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Mathematics Problem Collection

Welcome to our Mathematics Problem Collection. Here you can find problems and solutions organized by different topics.

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• VN-ENG MATH

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Given that \[ x^3+\frac{1}{x^3}=4, \] find the value of \[ P=x^6+\frac{4}{x^3}. \]

Solve the Algebraic Problem

Given that

\[ x^3+\frac{1}{x^3}=4, \]

find the value of

\[ P=x^6+\frac{4}{x^3}. \]

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Solution

Step 1. Introduce a New Variable

Let

\[ a=x^3. \]

Then the given equation becomes

\[ a+\frac{1}{a}=4. \]

Step 2. Form a Quadratic Equation

Multiply both sides by \(a\):

\[ a^2+1=4a. \]

Hence,

\[ a^2-4a+1=0. \]

Therefore,

\[ a^2=4a-1. \]

Step 3. Evaluate the Required Expression

Since

\[ P=x^6+\frac{4}{x^3}=a^2+\frac{4}{a}, \]

and

\[ \frac{1}{a}=4-a, \]

we obtain

\[ \begin{aligned} P &=a^2+4(4-a)\\ &=a^2-4a+16. \end{aligned} \]

Substitute \(a^2=4a-1\):

\[ \begin{aligned} P &=(4a-1)-4a+16\\ &=15. \end{aligned} \]

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Final Answer

\[ \boxed{15} \]

Solve the Equation

Solve the following equation:

\[ \frac{x^3+6x^2+15x+13}{x^2+4x+7} = \sqrt{x+9} - \frac{1}{x+12}. \]

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Solution

Step 1. Determine the Domain

Since the square root is defined only when

\[ x+9\ge0, \]

we must have

\[ x\ge-9. \]

Next, rewrite the numerator:

\[ x^3+6x^2+15x+13 = (x+2)(x^2+4x+7)-1. \]

Therefore,

\[ \frac{x^3+6x^2+15x+13}{x^2+4x+7} = (x+2)-\frac{1}{x^2+4x+7}. \]

Notice that

\[ x^2+4x+7=(x+2)^2+3, \]

and similarly,

\[ x+12=(\sqrt{x+9})^2+3. \]

Hence the equation becomes

Problem. Find the function \(f(x)\) satisfying $$ f(x)f(y)-f(xy)-90=\frac{10(x+y)}{xy}. $$

Problem. Find the function \(f(x)\) satisfying

$$ f(x)f(y)-f(xy)-90=\frac{10(x+y)}{xy}. $$

Solution.

Rewrite the equation as $$ f(x)f(y)-f(xy)=90+\frac{10}{x}+\frac{10}{y}. $$ Assume $$ f(x)=a+\frac{b}{x}. $$ Then

Math 2nd Day Phnom Penh Cambodia 20/04/2023

• Phnom Penh is a capital city of Cambodia where many clever students, great teachers which make students from Phnom Penh are mostly good at Math If compare to countryside students. In 2023, For 1st day of out standing student examination, I just got problems paper on social media and I will share with all of you here. Some of those problems I used to do when I was in high school.

Math Out Standing Problem 2012, 2013, 2014

•  Math out standing student whole country in Cambodia 2012, 2013, 2014 .

Math 1st Day Phnom Penh Cambodia 19/04/2023

•  Phnom Penh is a capital city of Cambodia where many clever students, great teachers which make students from Phnom Penh are mostly good at Math If compare to countryside students. In 2023, For 1st day of out standing student examination, I just got problems paper on social media and I will share with all of you here. Some of those problems I used to do when I was in highschool.

Prove that \[ \frac12\cdot\frac34\cdot\frac56\cdots\frac{2n-1}{2n} \leq \frac1{\sqrt{3n}} \]

•  This is the problem that I picked up from book : 101 Solved Problem in Algebra from The USA IMO with the 43th problem.

What is the coefficient of \(x^2\ when (1+x)(1+2x)(1+2^2x)\cdots(1+2^n x) \) is expanded?

•  This is the problem that I picked up from book : 101 Solved Problem in Algebra from The USA IMO with the 32th problem. 

Vietnamese Mathematical Olympiad 2017

•  This is the 2007 Vietnamese Mathematical Olympiad Problem number 6. This is kind of problem that you need to use Permutation Formular . Moreover, you have to know about Sum of Sigma as well. 

Math For Out-Standing Student 2023

• This is the sequence combined with logarithm function which lead us to be understood of method to find the general formula of sequence. It is the basic of method statement for you to know how to find the general form of sequence when you have the power of number in our sequence. 

Prove that \[ \left(a_n+\frac{3}{2^{n+2}}\right)^{\frac{1}{n}} \left(m-\left(\frac{2}{3}\right)^{\frac{n(m-1)}{m}}\right) < \frac{m^2-1}{m-n+1} \]

• This is the problem that I picked up from Math Book Around The World which is written by  Mr. Lim Phalkun And Mr. Sen Piseth.  This is the problem which combined many methods, many theory such  Bernualli.

Prove that \[ \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots+\frac{1}{a_{20}} \] is a natural number.

Find the function \(f(x)\) satisfying: \[ f(x)f(y)-f(xy)-90=\frac{10(x+y)}{xy} \]

Find the function \(f(x)\) satisfying:

\[ f(x)f(y)-f(xy)-90=\frac{10(x+y)}{xy} \]

Solution:

Put \(y=1\), and let \(f(1)=a\).

\[ af(x)-f(x)-90=\frac{10(x+1)}{x} \]

\[ (a-1)f(x)=100+\frac{10}{x} \]

Therefore,

\[ f(x)=A+\frac{B}{x} \]

 

Math PreyVeng Cambodia 2023

• Day 01

Math Chum Kiri District, Kampot 24/03/2023

• Solution Here:  Download PDF

Math Out Standing 2023 Provincial, Cambodia

Solution 

If \(x_1,x_2\) are the roots of the equation \[ x^2-x-3=0 \] Find the value of \[ A=7x_1^5+19x_2^4 \]

Solution:

Since \(x_1,x_2\) are roots of

\[ x^2-x-3=0 \]

we have:

\[ x^2=x+3 \]

For any root \(x\):

\[ x^3=x(x^2)=x(x+3)=x^2+3x=4x+3 \]

\[ x^4=x(4x+3)=4x^2+3x=7x+12 \]

\[ x^5=x(7x+12)=7x^2+12x=19x+21 \]

Math Problem and Solution Video

 

Problem 000

Prove that \[ a^2+b^2+c^2\geq a^{\frac{4}{3}}+b^{\frac{4}{3}}+c^{\frac{4}{3}} \]