- 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 Book Cambodia
Friday, April 21, 2023
Math 2nd Day Phnom Penh Cambodia 20/04/2023
Thursday, April 20, 2023
Wednesday, April 19, 2023
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.
Tuesday, April 18, 2023
Prove that `(1/2).(3/4)......((2n-1)/(2n))\leq1/sqrt(3n+1)`
- This is the problem that I picked up from book : 101 Solved Problem in Algebra from The USA IMO with the 43th problem.
Monday, April 17, 2023
What is the coefficient of `x^2` when `(1+x)(1+2x)...(1+2^nx)` 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.
Wednesday, April 12, 2023
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.
Saturday, April 8, 2023
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.
Monday, April 3, 2023
Prove that `(a_n+3/2^(n+2))^(1/n)(m-(2/3)^(n(m-1)/m))<(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.
Saturday, April 1, 2023
Friday, March 31, 2023
Saturday, March 25, 2023
Friday, March 24, 2023
Thursday, March 23, 2023
Wednesday, March 22, 2023
Tuesday, March 21, 2023
Monday, March 20, 2023
Find all number which its square has 4 digits number and divisible 33
- In order to solve type of this problem, you have to understand about divisible theory of natural number. As GCD (Greatest Common Divisor) and LCM (Least Common Divisor). When you know about GCD and LCM, you surely can simplify the problem to be more easier to solve.
- As example of our problem, `33=3.11` and `GCD(3;11)=1` Then, our problem can be found as divisible with `3` and `11`
Saturday, March 18, 2023
Solve the equation `sin^2012x+cos^2012x=1/2^1005`
Solve the equation `sin^2012x+cos^2012x=1/2^1005`
Friday, March 17, 2023
Vietnamese Mathematics Provincial 2011
Vietnamese Mathematics Provincial 2011
- This is the problems of Vietnamese Mathematics Provincial examination in 2011. I just picked problem number 04 to share all of you which is related to sequence. As you know, sequences are the problems which need more strategy to solve where you have to combine all your understanding.
Thursday, March 16, 2023
Cambodia National Math 2019, 22/04/2019 Day 02
Cambodia National Math 2019, 22/04/2019 Day 02
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Math Cambodia 2019 Day 02 |
- This was the problem that released for Out Standing Student in Cambodia in 2019 for 2nd day of testing.
- There were two days of the testing. This is the first day of exam.
- You all can Click here to download
Sunday, March 12, 2023
Cambodia Grade 12, 22/04/2019 Day 01
Cambodia National Math 2019, 22/04/2019
- This was the problem that released for Out Standing Student in Cambodia in 2019.
- There were two days of the testing. This is the first day of exam.
- You can watch solution here:
Tuesday, February 28, 2023
Mathematics Out Standing Student Phnom Penh 2020, Cambodia
Mathematics Out Standing Student Phnom Penh 2020, Cambodia
- This was the problem that released for Out Standing Student in Phnom Penh, Cambodia in 2020.
- There were two days of the testing. This is the first day of exam.
Friday, December 23, 2022
Prove that the polynomial `x^9999+x^8888+x^7777+...+x^1111+1` is divisible by `x^9+x^8+x^7+....+x+1`
Problem: 01
Solution
Thursday, December 22, 2022
Find all function `f(x)` if `(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)`
Find all function `f(x)` if `(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)`
Solution
Sunday, September 11, 2022
If `x+1/x=2` Find the value of `x^5+1/x^5`
If `x+1/x=2` Find the value of `x^5+1/x^5`
As we had: `x+1/x=2` `rightarrow(x+1/x)^2=4` `leftrightarrowx^2+1/x^2=2`
We continue with `(x^2+1/x^2)(x+1/x)=4` `leftrightarrowx^3+x+1/x+1/x^3=4`
`leftrightarrowx^3+1/x^3=2`
Friday, September 9, 2022
Vietnamese Olympiad 2022: Find the value of `1/(a^2023)+1/(b^2023)+1/(c^2023)`
If `a+b+c=2022` and `1/a+1/b+1/c=1/2022` Find the value of `1/(a^2023)+1/(b^2023)+1/(c^2023)`
Solution
Sunday, August 29, 2021
Find all polynomials `P(x)` such that: `P(x-1).P(x+1)=P(x^2-1)`
Find all polynomials `P(x)` such that: `P(x-1).P(x+1)=P(x^2-1)`
Solution
Suppose that `\alpha` is a root of `P(x)` then, `P(\alpha)=0`
Therefore, `P((\alpha+1)-1)=0`
Saturday, August 28, 2021
Find the last two digits of : `N=(1!+2!+3!+.......+101!)^101`
Find the last two digits of : `N=(1!+2!+3!+.......+101!)^101`
Note: This is equivalent to finding `N(mod11)`.
ie: The remainder when dividing `N` by `100`.
Observation: `10!\equiv0(mod100)` Because, `10!=10...5...2`
Therefore, `N\equiv(1!+2!+3!+......+9!)^101(mod100)`
`N\equiv(1+2+6+24+20+20+40+20+80)^101(mod100)`
`N\equiv13^101(mod100)`
Thursday, July 15, 2021
Prove that: `A_n=3^(n+3)-4^(4n+2)` is divided by `11`
Prove that: `A_n=3^(n+3)-4^(4n+2)` is divided by `11`
Solution
We will prove by Mathematics Induction:
If `n=0` then `A_n=3^3-4^2=27-16=11` It is true that `A_n` is divided with `11`
Assuming that: `A_n` is divided with `11` for `n=k` `(1)`
We will prove it is true for `n=k+1` then `A_(k+1)` is divided with `11`
Wednesday, July 14, 2021
Prove that: `forall x\inR` : `:|acosx+bsinx|\leqsqrt(a^2+b^2)`
a. Prove that: `forall x\inR` : `:|acosx+bsinx|\leqsqrt(a^2+b^2)`
b. Find the maximum and minimum of `f(x)=20cosx+21sinx+27`
Solution
a. Prove `|acosx+bsinx|\leqsqrt(a^2+b^2)` `forall x\inR`
Assuming we have:
`acosx+bsinx=sqrt(a^2+b^2)(a/sqrt(a^2+b^2)cosx+b/sqrt(a^2+b^2)sinx)`
`X_1;X_2` are the root of : `X^2-(2cost+3sint)X-11sin^2t=0` Find the minimum of `A=X_1^2+X_1X_2+X_2^2`.
`X_1;X_2` are the root of : `X^2-(2cost+3sint)X-11sin^2t=0`
Find the minimum of `A=X_1^2+X_1X_2+X_2^2`.
Solution
We can see that: `A=X_1^2+X_1X_2+X_2^2`
`=(X_1+X_2)^2-X_1X_2`
Following Vieta's Formulas `X^2-SX+P=0` `(1)`
Monday, July 12, 2021
Prove that: `tan^3x/(1-3tan^2x)=1/8(tan3x-3tanx)`
a. Prove that: `tan^3x/(1-3tan^2x)=1/8(tan3x-3tanx)`
As we knew: `tan3x=(3tanx-tan^3x)/(1-3tan^2x)`
`=(3tanx-9tan^3x+8tan^3x)/(1-3tan^2x)`
`=(8tan^3x+3tanx(1-3tan^2x))/(1-3tan^2x)`
`=(8tan^3x)/(1-3tan^2x)+3tanx`
Then, `tan3x-3tanx=(8tan^3x)/(1-3tan^2x)`
Find the limit of `S_n=2/(1.3)+2/(3.5)+2/(5.3)+.......+2/((2n+1)(2n+3))`
Find the limit of `S_n=2/(1.3)+2/(3.5)+2/(5.3)+.......+2/((2n+1)(2n+3))`
Solution
We can see that general term of this sequence is:
`2/((2k+1)(2k+3))=1/(2k+1)-1/(2k+3)`
Thursday, July 8, 2021
If `g(x)=f(x)+1-x` , find the value of `g(2020)`
It is given function `f(x)` determine on `R` , satisfied that:
`f(1)=1`
`f(x+5)\geqf(x)+5`
`f(x+1)\leqf(x)+1`
If `g(x)=f(x)+1-x` , find the value of `g(2020)` `\forallx,y\inR`
If `(1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20)` then find the coefficients of `a_1` ; `a_2` ; `a_3` and `a_20`
- If `(1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20)` then find the coefficients of `a_1` ; `a_2` ; `a_3` and `a_20`
Solution
From our hypothesis we already had:
`(1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20)`
We will find the coefficients of `a_1` ; `a_2` ; `a_3` and `a_20`
Which meant they are the coefficients of `x` ; `x^2` ; `x^3` and `x^(20)` .
We can rewrite `(1+2x+3x^2)^10=[1+x(2+3x)]^10`
We will use the Newton's Formula
`(a+b)^n`=`\sum_{i=0}^nC(n,i)``a^(n-i).b^i`
Then, `[1+x(2+3x)]^10=C(10,0)+C(10,1)x(2+3x)+C(10,2)x^2.(2+3x)^2`
`+C(10,3)x^3.(2+3x)^3+....+C(10,10)x^10.(2+3x)^10`
We observe that:
`Coef(x)=2C(10,1)=20` then, `a_1=20`
`Coef(x^2)=2^2C(10,2)=4.45=180` then, `a_2=180`
`Coef(x^3)=12C(10,2)+8C(10,3)=45.12+120.8=1500` then `a_3=1500`
`Coef(x^20)=C(10,10).3^10=3^10` then, `a_20=3^10`
Wednesday, July 7, 2021
Prove `(21n+4)/(14n+3)` Is Irreducible For Every Natural Number `n`
Solution
Solution 01
Denoting the greatest common divisor (GCD) of `a` and `b` `(a,b)` and we will use Euclidean Algorithm Theory.
`(21n+4,14n+3)=(7n+1,14n+3)=(7n+1,1)=1`
It follows that `(21n+4)/(14n+3)` is irreducible. Q.E.D
Solution 02
Tuesday, July 6, 2021
Vietnam Math Out Standing Student 2012-13
Vietnam Math Out Standing Student 2012-13
Solution
Monday, July 5, 2021
Find all functions `f:R\rightarrow R` such that: `f(x)f(y)=f(xy-1)+xf(y)+yf(x)` `\forall` `x,y` `\inR`
Find all functions `f: R \rightarrow R` Such that:
`f(x)f(y)=f(xy-1)+xf(y)+yf(x)` `\forall` `x ,y ``\inR`
Solution
Sunday, April 4, 2021
1970 IMO Problems And Solutions
Problem 01
Let be a point on the side
of
. Let
, and
be the inscribed circles of triangles
, and
. Let
, and
be the radii of the exscribed circles of the same triangles that lie in the angle
. Prove that
.
Solution
Wednesday, March 31, 2021
1979 IMO Problems And Solutions
Problem 01
![$p$](https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png)
![$q$](https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png)
![\[\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319},\]](https://latex.artofproblemsolving.com/f/e/7/fe7a47dd9fd28d623ff2b2b7186626d70073ea29.png)
![$p$](https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png)
![$1979$](https://latex.artofproblemsolving.com/7/b/6/7b6cdf62a4c800756c74d0fac1e13ec127b1d146.png)
Solution
Monday, March 29, 2021
1963 IMO Problems And Solutions
Problem 01
Find all real roots of the equation
![$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$](https://latex.artofproblemsolving.com/4/c/5/4c50288029a0bd6789e2dcec7a9350ad9a6c605f.png)
where is a real parameter.
Solution
2007 IMO Problems And Solutions
Problem 01
Real numbers are given. For each
(
) define
and let
.
(a) Prove that, for any real numbers ,
(b) Show that there are real numbers such that equality holds in (*)