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`

With Mathematics Induction, Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11` for all natural number `n`

 With Mathematics Induction, Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11` for all natural number `n` .

Solution

Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11`

• If `n=0` then `A_0=2+9=11` It is true that `A_0 \equiv0(mod11)`
• If `n=1` then `A_1=2^7+9^2=209=11.19` It is true that `A_1 \equiv0(mod11)`

We assuming that: `A_k \equiv0(mod11)` for all natural number `k`

Let `(a_n)` be a sequence of real number satisfied that: `a_0=1` and `a_(n+1)=a_0a_1......a_n+4` for all natural number n. Prove that: `a_n-\sqrt{a_{n+1}}=2` for all `n>0`

 Let `(a_n)` be a sequence of real number satisfied that: `a_0=1` and 

`a_(n+1)=a_0a_1......a_n+4` for all natural number n.

Prove that: `a_n-\sqrt{a_{n+1}}=2` for all `n>0`.

Solution

We observation that `a_k>0` for all natural number `n`

And from `a_(n+1)=a_0a_1......a_n+4`

It is give function `f(x)` for all real number of `x` such that: `x(2x+1)f(x)+f(1/x)=x+1`

It is give function `f(x)` for all real number of `x` such that:
`x(2x+1)f(x)+f(1/x)=x+1`.

Find the sum of: `S=f(1)+f(2)+f(3)+.......+f(2022)`

Solution

From `x(2x+1)f(x)+f(1/x)=x+1` (1)then we replace `x=1/x` so will get new relation :
            `1/x(2/x+1)f(1/x)+f(x)=1/x+1`

            `f(1/x)+(x^2/(x+2))f(x)=(x^2+x)/(x+2)` (2)

It is given polynomial `P(x)=(xsina+cosa)^n` for natural number n. Finding the remaining when `P(x)` divide to `(x^2+1)`

 It is given polynomial `P(x)=(xsina+cosa)^n` for natural number n.

Finding the remaining when `P(x)` divide to `(x^2+1)`

Solution

Let `R(x)` be the remainder of division between `P(x)` and `(x^2+1)`

• If `n=1` then `P(x)=xsina+cosa`

Therefore, the remainder `R(x)=xsina+cosa`

• If `n` ≥ `2` then `P(x)=(x^2+1)Q(x)+R(x)`

which `Q(x)` is the result and `R(x)=Ax+B`

Then, `(xsina+cosa)^n=(x^2+1)Q(x)+Ax+B`

If `x=i` then `cosna+isinna=B+iA`

Therefore, `A=sin(na) ;B=cos(na)`

Hence, the remainder is `R(x)=sin(na)x+cons(na)` .

Solution by: Thin Sokkean 

It is given three positive real number `x,y,z` such that : `cos x+cosy+cosz=0` and `cos3x+cos3y+cos3z=0`

 It is given three positive real number `x,y,z` such that : `cos x+cosy+cosz=0` and `cos3x+cos3y+cos3z=0`.

Prove that: `cos 2x.cos 2y.cos 2z`≤ 0

    Solution

Prove that : `cos 2x.cos 2y.cos 2z`≤ 0

Following formula : 

`\cos3a=4\cos^3a-3\cos a` then we can see that:

Finding the sum of following problem: `S_n=1.1!+2.2!+3.3!+.....+n.n!` which `(n!)=1.2.3....n`

 Finding the sum of following problem:
`S_n=1.1!+2.2!+3.3!+.....+n.n!` which `(n!)=1.2.3....n`

Solution

In order to solve kind of these problem, you need to start with the general term of the sequence: 

Exactly, the general term of our problem here is `k.k!`

We can rewrite such that: `k.k! = (k+1-1)k! =(k+1)k!-k! =(k+1)!-k!`

From that we can replace the value of k following our main problem:

It is given two positive real numbers `x,y` which are satisfied that `4x+3y=11`. Find the maximum value of the following function: `f(x,y)=(x+6)(y+7)(3x+2y)`

 It is given two positive real numbers `x,y` which are satisfied that `4x+3y=11`. 

Find the maximum value of the following function: 

`f(x,y)=(x+6)(y+7)(3x+2y)`

Solution

Find the maximum value of the following function: 

It is give positive number `x,y,z` which satisfied that `x^2+y^2=z^2`. Prove that: `xyz` is divided by `x+y+z`.

 It is give positive number `x,y,z` which satisfied that `x^2+y^2=z^2`. Prove that: `xyz` is divided by `x+y+z`.

Solution

From the following hypothesis: `x^2+y^2=z^2` we can rewrite such as: `x^2+y^2+2xy=z^2+2xy` 

                                            `(x+y)^2=z^2+2xy`

                                            `2xy=(x+y+z)(x+y-z)`

We can see that: `(x+y+z)` or `(x+y-z)` is divided by 2 which `(x+y-z)` gives remain same to `(x+y+z)` then `(x+y-z)` is dived by 2 exactly.
Let, `x+y-z=2k` then `xy=k(x+y+z)`then `xyz` is dived by `x+y+z`.

Solution by: Thin Sokkean

Best Resources For Preparing IMO

    In fact, If you want to join any competition, you need to train again and again with kind of those thing. Mathematics as well, You need to take time, spend time, work hard, keep patience. But, the most important things are resources. 

You need to find the sample tests, sample form of the test so that you can get more understand and also more effective than someone who has no the resources.

Here are some of the resources that I used to find when I prepared for IMO:

• Art of Problem Solving : an online forum for mathematical enthusiasts. It has an extensive contest section where you can always find problems to challenge yourself. In particular, the best for preparation for the IMO are past IMO and IMO shortlist problems, but also some other national or international Olympiads (EGMO, MEMO, USA, Canada, Russia …)
• Yufei Zhao : handouts are also very good.
• Evan Chen's Website : with his awesome handouts.
• Euclid Geometry For IMO : from Evan Chen has recently been published. This book provides a very solid background in geometry. It assumes no previous knowledge
• 14 Number Theory Problems : You can find It in Amazon.
• 102 Combination Problem

I believe the most efficient way to prepare for the IMO is to do as many problems as you can. This needn’t be limited to the problems from one resource so don’t feel obliged to go through one book systematically. (Also, all mentioned books are available on the internet, but I am not sure if legally.)

Moldova Mathematical Olympiad | 2000 Q2

    This problem is being picked up from the Moldova Mathematical Olympiad in 2002 (Question 02) . In this following page, you will see how to find the root of non-linear equation system with more power of variables.

Compare these number : 50^99 and 99!

 In other to compare two of number, we have several methods to prove . In here, I will use the division operation to see which number is bigger. 

If A is bigger than B, we can write in the fraction A/B is bigger than 1. 

New Zealand Mathematical Olympiad 2019 Question 5

 Noticed that: All IMO Problems are not too difficult for you to solve, It is just because you either rarely study or never face with kind of those problems. For my opinion, you should research more about those problems to make sure that you have ability to join to contest

2015-2016 Slovakian Math Olympiad

 Slovakian Math Olympiad | 2015-2016

Noticed that: All IMO Problems are not too difficult for you to solve, It is just because you either rarely study or never face with kind of those problems. For my opinion, you should research more about those problems to make sure that you have ability to join to contest

1985 International Mathematics Olympiad

 This is the problem that I picked up from 1985 International Mathematics Olympiad Long List Problems. I will make the solution for you by my Microsoft Word and convert it to images for you to see my solution. 

Noticed that: All IMO Problems are not too difficult for you to solve, It is just because you either rarely study or never face with kind of those problems. For my opinion, you should research more about those problems to make sure that you have ability to jion to contest.

 

Solution By Thin Sokkean