S = (2 + 1)(2² + 1)(2⁴ + 1)...(2²⁰ + 1)

   = (2 - 1)(2 + 1)(2² + 1)(2⁴ + 1)...(2²⁰ + 1)

   = (2² - 1)(2² + 1)(2⁴ + 1)...(2²⁰ + 1)

   = (2⁴ - 1)(2⁴ + 1)...(2²⁰ + 1)

..........................................................

   = (2²⁰ - 1)(2²⁰ + 1)

   = 2⁴⁰ - 1

S = (2 + 1)(22 + 1)(24 + 1)...(220 + 1)

   = (2 - 1)(2 + 1)(22 + 1)(24 + 1)...(220 + 1)

   = (22 - 1)(22 + 1)(24 + 1)...(220 + 1)

   = (24 - 1)(24 + 1)...(220 + 1)

..........................................................

   = (220 - 1)(220 + 1)

   = 240 - 1

S = (2 + 1)(22 + 1)(24 + 1)...(220 + 1)

   = (2 - 1)(2 + 1)(22 + 1)(24 + 1)...(220 + 1)

   = (22 - 1)(22 + 1)(24 + 1)...(220 + 1)

   = (24 - 1)(24 + 1)...(220 + 1)

   = (220 - 1)(220 + 1)

   = 240 - 1

We have:

 \(a_9x^9+a_8x^8+...+a_1x+a_0=\left(x+2\right)\left(x+3\right)...\left(x+9\right)\left(x+10\right)\)

With x = 1, we have:

  ​ ​​ ​\(a_9+a_8+a_7+...+a_1+a_0=3.4...10.11\)       (1)

With a = -1, we have:

  ​ ​\(-a_9+a_8-a_7+...+a_2-a_1+a_0=1.2.3...8.9\)   (2)

subtracting (1) and (2), we have:

  \(2\left(a_9+a_7+..+a_3+a_1\right)=3.4...10.11-1.2...8.9\)

 \(2\left(a_1+a_3+a_5+a_7+a_9\right)=3.4...8.9\left(10.11-1.2\right)\)

  \(2\left(a_1+a_3+a_5+a_7+a_9\right)=3.4..8.9.108\)

 \(a_1+a_3+a_5+a_7+a_9=3.4...8.9.54=1.2.3...8.9.27=9!27\)

The answer is (c),but I don't know the solution.

Is this problem from USC Mathematics Contest 3/12/1994 ?

Visit here : http://www.math.sc.edu/math-contest-1994-answer-key

Let C and P be the price of each computer and printer respectively.We have :

\(\left\{{}\begin{matrix}10C+10P=10000\\12C+2P=10000\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}C+P=1000\\6C+P=5000\end{matrix}\right.\)

\(\Rightarrow5C=4000\Rightarrow C=800\)

So,a computer costs 800$

P/S : I answered a similar question.

Apply Pythagorean theorem , we have :

\(\text{Hypotenuse}=\sqrt{6^2+7^2}=\sqrt{85}\left(cm\right)\)

I don't know this question

But i think Kaya Renger Coordinator is doing correct

Sorry very much

One-half of rectangular sports field is:

880: 2 = 440 [m]

The length of rectangular sports field is:

440 : \(\left(3+1\right)\cdot3=330\left(m\right)\)

The width of rectangular sports field is:

440 - 330 = 110 [m]

So the area of the field is:

330.110 = 36300 [m²]

The first day of this month is not Tuesday. So may be first Tuesday is 3^rd. We let's try: 

Tuesday: 3              10              17              24              31

but 31^st is last day of a month has 31days => Unsatisfactory

So first Tuesday of this month is 2^nd

We let's try too.

Tuesday: 2               9               16               23               30

So the last Tuesday of this month is 30^th may be last day of this month. But the topic talk last day isn't Tuesday. Angain have in a year will has 7 months have 31 days.

So the last day of the month is 31 

The last day of the month is 31.

Please for me! 

I'd add the answer more from Dao Trong Luan's answer:

Because 30 is Tuesday (1) => The month has 31 days. (2)

From (1) and (2) => The last day of the month is Wednesday.

A day the man floats \(3-2=1\) (mi north)

So until 47 days the man can go 47 mis north ( Cause 1 mi / day).

In the last day (day 48) the man floats 3 mi north so the distance he went is 47 + 3 = 50 (mi north).

So it will take him 48 days to reach the location.

The least possible number is the least number divisible by 2 of the following statements. Since  \(60⋮15\) in the 2 following statements e and a so the least possible number of k is 60.