 AM, rocdocmac said:
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116 seconds (or possibly even 106 seconds) presuming first two pieces are already in place.

Toasting alone takes 90 seconds for slices numbered 1, 2 and 3 with sides A and B.

Stages: (1) 1A  and 2A under grill (30 sec), followed by (2) 1B and 3A under grill (30 sec); finally (3) 2B and 3B under grill (30 seconds)

Normal additional time for single movements:

End of 1st stage: Slice 1 flipped from 1A to 1B (3 sec), 2A removed (5 sec), 3A inserted (5 sec)

End of 2nd stage: Slice 1 removed (5 sec.), slice 3 flipped from 3A to 3B (3 sec), slice 2 (2B) re-inserted with flip side up (5 sec)

End of 3rd stage: Done

Normal additional time = 26 seconds

Total time = 116 seconds.(1:56)

Exchange in one go:

If one is allowed to do simultaneous movements (exchanging in one switch of 5 sec), 10 seconds can be deducted, i.e. a total time of 106 seconds (1:46)

14 hours ago, Pirate8 said:

 Left side Right side 1-5 A1 in 6-10 B1 in 6-35 A1 toast 11-40 B1 toast 36-38 A flip 41-45 B out 39-68 A2 toast 46-50 C in 69-74 A out 51-80 C1 Toast 75-79 B2 in 81-83 C flip 80-109 B2 toast 84-113 C2 toast 110-114 B out 114-116 C out Manual operations must not overlap.

14. ### Bringing back an old one.... Making Toast

Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out and 3 seconds to turn a piece over. What is the shortest time in which the three slices can be toasted?

15. ### Fair division of sausages

2 hours ago, Thalia said:

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12 cuts, 42 pieces? Unless you want to line up the sausages and do one giant cut...

You have the right idea with your comment. I an designing a cut as a swipe of the knife,  so lining them up and then cutting with one swipe would  count as one cut.

16. ### Fair division of sausages

I wish to share 30 identical individual sausages equally amongst 18 people. What is the minimum number of cuts I need to make? What is the minimum number of pieces I need to create?

17. ### Impoo

Assume that only four buttons work on your calculator: 5, 7, enter, and plus.  What whole numbers can you not use your calculator to make?

18. ### Arc length = area

20 hours ago, EventHorizon said:

I'm sure I know what BMAD was saying.  Yes, ignore the units / squared units difference.  The problem doesn't involve a circle though.  You need to find a function f(x) such that the area under the curve between two points is the same as the arc length between those two points along the function.  Here the arc length is like the length of a string representing the function cut at the two points.

How to find the arc length of a function:

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If the function was a straight line, you could use the Pythagorean theorem to find it.

sqrt( (y2-y1)^2 + (x2-x1)^2 )

If you assume the arc length was a straight line, and slowly bring the points on the function closer together, eventually you get the change in y approaching the slope of the line times the difference in x's.

The arc length for that infinitesimally small section is then sqrt( ( f'(x)dx )^2 + dx^2 ) = sqrt( dx^2 * (f'(x)^2 + 1 ) = sqrt(   f'(x)^2 + 1 ) dx.  Where f'(x) is the derivative of f(x).

Integrate that between two points to get the arc length between those two points.

Example: f(x)=2x

f'(x) = 2

integrate( sqrt( 2^2 + 1 ) dx ) = integrate( sqrt(5) dx) = x*sqrt(5) + c

If you evaluate the integral between 0 and 1 you end up with arc length =  sqrt(5).  Which is the same as if you just used the Pythagorean theorem.  Yeah, I'm too lazy right now to find a function that ends up with a nice arc length function that isn't just a straight line.

Restatement of problem using previous spoiler:

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Find a function f(x) such that F(x) = integrate( sqrt( f'(x)^2 + 1) ), where F(x) is the indefinite integral of f(x) and f'(x) is the derivative of f(x).

One step further:

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Taking the derivative of both sides gives

f(x) = sqrt( f'(x)^2 + 1)

One answer is simple, the other is less so.

yes, this is what I mean.

19. ### distance vs time

James and Mike are running in a race.  They both walked and ran for part of the rate.  They each walked and ran at the same speed.  James ran for half the distance and walked for half the distance.  Mike ran for half of his time and walked for half of his time.  Who finished first?

20. ### Squares on a plywood

On 12/15/2018 at 4:16 PM, CaptainEd said:

I think not: please check my counting

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In what follows, the rubber band’s path goes on the outside of the nails marked “x

o o o o o o o
o o o o o o o o x
o o o o o o o o
o o o o o o o
o o o o o o
o o o o o x
N = 43
P = 21 + sqrt(5)
S = 62: (5x5:1, 4x4:4, 3x3:10, 2x2:18, 1x1:29)

Compare with

o o o o o o o
o o o o o o o o x
o o o o o o o o
o o o o o o o
o o o o o o
o o o o o x
N = 43
P = 21 + sqrt(5)
S = 56: (4x4:2, 3x3:8, 2x2:17, 1x1:29)

All you showed is the bound is wider than one thought.

21. ### Squares on a plywood

On 12/12/2018 at 4:52 PM, CaptainEd said:

I question whether the information given about a figure is sufficient to uniquely determine the number of included squares:

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If I understand right, these two figures have the same N and P, but don’t have the same number of squares.

o o o o
o o o o o
o o o
Nails = 12, perimeter = 7+sqrt2 + sqrt5
Squares =5

o o o o
o o o o o
o o o
nails=12, perimeter = 7 + sqrt2 + sqrt5
Squares = 6

So then if we know N and P we should be able to bound  squares x by a two values. Are those values always consecutive?

22. ### the distinguished matrix

On 12/11/2018 at 9:00 PM, CaptainEd said:

While this doesn’t answer the question with randomly placed 1s, I think it does give an upper bound to the number of columns that can be removed.

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What is the smallest number of remaining columns that would allow (a) the 30 rows to be distinct, (b) each row to have 5 1-bits, and (c) all remaining columns to be distinct?

7 columns: 7!/2!5! = 21, too few distinct rows
8 columns: 8!/3!5! = 56, that’s enough.

The first 12 columns in each of the 30 rows should be one of these patterns:
11000 00000 00
01100 00000 00
00110 00000 00
00011 00000 00
00001 10000 00

00000 11000 00
00000 01100 00
00000 00110 00
00000 00011 00
00000 00001 10

00000 00000 11
10000 00000 01

The last 8 columns should be chosen from the C(8,5) possible patterns.

The resulting matrix satisfies the requirements because:
(a) the last eight columns differentiate the 30 rows
(b) the first twelve columns are distinct by design
(c) the last eight columns are distinct because of the distinct patterns chosen from the C(8,5)
(d) all row sums are 7 before column removal, and 5 afterwards

I agree

23. ### Arc length = area

Find a function where the arc lenth and area between any two randomly defined points is the same.  There are two.

24. ### infinite powers

Say we have the function:

y=x^x^x^x^x.....

Find an x value for which the derivative of this function converges.

If you are really clever you'll find the interval that converges.

25. ### An interesting limit

On 11/16/2018 at 6:48 AM, EventHorizon said:
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e/(e-1)

That is an interesting limit :-)

Hmmmm, my answer was the reciprocal of yours.  Maybe I am wrong.  Can you support your answer?

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