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Do not post A O P S homework.

They will each get

\(\frac{1}{4}\;\;of\;\;\frac{1}{2}\)

what symbol is used for  'of'

I mean is it +  or  -  or  times or divide ?

Let $P(x)$ be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers $n$  such that $P(n)$ is composite.

How about putting a little effort into presenting your question properly.

Note that \(2^a2^b2^c....=2^{a+b+c...}\)

In your case \(a+b+c\) forms an infinite geometric series, ie. \(b = ar, c = ar^2, ...etc\)

The limit of the infinite geometric series here is \(\frac{a}{1-r}\) or \(\frac{1/3}{1-1/3}\) 

You can complete the calculation.

edited, original answer was in error.   Thanks Alan.

3^1=3

3^2=9

3^3=27

3^4=81

None above are equivalent to 5 (mod7)

3^5 = 243 which is equivalent to 5 (mod7)    So    i=5

Go through the same process to find j.   

What is j?    Your turn. 

Find the value of n.

Hello Guest!

\(\frac{1}{27^{-4}}\cdot 9=81^n\\ 27^4\cdot9=81^n\\ 3^{3\cdot 4}\cdot 3^2=3^{4n}\\ 3^{3\cdot 4+2}=3^{4n}\\ 3\cdot 4+2=4n\\ 4n=14\)

\(n=3.5\)

  !

If  

F=4

R=0

and all the rest are 9

then the sum would be   9989

a/45=N+37/45               b/30=K+9/37                             (a+b)/15=what remainder         

a (mod45)=37     so     a mod15= 7     

b(mod30)=9        so     b mod 15 = 9

a+b (mod15) = (7+9)mod15    =   1 mod 15

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If people copy incorrect answers then it serves them right.

Let the number of fruits be x,

the number of pears be y the number of oranges be z.

y:z = 2:7

                           => y = 2/9 (3/5x)
y + z = x (1 - 2/5)     z = 7/9 (3/5x)

Apples : 2/5x

Pears: 2/15x

Oranges: 7/15x

2/5x: 2/15x: 7/15x = 6:2:7

See the solution here:  https://web2.0calc.com/questions/modular-number-theory#r6

                     Box A              Box B

Before transferred 563            187

After transferred  563 - x        187 + x

                (563 - x) - (187 + x) = 48

                             2x = 328

                                x = 164

So, in the end

        Box A                                      Box B

  563 - x = 563 - 164 = 199       187 + 164 = 351

Suppose the full price cost of a bench is $x.

then x + 2 * (1 - 25%)x + 2 = (1 - 50%)x = 385

        x + 2 * 0.75x + 2 * 0.50x = 385

                 x + 1.5x + x = 385

                               3.5x = 385

                                    x = 110 dollars

We can rewrite \(\large{{u \over v} + {v \over u} }\) as \(\large{{u^2 \over xy} + {v^2 \over xy}}\). Because these have common denominators, we can simplify furthur: \(\large{{x^2+y^2} \over {xy}}\). Now recall the identity: \(a^2+b^2=(a+b)^2-2ab\). We can then apply this here, to get: \(\large{{(u+v)^2-2uv} \over uv}\).

Using Vieta's, we know that \(u+v= -{ b\over a} = -{5 \over 3}\) and \(uv = {c \over a} = {11 \over 3} \)

Now, we have to substitute these values and simplify. 

Can you take it from here?

The answer was 1154.

By the intersecting chord theorem, we have this equation: \(AQ \times BQ = CQ \times DQ\)

Substituting what we know, we have: \(6 \times 12 = DQ \times CQ\)

Let \(CQ = x\) and \(DQ = y\). 

We can form the system: \(xy = 72\) and \(x+y = 30\)

Now, we have to solve for and take the smaller option for x. 

Can you do it from here?

Hint: You will substitute into a quadratic, and you will have to use the formula: \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

That was incorrect. Are you sure?

For the equation to have a double root, the discriminant (\(b^2-4ac\)) must equal 0. Substituting what we know, we have: \(b^2-4 \times 16 = 0 \)

Now, we have to solve for b. 

Can you take it from here?

angle b=60 degrees

tan(b)=√3

A) The largest possible degree of f(x) + ag(x) is 8.

B) The smallest possible degree of f(x) + bg(x) is 2.