Math Book Cambodia: 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

$a$ ;

$b$ ;

$b$ 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

$xy^2+y+7$

Math Book Cambodia

Pages

Saturday, November 28, 2020

Mathematics Problem Everyday

No comments:

Post a Comment

Popular