Mathematics Problem Everyday

Mathematics Problem Everyday

1- It is given that : `E_n=831^n+709^n-743^n-610^n` for all natural number `n`.

    Prove that: `E_n` is divided by `189` for all natural number `n`.

    Hint: Using modulo formula and `gcd(9,21)=189`

2- Prove that for all natural number `n` we have the following inequation:
           `1+1/\sqrt{2}+1/\sqrt{3}+....+1/sqrt{n+1}<2sqrt{n+1}` 

    Hint: Using Mathematics Induction.

3- It is given natural real sequence satisfied that: 

                    `U_0=\sqrt{2}` and `U_(n+1)=sqrt{2+U_n}`

    a. Find the `U_n` which is functioned to `n`.

    b. Find the product of `P_n=U_0.U_1.U_2......U_n`

4- There is a 4 digits number with every single digit is `a` ; `a` ; `b` ; `b` correct order. 

    Find those number If It is a perfect square.

5- It is given that : `33^2=1089` , `333^2=110889` , `3333^2=11108889` , `33333^2=1111088889`.

    From the following given, let find the general term of it and prove.

6- a. Prove that: `1+1/cosx=cot(x/2)/cotx`.

    b. Calculate the product of :
    `P_n=(1+1/cosa)(1+1/cos(a/2))(1+1/cos(a/2^2))......(1+1/cos(a/2^n))`.

7- Calculating the value of : `S=cos^3(\pi/9)-cos^3(4\pi/9)+cos^3(7\pi/9)`.

8- Calculating the sum of : `S_n=9+99+999+....+999` which is `n` times of number `9`.

9- Find all pair of integer `(m,n)>2`satisfied that: for all positive integer 

                    `a`  we got `(a^m+a-1)/(a^n+a^2-1)` be a integer. 

        Solution is `(m,n)=(5,3)`

10- It is given `3` positive integer `a,b,c` satisfied that `a+b+c=10`.

       Find the minimum value of `P=a\timesb\timesc`.

        Solution is `P=36`.

11- Find the exact value of `sin(\pi/10)` and `cos(\pi/10)`.

12- It is given two positive real number `a` and `b`. Prove that: `(1+a)(1+b)\geq(1+\sqrt{ab})^2`

       From the proven, let's minimum of function :
                `f(x)=(1+4^(sin^2x))(1+4^(cos^2x))` for all  real number `x`.

13- It is given three real numbers `a,b,c` .

        Prove that: `a^2+b^2+c^2` ≥ `ab+bc+ac`

14- It is given `n` positive real numbers `a_1;a_2;a_3;.....;a_n` satisfied that
        `a_1.a_2.a_3......a_n=1`.

       Prove that:  `(1+a_1)(1+a_2)(a+a_3)........(1+a_n)` ≥ `2^n`.

15- It is given `m,n` be the positive integers . Prove that for all real positive number `x` 
             `(x^(mn)-1)/m` ≥`(x^n-1)/x`

16- For all real number `x` prove that: 

            `(1+sinx)(1+cosx)` ≤ `3/2+\sqrt{2}`

17- It is given `x_n=2^(2^n)+1` for `n=1,2,3......`

       Prove that: `1/x_1+2/x_2+2^3/x_3+......+2^(n-1)/x_n` < `1/3`.

18- It is given function `y=(x^2+2mx+3m-8)/(2(x^2+1)` which `x` is real number and `m` is a 

       parameter of the problem. Is It possible that we can find the value of `m` to make 

       function `y` be the value of `cos` of one single angle ?

19- It is given two variables function:
            `f(x,y)=((x^2-y^2)(1-x^2y^2))/((1+x^2)^2(1+y^2)^2) ;x,y ` be real number.

       Prove that: `|f(x,y)| \leq1/4`.

20- It is given `\theta` be a real number such that `0<\theta<\pi/2`.

       Prove that: `(sin\theta)^(cos\theta)+(cos\theta)^(sin\theta)>1`.

21- There are three real number: `a>0 ; b> ; c>0` Prove that:
        `ab(a+b)+bc(b+c)+ac(a+c)\geq6abc`

22- Solve the equation below:
       `9^((x^2-x))+3^((1-x^2))=3^((x-1)^2)+1`

23- Let's finding both functions `f(x)` and `g(x)` which satisfied that: 

               `f(2x-1)+2g(3x+1)=x^2`

                `f(4x-3)-g(6x-2)=-2x^2+2x+1`

23- Find the sum of:
       `S_n=tana+1/2tan(a/2)+1/2^2tan(a/2^2)+....+1/2^ntan(a/2^n)`

24- It is `3rd` degree polynomial `P(x)` which is satisfied that:
                    `P(x)+2` is divided by `(x+1)^2`

                    `P(x)-2` is divided by `(x-1)^2`

      Determind the polynomial `P(x)` ?

25- Let `A=(1/\sqrt{3}+i)^n-(1/\sqrt{3}-i)^n` for all natural number `n`.

       Prove tha: `A=i.2^(n+1)/(\sqrt{3})^n.sin(n\pi/3)`.

26- Solve the equation in integer set :

                    `47x+29y=1`

27- Find all function `f(x)` possible which is satisfied that:
                    `f(x+\sqrt{x^2-2x+1})=(x^2-1)/(x^2+1)`

28- It is given sequence of real number `(a_n): n\geq1`satisfied that: `a_1=1;a_2=3`

       and `a_(n+2)=(n+3)a_(n+1)-(n+2)a_n` ; `\forall n \in R`.

      Evaluate the value of `n` if `a_n\equiv0(mod11)`.

29- Prove that for all positive integer `n` , `3^n+n^3` is divided by `7` if only if `3^n n^3+1` is divided by `7`.

30- Find the sum of:
          `S_n = 3/(1!+2!+3!) +4/(2!+3!+4!) +.....+(n+2)/(n!+(n+1)!+(n+2)!)`

31- Prove that: `16<\sum_{k=1}^{80}``1/sqrt{k}``<17`. (China 1992) 

32- Find all real number `x` satisfied the equation below:
                           `2^x+3^x-4^x+6^x-9^x=1`     (Korean 2000)

33- It is given `X_1 , X_2 , X_3 , ......., X_n` be positive real number satisfied that:
       `\sum_{i=1}^{n}X_i =1`. Prove that :
            `(\sum_{i=1}^{n}\sqrt{X_i})(\sum_{i=1}^{n}1/\sqrt{1+X_i})\leqn^2/\sqrt{n+1}`

(China Team Selectioin Test 2006)

34- There are `a, b, c` be nonnegative real number which `ab+bc+ac=1/3`

        Prove that: `1/(a^2-bc+1)+1/(b^2-ac+1)+1/(c^2-ab+1)\leq1/3`

(China Team Selection Test 2005)

35- It is given sequence of real number such that: `a_1=1` `a_2=5` and
            `a_(n+1)=(a_na_(n-1))/\sqrt{(a_n)^2+(a_(n-1))^2+1} \forall ngeq2`

       Determine the general term of `(a_n)`.

(China 2002)

36- Find all function `f(x)`:`R\rightarrowR` which is satisfied the equivalent that
        `f(\left[x\right]y)=f(x)\left[f(y)\right]` is true for all `x,y\inR`.

    `[a]` is the greatest integer `\leqa`.

(IMO 2010)

37- It is given function `f(x)=(x+4)/(x+1)` which is `x is not belong to -1` 
       Evaluate the value of function : `f_n[f[....f[f(x)]....]]`.

38- It is given function with the relative of : 
       `2f(\pi/2-x)+f(\pi/2+x)=sinx+3\sqrt3cosx`

       Find the value of `\theta` and `r` if `f(x)=rsin(x+\theta)`

39- It is given equation of real number : `x^3-ax^2+bx-c=0` has three roots.
       Find the possible minimum value of `(1+a+b+c)/(3+2a+b)-c/b` .

40- Find all pair of integer `(a,b)` which are satisfied that `x^2y+x+y` is dived by `xy^2+y+7`