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Thursday, July 15, 2021

Prove that: An=3n+3-44n+2 is divided by 11

 Prove that: An=3n+3-44n+2 is divided by 11

Solution

We will prove by Mathematics Induction:

If n=0 then An=33-42=27-16=11 It is true that An is divided with 11

Assuming that: An is divided with 11 for n=k        (1)

We will prove it is true for n=k+1 then Ak+1 is divided with 11

Wednesday, July 14, 2021

Prove that: xR : |acosx+bsinxa2+b2

a. Prove that: xR : |acosx+bsinxa2+b2

b. Find the maximum and minimum of f(x)=20cosx+21sinx+27

Solution

a. Prove |acosx+bsinx|a2+b2 xR

Assuming we have: 

acosx+bsinx=a2+b2(aa2+b2cosx+ba2+b2sinx)

X1;X2 are the root of : X2-(2cost+3sint)X-11sin2t=0 Find the minimum of A=X21+X1X2+X22.

X1;X2 are the root of : X2-(2cost+3sint)X-11sin2t=0 

Find the minimum of A=X21+X1X2+X22.

Solution

We can see that: A=X21+X1X2+X22

                                 =(X1+X2)2-X1X2

Following Vieta's Formulas X2-SX+P=0    (1)

Monday, July 12, 2021

Find all Polynomials P(x) which is satisfied that: (x-2010)P(x+67)=xP(x) (2010 Baltic Way)

 Find all Polynomials P(x) which is satisfied that:

(x-2010)P(x+67)=xP(x)

(2010 Baltic Way) 

Prove that: tan3x1-3tan2x=18(tan3x-3tanx)

 a. Prove that: tan3x1-3tan2x=18(tan3x-3tanx)

As we knew: tan3x=3tanx-tan3x1-3tan2x

                                  =3tanx-9tan3x+8tan3x1-3tan2x

                                  =8tan3x+3tanx(1-3tan2x)1-3tan2x

                                  =8tan3x1-3tan2x+3tanx

Then,             tan3x-3tanx=8tan3x1-3tan2x

Find the limit of Sn=21.3+23.5+25.3+...

 Find the limit of S_n=2/(1.3)+2/(3.5)+2/(5.3)+.......+2/((2n+1)(2n+3))

Solution

We can see that general term of this sequence is:

                    2/((2k+1)(2k+3))=1/(2k+1)-1/(2k+3)

Thursday, July 8, 2021

If g(x)=f(x)+1-x , find the value of g(2020)

 It is given function f(x) determine on R , satisfied that:

            f(1)=1

            f(x+5)\geqf(x)+5

            f(x+1)\leqf(x)+1

If g(x)=f(x)+1-x , find the value of g(2020)   \forallx,y\inR

If (1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20) then find the coefficients of a_1 ; a_2 ; a_3 and a_20

  1. If (1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20) then find the coefficients of a_1 ; a_2 ; a_3 and a_20

Solution

        From our hypothesis we already had:

                        (1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20)

        We will find the coefficients of a_1 ; a_2 ; a_3 and a_20

        Which meant they are the coefficients of x ; x^2 ; x^3 and x^(20) .

        We can rewrite (1+2x+3x^2)^10=[1+x(2+3x)]^10

        We will use the Newton's Formula  

            (a+b)^n=\sum_{i=0}^nC(n,i)a^(n-i).b^i

        Then, [1+x(2+3x)]^10=C(10,0)+C(10,1)x(2+3x)+C(10,2)x^2.(2+3x)^2

                                            +C(10,3)x^3.(2+3x)^3+....+C(10,10)x^10.(2+3x)^10

        We observe that: 

            Coef(x)=2C(10,1)=20 then, a_1=20

            Coef(x^2)=2^2C(10,2)=4.45=180 then, a_2=180

           Coef(x^3)=12C(10,2)+8C(10,3)=45.12+120.8=1500 then a_3=1500

            Coef(x^20)=C(10,10).3^10=3^10 then, a_20=3^10     

Wednesday, July 7, 2021

Prove (21n+4)/(14n+3) Is Irreducible For Every Natural Number n

Prove (21n+4)/(14n+3) Is Irreducible For Every Natural Number n
Solution

 Solution 01

Denoting the greatest common divisor (GCD) of a and b (a,b) and we will use Euclidean Algorithm Theory.

                (21n+4,14n+3)=(7n+1,14n+3)=(7n+1,1)=1

It follows that (21n+4)/(14n+3) is irreducible. Q.E.D

 Solution 02

Tuesday, July 6, 2021

Vietnam Math Out Standing Student 2012-13

 

Vietnam Math Out Standing Student 2012-13

Problem 01: Solve the equation of: x^(4n)+\sqrt{x^{2n}+2012}=2012    for all n\in N.
Problem 02: It is given (U_n) which is determined by: 
                    U_1=3
                    U_(n+1)=1/3(2U_n+3/U_n^2)    for all n\in N.
                    Let's finding the limit of: \lim_{n\rightarrow\infty}(u_n).
Problem 03: It is given three non-negative real numbers x,y,z , Prove that:
                    1/x+1/y+1/z>36/(9+x^2y^2+y^2^2+z^2x^2).
Problem 04: Find roots positive numbers of equation:
                    \sqrt{x+2\sqrt3}=\sqrt y+\sqrt z

Solution

Monday, July 5, 2021

Find all functions f:R\rightarrow R such that: f(x)f(y)=f(xy-1)+xf(y)+yf(x) \forall x,y \inR


Find all functions f: R \rightarrow R Such that:
f(x)f(y)=f(xy-1)+xf(y)+yf(x)    \forall x ,y \inR

Solution

Replace : y=0 we will see that: f(x).f(0)=f(-1)+x.f(0)
Notice that: f(0) not equal to 0 then, f(x)=x+c     which c is a constant. We replaced to our hypothesis unsatisfactory. Then, f(0)=0 from that we can see that: f(-1)=0.