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The smallest possible N == 1,166

1,166 mod 9==5

1,166 mod 7==4

1,166 mod 5==1

Hi Mobius,

You are correct. There is a second answer to the question, but does it fit in the range?

Best,

MPS

You know the sum beetween 1 and 9 is 45, multiply this by 10 (because there are 11 and 19, 21 and 29, ... 91 and 99) and that equals 450. The first digit could be all digits (except 0) and the second digit too, but it could be 0, but x + 0 = is still x. So we double the 450 and that is (450 * 2 =) 900 and now + 1 because of 100 (1 + 0 + 0 = 1). So (900 + 1 =) 901 is the solution.

Straight

Adding them will not change the dgree unless terms 'cancel out'       As there is no 5th degree in f(x) the 5th degree term in g(x) will remain

Degree 5

This is very simple...

AB = 12

BC = 300/12

BW = BC - CW

∠CBD = tan^-1(12/25)

∠BWZ = tan^-1(12/13)

Now you can solve triangle BQW using the law of sines, and using Heron's formula, you can find the area of this triangle.

btw., [BQW] = 57 square units

I'm still a bit confused. So I would then know that one of the answers in pi/3. But I don't know what other answers there are...

$\sec(x) = \frac{1}{\cos(x)}$

\(\begin{align*} \tan^2(x) &= \left(\frac{\sin x}{\cos x} \right)^2 \\ &= \frac{\sin^2 x}{\cos^2 x} \\ &= \frac{\sin^2 x}{\cos^2 x} + \frac{\cos^2 x}{\cos^2 x} - 1\\ &= \frac{\sin^2 x + \cos^2 x}{\cos^2 x} - 1\\ &= \frac{1}{\cos^2 x} - 1 \end{align*}\)

$\frac{5}{\cos(x)} = \frac{3}{\cos^2 x} - 2$

$5 \cos(x) = 3 - 2 \cos^2 x$

$2 \cos^2 x + 5 \cos x - 3 = 0$

$(\cos x + 3) (2 \cos x - 1) = 0$

$\cos x = -3$

$\cos x = \frac{1}{2}$

I think you can take it from here.

x+3x+2x=180

6x=180

x=30

By the Chinese Remainder Theorem, we have $N \equiv 221 \pmod {315}$

The least such integer is $\boxed{221}$

Ugly problem with solutions. I tried and got a very complex radical expression. Are you sure the question is stated correctly?

Hello Guest,

from the sum beetween 1 and 9 is (\(1+9*(\frac{9}{2})=\)) 45, the sum beetween 10 and 19 is (45+10 =) 55, the sum beetween 20 and 29 is (45+20 =) 65, do this trick every time and you'll get the answer. May you response and tell me the answer?

Straight

is it 901?

Actually putting your mind to it would help.

In a certain sequence of numbers, each number after the first is 3 less than twice the previous.  If the third number in the sequence is 51, what is the first number of the sequence?     

To solve this problem, add 3 to 51 to get 54, then halve it to get 27.  

                           then add 3 to 27 to get 30, then halve it to get 15.  

What is angle AHC?

Hello Guest!

Let the base points of the heights from corners A, B, C of the triangle be a, b, c.

\(ACH=90-CAB=90-43=47\\ HCB=ACB-ACH=70-47=23\\ CHb=90-ACH=90-47=43\\ CHa=90-HCB=90-23=67\\ AHC=180-CHa=180-67=113 \)

\(AHC=113°\)

The angle AHC is 113°.

  !

Hello Guest,

как называется число с 83 нулями после числа

what is the number with 83 zeros after the number,

the number called 100 Tredecillion. (in German 100 Tredezilliarde and in Russian 100 тридециллионов.) I translated it with DeepL.

Straight

Russian:

Здравствуйте, гость,

число называется 100 Тредециллионов. (на немецком 100 Tredezilliarde и на русском 100 тридециллионов.) Я перевел его с помощью DeepL.

Straight

How did you come up with those Alan?  

Is there some technique you can show us?

You got it   

Thanks catmg

Use a contour probability map to solve.

Area of little (desirable ) triangle = 0.5

Area of large (sample space) triangle = 0.5*5*5 = 12.5

probability of  that the yare all less than 2 units =  0.5 / 12.5  = 5/125  = 1/25   or  0.04   or  4%

quadratic in terms of cos(x)?

The following also work:

  \(a=\frac{2}{\sqrt3}, b=\frac{1}{\sqrt3}\\a=-\frac{2}{\sqrt3}, b = -\frac{1}{\sqrt3}\)

Hello Guest,

why you had multiply 11424 by 8/7? I thought you would do it with 7/8.

Straight

This was answered yesterday or the day before.  Go look for it.

if the angle is acute then

sin(x)=cos(pi/2 - x)

You can draw a right-angled triangle to convince yourself that this is true.

use that.