Mathematics Problem Everyday
1- It is given that : En=831n+709n-743n-610n for all natural number n.
Prove that: En 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√2+1√3+...
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)>2satisfied 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\geq1satisfied 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