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I think the guest meant {} as fractional part

BTW the answer is 6.

1) do the braces mean ceiling or fractional part or neither?

2) clearly x = z + 0.8

a little tinkering shows x = 1.8

2 + 1 + 1.8 = 4.8

11 possible each pick, 4 picks, total of (11)⁴ = 14641 arrangements

Where did the word 'living' get mentioned??? 

L = A  + (B-A)t = (1+t, 1+2t, 2+2t)

a=1, b=1, c=2, d=1, e=2, f=2

(c/a, e/a) = (2,2)

\(\text{Just by looking at it I maintain the answer is 3, but we'll do it formally}\\ \nabla\left(\dfrac a b + \dfrac b c + \dfrac c a\right) = 0\\ \dfrac 1 b - \dfrac{c}{a^2} = 0\\ \dfrac 1 c - \dfrac{a}{b^2} = 0\\ \dfrac 1 a - \dfrac{b}{c^2} = 0\)

\(\text{The only solution of all positive numbers is $a=b=c$}\\ \text{and the extrema value is 3}\\ \text{Suppose $c=2a$}\\ \dfrac a b + \dfrac b c + \dfrac c a = 1+\dfrac 1 2 + 2 = \dfrac 7 2 > 3\\ \text{we know 3 is either a min or a max and thus as $3 < \dfrac 7 2$ it must be a min}\)

\(\sqrt{x y} = \sqrt{3}\\ \dfrac{2}{\dfrac 1 x + \dfrac 1 y}=\dfrac 3 2\\ \dfrac{2xy}{x+y}=\dfrac 3 2\\ \dfrac{6}{x+y}=\dfrac 3 2\\ x+y = 4\\ xy=3\\ x+\dfrac 3 x = 4\\ x^2-4x+3=0\\ (x-3)(x-1)=0\\ x=3,1\\ (x,y) = (1,3), (3,1)\)

how do you donate 1/3 of a living cow? just cuirous.

The dollar signs are there because it is how people write in Latex code.

(x * 1.25)  * (1- d) =   x

1.25(1- d) =1

d=1-1/1.25

d = .2 = 20 %

1 in base 6 = 1

45 in base 6 = (4x6)+5 = 29

Sum of arithmetic sequence   = n (a1+an)/2 = 29 (30/2) = 435

435 in base 6 =   2003₆

The second solution assumes that Company A would take 10 hours longer than the 2 Companies working together. But that is NOT what the question says. It says Co. A takes 10 hours longer than Co. B, when working alone! So the solution looks like this:

1/B + 1/(B+10) = 1/26

B =47.48 hours - for Co. B working alone to clear the land.

A =47.48 + 10 =57.48 hours - for Co. A to clear the land working alone.

So that: 1/47.48  +  1/57.48 =1/26, or 26 hours when both Companies work together.

oh I get the first part now. what about the second part?

Actually, this is a written contest where you have to do 30 questions in 40 minutes by hand, no calculator.

Thanks rom :)

Company A  = A

Company B  = B

Company A's hourly rate =1 / A = 1 / [26 + 10] =1 / 36

1 / A  +  1 / B = 1 / 26

1 / 36  +  1 / B = 1 / 26

1 / B = 1 / 26  - 1 / 36 [The LCM of 26 and 36 =468

1 / B = 18 / 468  -  13 / 468 =5 / 468 Company B's rate. Therefore, it would take Company B:

B =468/5 =93.6 hours to finish the job working alone! THIS IS THE CORRECT ANSWER!.

Check: 1/A + 1/B = 1/26

            1/36  + 1/93.6 = 1/26

                  1 / 26 = 1 / 26

A clock each time it increased 20 minutes , it stopped for 10min.
How long will it take the minutes arrow to do 2 full rounds ?

\(\begin{array}{|rcll|} \hline \text{Time} &=& {\color{red}20}\ \text{min.} + 10\ \text{min.} +{\color{red}20}\ \text{min.} + 10\ \text{min.} + {\color{red}20}\ \text{min.} \quad | \quad \text{minutes arrow first round } 60\ \text{min.}\\ && + 10\ \text{min.}+{\color{red}20}\ \text{min.} + 10\ \text{min.} + {\color{red}20}\ \text{min.} + 10\ \text{min.}+ {\color{red}20}\ \text{min.} \quad | \quad \text{minutes arrow second round } 120\ \text{min.}\\ \text{Time} &=& 170\ \text{min.} \\ \hline \end{array}\)

Help with absolute value

1)

Find all real numbers x that satisfy the equation \(\large{\mathbf{|x+4| + |x-7| = |2x-1|}}\).

There are three discontinuities at \(x=-4,\ x=7,\ x=\dfrac{1}{2}\).
These digits divide the number range from \(-\infty\) to \(+\infty\) into the four ranges \(x<-4\),   \(-4\leq x < \dfrac{1}{2}\),   \(\dfrac{1}{2}\leq x <7\)  and  \(x\geq 7\).

\(\begin{array}{lrcll} x<-4 :& -(x+4)-(x-7)+(2x-1) &=& 0 \\ & -x-4-x+7+2x -1 &=& 0 \\ &6 &=& 0 \quad | \quad \text{no solution!} \\ -4\leq x < \dfrac{1}{2} : & (x+4)-(x-7)+(2x-1) &=& 0 \\ & x+4-x+7+2x-1 &=& 0 \\ & 2x+10 &=& 0 \\ & 2x &=& -10 \\ & x &=& -5 \quad | \quad \text{no solution! Because }-4\leq x < \dfrac{1}{2} \\ \\ \dfrac{1}{2}\leq x <7 : & (x+4)-(x-7)-(2x-1) &=& 0 \\ & x+4-x+7-2x+1 &=& 0 \\ &-2x+12 &=& 0 \\ & 2x &=& 12 \\ & \mathbf{x} &=& \mathbf{6} \\ \\ x\geq 7 : & (x+4)+(x-7)-(2x-1) &=& 0 \\ & x+4+x-7-2x+1 &=& 0 \\ & -2 &=& 0 \quad | \quad \text{no solution!} \\ \end{array}\)

\(\mathbf{x = 6}\)  satisfy the equation \(\large{\mathbf{|x+4| + |x-7| = |2x-1|}}\).

how to solve this

\(\sin^4 ( A ) + \sin^2 ( A ) \cos^2 ( A ) + \cos^2 ( A ) = 1\)

\(\begin{array}{|rcll|} \hline \sin^4 ( A ) + \sin^2 ( A ) \cos^2 ( A ) + \cos^2 ( A ) &=& 1 \\ \sin^2 ( A )\sin^2 ( A )+ \sin^2 ( A ) \cos^2 ( A ) + \cos^2 ( A ) &=& 1 \\ \sin^2 ( A ) \left(\sin^2 ( A )+ \cos^2 ( A )\right) + \cos^2 ( A ) &=& 1 \\ \sin^2 ( A ) \left(\underbrace{\sin^2 ( A )+ \cos^2 ( A )}_{=1} \right) + \cos^2 ( A ) &=& 1 \\ \sin^2 ( A )*1 + \cos^2 ( A ) &=& 1 \\ \underbrace{\sin^2 ( A ) + \cos^2 ( A )}_{=1} &=& 1 \\ \hline \end{array} \)

It will take 30 minutes to turn 1/3 of a round,

So it will take 6(30) = 180 minutes to turn 6(1/3) = 2 rounds.

Since the 2 companies together take 26 hours to finish.

Therefore:

A+B=26

"Company A takes 10 more hours than company B)

Therefore:

B-10(hours)=A 

Subsitute A with B-10 in A+B=26

We get: 

(B-10)+B=26

2B-10=26   

(Add 10 to both sides)

2B=36 

(Divide by 2)

B=18 hours

To get A= A+18=26

Subract 18

A=8 hours

Final answer:

It would take company B 18 hours to finish alone.

It would take company A 8 hours to finish alone.

My answer was 180 minutes