If \(n=0\), then: \(A_0=2^{6(0)+1}+9^{0+1}=2+9=11\equiv0\pmod{11}\) Therefore, \(A_0\) is divisible by \(11\).

Proof by Mathematical Induction

We prove that \(A_n=2^{6n+1}+9^{n+1}\) is divisible by \(11\) for all natural numbers \(n\).

Step 1: Base case

If \(n=0\), then:

\(A_0=2^{6(0)+1}+9^{0+1}=2+9=11\equiv0\pmod{11}\)

Therefore, \(A_0\) is divisible by \(11\).

Step 2: Induction hypothesis

Assume that for \(n=k\):

\(A_k=2^{6k+1}+9^{k+1}\equiv0\pmod{11}\)

Step 3: Prove for \(n=k+1\)

We have:

\[ A_{k+1}=2^{6k+7}+9^{k+2} \]

\[ A_{k+1}=2^6(2^{6k+1})+9^{k+1}\cdot9 \]

\[ A_{k+1}=64(2^{6k+1}+9^{k+1})+9^{k+1}(9-64) \]

\[ A_{k+1}=64A_k-55\cdot9^{k+1} \]

Since \(A_k\equiv0\pmod{11}\) and \(55\equiv0\pmod{11}\), we get:

\[ A_{k+1}\equiv64(0)-0\cdot9^{k+1}\equiv0\pmod{11} \]

Therefore, \(A_{k+1}\) is divisible by \(11\).

Hence, by mathematical induction, \(A_n=2^{6n+1}+9^{n+1}\) is divisible by \(11\) for all natural numbers \(n\).

Practice Problem For You

Mathematics Problems

Problem 01

It is given that:

\(E_n=831^n+709^n-743^n-610^n\)

for all natural numbers \(n\). Prove that \(E_n\) is divisible by \(189\) for all natural numbers \(n\).

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

Problem 02

Prove that for all natural numbers \(n\), we have:

\[ 1+\frac1{\sqrt2}+\frac1{\sqrt3}+...+\frac1{\sqrt{n+1}}<2\sqrt{n+1} \]

Hint: Using Mathematical Induction.

Problem 03

It is given a natural real sequence satisfied that:

\[ U_0=\sqrt2 \]

\[ U_{n+1}=\sqrt{2+U_n} \]

a. Find \(U_n\) as a function of \(n\).

b. Find the product:

\[ P_n=U_0U_1U_2...U_n \]

Problem 04

There is a 4-digit number with every single digit arranged as:

\(aabb\)

Find those numbers if they are perfect squares.

Problem 05

It is given that:

\[ 33^2=1089 \]

\[ 333^2=110889 \]

\[ 3333^2=11108889 \]

\[ 33333^2=1111088889 \]

From the given examples, find the general term and prove it.

Problem 06

a. Prove that:

\[ 1+\frac{1}{\cos x}=\frac{\cot(x/2)}{\cot x} \]

b. Calculate the product:

\[ P_n=(1+\frac1{\cos a})(1+\frac1{\cos(a/2)}) (1+\frac1{\cos(a/2^2)})\cdots (1+\frac1{\cos(a/2^n)}) \]

Problem 07

Calculate the value of:

\[ S=\cos^3(\frac{\pi}{9})-\cos^3(\frac{4\pi}{9}) +\cos^3(\frac{7\pi}{9}) \]

Problem 08

Calculate the sum:

\[ S_n=9+99+999+\cdots+\underbrace{99\cdots9}_{n\text{ digits}} \]

Problem 09

Find all pairs of integers \((m,n)>2\) satisfied that for all positive integers \(a\):

\[ \frac{a^m+a-1}{a^n+a^2-1} \]

is an integer.

Solution: \((m,n)=(5,3)\)

Problem 10

It is given three positive integers \(a,b,c\) satisfied that:

\[ a+b+c=10 \]

Find the minimum value of:

\[ P=a\times b\times c \]

Solution: \(P=36\)

Problem 11

Find the exact value of:

\[ \sin(\frac{\pi}{10}) \quad \text{and} \quad \cos(\frac{\pi}{10}) \]

Problem 12

It is given two positive real numbers \(a\) and \(b\). Prove that:

\[ (1+a)(1+b)\geq(1+\sqrt{ab})^2 \]

From the result above, find the minimum value of the function:

\[ f(x)=(1+4^{\sin^2x})(1+4^{\cos^2x}) \]

for all real numbers \(x\).

Problem 13

It is given three real numbers \(a,b,c\). Prove that:

\[ a^2+b^2+c^2\geq ab+bc+ac \]

Problem 14

It is given \(n\) positive real numbers:

\[ a_1,a_2,a_3,\ldots,a_n \]

satisfied that:

\[ a_1a_2a_3\cdots a_n=1 \]

Prove that:

\[ (1+a_1)(1+a_2)(1+a_3)\cdots(1+a_n)\geq2^n \]

Problem 15

It is given \(m,n\) are positive integers. Prove that for all positive real numbers \(x\):

\[ \frac{x^{mn}-1}{m}\geq\frac{x^n-1}{x} \]

Mathematics Problems Collection

Problem 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 \(\gcd(9,21)=189\).

Problem 2

Prove that for all natural number \(n\): \[ 1+\frac1{\sqrt2}+\frac1{\sqrt3}+...+\frac1{\sqrt{n+1}} <2\sqrt{n+1} \]

Hint: Using Mathematical Induction.

Problem 3

It is given the real sequence: \[ U_0=\sqrt2 \] and \[ U_{n+1}=\sqrt{2+U_n} \]

a. Find \(U_n\) as a function of \(n\).

b. Find the product: \[ P_n=U_0U_1U_2...U_n \]

Problem 4

There is a 4 digit number with every single digit in the order: \[ aabb \] Find those numbers if they are perfect squares.

Problem 5

It is given:

\[ 33^2=1089 \]

\[ 333^2=110889 \]

\[ 3333^2=11108889 \]

\[ 33333^2=1111088889 \]

Find the general term and prove it.

Problem 6

a. Prove that: \[ 1+\frac1{\cos x}=\frac{\cot(x/2)}{\cot x} \]

b. Calculate: \[ P_n= (1+\frac1{\cos a}) (1+\frac1{\cos(a/2)}) (1+\frac1{\cos(a/2^2)}) ... (1+\frac1{\cos(a/2^n)}) \]

Problem 7

Calculate the value: \[ S=\cos^3(\frac{\pi}{9}) -\cos^3(\frac{4\pi}{9}) +\cos^3(\frac{7\pi}{9}) \]

Problem 8

Calculate the sum: \[ S_n=9+99+999+...+999 \] where the last number contains \(n\) digits of 9.

Problem 9

Find all pairs of integers \((m,n)>2\) such that for every positive integer \(a\):

\[ \frac{a^m+a-1}{a^n+a^2-1} \]

is an integer.

Problem 10

It is given three positive integers \(a,b,c\) satisfying: \[ a+b+c=10 \]

Find the minimum value of: \[ P=a\times b\times c \]

Mathematics Problems Collection

Problem 11

Find the exact value of:

\[ \sin(\frac{\pi}{10}) \] and \[ \cos(\frac{\pi}{10}) \]

Problem 12

It is given two positive real numbers \(a\) and \(b\). Prove that:

\[ (1+a)(1+b)\geq(1+\sqrt{ab})^2 \]

From the proven result, find the minimum value of:

\[ f(x)=(1+4^{\sin^2x})(1+4^{\cos^2x}) \]

for all real numbers \(x\).

Problem 13

It is given three real numbers \(a,b,c\).

Prove that:

\[ a^2+b^2+c^2\geq ab+bc+ac \]

Problem 14

It is given \(n\) positive real numbers:

\[ a_1,a_2,a_3,...,a_n \]

satisfying:

\[ a_1a_2a_3...a_n=1 \]

Prove that:

\[ (1+a_1)(1+a_2)(1+a_3)...(1+a_n)\geq2^n \]

Problem 15

It is given positive integers \(m,n\). Prove that for all positive real numbers \(x\):

\[ \frac{x^{mn}-1}{m}\geq\frac{x^n-1}{x} \]

Problem 16

For all real numbers \(x\), prove that:

\[ (1+\sin x)(1+\cos x) \leq \frac32+\sqrt2 \]

Problem 17

It is given:

\[ x_n=2^{2^n}+1 \]

for \(n=1,2,3,...\)

Prove that:

\[ \frac1{x_1} +\frac2{x_2} +\frac{2^3}{x_3} +... +\frac{2^{n-1}}{x_n} <\frac13 \]

Problem 18

It is given the function:

\[ y=\frac{x^2+2mx+3m-8}{2(x^2+1)} \]

where \(x\) is a real number and \(m\) is a parameter.

Is it possible to find a value of \(m\) to make the function \(y\) be the value of cosine of one single angle?

Problem 19

It is given the function:

\[ f(x,y)= \frac{(x^2-y^2)(1-x^2y^2)} {(1+x^2)^2(1+y^2)^2} \]

where \(x,y\) are real numbers.

Prove that:

\[ |f(x,y)|\leq\frac14 \]

Problem 20

It is given:

\[ 0<\theta<\frac{\pi}{2} \]

Prove that:

\[ (\sin\theta)^{\cos\theta} + (\cos\theta)^{\sin\theta} >1 \]

Mathematics Problems Collection

Problem 21

There are three real numbers: \[ a>0,\quad b>0,\quad c>0 \]

Prove that:

\[ ab(a+b)+bc(b+c)+ac(a+c)\geq6abc \]

Problem 22

Solve the equation:

\[ 9^{(x^2-x)}+3^{(1-x^2)} = 3^{(x-1)^2}+1 \]

Problem 23

Find both functions \(f(x)\) and \(g(x)\) satisfying:

\[ f(2x-1)+2g(3x+1)=x^2 \]

\[ f(4x-3)-g(6x-2)=-2x^2+2x+1 \]

Problem 24

Find the sum:

\[ S_n= \tan a+\frac12\tan\frac a2 +\frac1{2^2}\tan\frac a{2^2} +... +\frac1{2^n}\tan\frac a{2^n} \]

Problem 25

It is a third degree polynomial \(P(x)\) satisfying:

\[ P(x)+2 \] is divisible by \[ (x+1)^2 \]

and

\[ P(x)-2 \] is divisible by \[ (x-1)^2 \]

Determine the polynomial \(P(x)\).

Problem 26

Let:

\[ A= (\frac1{\sqrt3}+i)^n - (\frac1{\sqrt3}-i)^n \]

for all natural numbers \(n\).

Prove that:

\[ A= i\frac{2^{n+1}}{(\sqrt3)^n} \sin\frac{n\pi}{3} \]

Problem 27

Solve the equation in integer set:

\[ 47x+29y=1 \]

Problem 28

Find all possible functions:

\[ f(x) \]

satisfying:

\[ f(x+\sqrt{x^2-2x+1}) = \frac{x^2-1}{x^2+1} \]

Problem 29

It is given a sequence of real numbers:

\[ (a_n),\quad n\geq1 \]

satisfying:

\[ a_1=1,\quad a_2=3 \]

and

\[ a_{n+2}=(n+3)a_{n+1}-(n+2)a_n \]

Evaluate the value of \(n\) if:

\[ a_n\equiv0\pmod{11} \]

Problem 30

Prove that for all positive integers \(n\):

\[ 3^n+n^3 \]

is divisible by \(7\) if and only if:

\[ 3^n n^3+1 \]

is divisible by \(7\).

Mathematics Problems Collection

Problem 31

Find the sum:

\[ S_n= \frac{3}{1!+2!+3!} +\frac{4}{2!+3!+4!} +... +\frac{n+2}{n!+(n+1)!+(n+2)!} \]

Problem 32

Prove that:

\[ 16< \sum_{k=1}^{80}\frac1{\sqrt{k}} <17 \]

(China 1992)

Problem 33

Find all real numbers \(x\) satisfying:

\[ 2^x+3^x-4^x+6^x-9^x=1 \]

(Korean 2000)

Problem 34

It is given positive real numbers:

\[ X_1,X_2,X_3,...,X_n \]

satisfying:

\[ \sum_{i=1}^{n}X_i=1 \]

Prove that:

\[ \left(\sum_{i=1}^{n}\sqrt{X_i}\right) \left(\sum_{i=1}^{n}\frac1{\sqrt{1+X_i}}\right) \leq \frac{n^2}{\sqrt{n+1}} \]

(China Team Selection Test 2006)

Problem 35

There are \(a,b,c\) non-negative real numbers satisfying:

\[ ab+bc+ca=\frac13 \]

Prove that:

\[ \frac1{a^2-bc+1} + \frac1{b^2-ac+1} + \frac1{c^2-ab+1} \leq\frac13 \]

(China Team Selection Test 2005)

Problem 36

It is given the sequence of real numbers:

\[ a_1=1,\quad a_2=5 \]

and:

\[ a_{n+1} = \frac{a_na_{n-1}} {\sqrt{a_n^2+a_{n-1}^2+1}}, \quad n\geq2 \]

Determine the general term of \((a_n)\).

(China 2002)

Problem 37

Find all functions:

\[ f(x):\mathbb{R}\rightarrow\mathbb{R} \]

such that:

\[ f([x]y)=f(x)[f(y)] \]

is true for all \(x,y\in\mathbb{R}\).

where \([a]\) is the greatest integer less than or equal to \(a\).

(IMO 2010)

Problem 38

It is given:

\[ f(x)=\frac{x+4}{x+1} \]

where \(x\neq-1\).

Evaluate:

\[ f_n[f[...f[f(x)]...]] \]

Problem 39

It is given the function relation:

\[ 2f(\frac{\pi}{2}-x) + f(\frac{\pi}{2}+x) = \sin x+3\sqrt3\cos x \]

Find the values of \(\theta\) and \(r\) if:

\[ f(x)=r\sin(x+\theta) \]

Problem 40

Find all pairs of integers \((a,b)\) satisfying that:

\[ \frac{x^2y+x+y}{xy^2+y+7} \]

is an integer.