Monday, March 29, 2021

1960 IMO Problems And Solutions

Problem 01

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$ .

Solutions

Solution 1

Let $N = 100a + 10b+c$ for some digits $a,b,$ and $c$ . Then $100a + 10b+c = 11m$ for some $m$ . We also have $m=a^2+b^2+c^2$ . Substituting this into the first equation and simplification, we get

Problem 01 (1959 IMO)

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number.

Solution 01 (Euclidean Algorithm)

Cambodian Olympiad Math

Cambodian Olympiad Math is written by Lim Sovanvichet . This book is shared on social media by many students which is very important document for all Khmer students. In this book, there are many International Math Problems which he translated from other foreign books and wrote them in to Khmer language.

Download This Book Here:

Vietnames's Sequence Math

This is a document that I research from the internet and rewrite it to upload at my website, I think It can help you do Math with different language which makes you to be better at Math. Why Vietnames students can challenge with other students around the world.

Here you can check out my collection.

Monday, March 22, 2021

When is `n^2+2021n` a perfect square?

Solution

As we knew, `2021=43\times47`

Let find `m;n\inN` such that: `n^2+2021n=m^2`

Irish Math Olympiad 2009

Find all positive integer `n` such that `n^8+n+1` is a prime.

Irish Math Olympiad 2009

2001 Dutch Math Olympiad

Suppose for all `x,y\inR` we have `f(x+y)=f(x)+f(y)+xy` and `f(4)=10`.

Let find the value of `f(2001)`

Solution

Prove that: `1/2+1/5+1/8+1/11+1/20+1/40+1/110+1/1640=1`

1. Prove that: `1/2+1/5+1/8+1/11+1/20+1/40+1/110+1/1640=1`

Solution

We observe that:

`1/2+1/5+1/8+1/11+1/20+1/40+1/110+1/1640`

`=1/2+1/20+1/5+1/8+1/11+1/110+1/20+1/1640`

`=22/40+13/40+11/110+41/1640`

`=22/40+13/40+4/40+1/40`

`=40/40=1`

Hence, `1/2+1/5+1/8+1/11+1/20+1/40+1/110+1/1640=1`

Sequence Book For Grade 12

This book is related to Sequence Problem, the tips to solve Sequence problem, which is written in Khmer language. I hope this book is used for all of you.

IMO 2019 in South Africa

Let Z be the set of integers. Determine all function `f: Z \rightarrow Z` such that, for all integers `a` and `b`

`f(2a)+2f(b)=f(f(a+b))` `(1)`

Answer: The solution are `f(n)=0` and `f(n)=2n+k` for any constant `k\in Z`

Substituting `a =0, b= n+1 ` gives `f(f(n+1))=f(0)+2f(n+1)`.

Substituting `a =1, b= n` gives `f(f(n+1))=f(2)+2f(n)` .

In particular, `f(0)+2f(n+1)=f(2)+2f(n)`, and so `f(n+1)-f(n)=1/2(f(2)-f(0))` .

Evaluate the value of : `A=3/(1!+2!+3!)+(4!)/(2!+3!+4!)+......+n/((n-2)!+(n-1)!+n!)`

Evaluate the value of :

1. `A=3/(1!+2!+3!)+(4!)/(2!+3!+4!)+......+n/((n-2)!+(n-1)!+n!)`

Solution:

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` ; `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`

Friday, November 27, 2020

With Mathematics Induction, Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11` for all natural number `n`

With Mathematics Induction, Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11` for all natural number `n` .

Solution

Prove that: `A_n=2^(6n+1)+9^(n+1)` is divided by `11`

If `n=0` then `A_0=2+9=11` It is true that `A_0 \equiv0(mod11)`
If `n=1` then `A_1=2^7+9^2=209=11.19` It is true that `A_1 \equiv0(mod11)`

We assuming that: `A_k \equiv0(mod11)` for all natural number `k`

Let `(a_n)` be a sequence of real number satisfied that: `a_0=1` and `a_(n+1)=a_0a_1......a_n+4` for all natural number n. Prove that: `a_n-\sqrt{a_{n+1}}=2` for all `n>0`

Let `(a_n)` be a sequence of real number satisfied that: `a_0=1` and

`a_(n+1)=a_0a_1......a_n+4` for all natural number n.

Prove that: `a_n-\sqrt{a_{n+1}}=2` for all `n>0`.

Solution

We observation that `a_k>0` for all natural number `n`

And from `a_(n+1)=a_0a_1......a_n+4`

It is give function `f(x)` for all real number of `x` such that: `x(2x+1)f(x)+f(1/x)=x+1`

It is give function `f(x)` for all real number of `x` such that:
`x(2x+1)f(x)+f(1/x)=x+1`.

Find the sum of: `S=f(1)+f(2)+f(3)+.......+f(2022)`

Solution

From `x(2x+1)f(x)+f(1/x)=x+1` (1)then we replace `x=1/x` so will get new relation :
`1/x(2/x+1)f(1/x)+f(x)=1/x+1`

`f(1/x)+(x^2/(x+2))f(x)=(x^2+x)/(x+2)` (2)

It is given polynomial `P(x)=(xsina+cosa)^n` for natural number n. Finding the remaining when `P(x)` divide to `(x^2+1)`

It is given polynomial `P(x)=(xsina+cosa)^n` for natural number n.

Finding the remaining when `P(x)` divide to `(x^2+1)`

Solution

Let `R(x)` be the remainder of division between `P(x)` and `(x^2+1)`

If `n=1` then `P(x)=xsina+cosa`

Therefore, the remainder `R(x)=xsina+cosa`

If `n` ≥ `2` then `P(x)=(x^2+1)Q(x)+R(x)`

which `Q(x)` is the result and `R(x)=Ax+B`

Then, `(xsina+cosa)^n=(x^2+1)Q(x)+Ax+B`

If `x=i` then `cosna+isinna=B+iA`

Therefore, `A=sin(na) ;B=cos(na)`

Hence, the remainder is `R(x)=sin(na)x+cons(na)` .

Solution by: Thin Sokkean

It is given three positive real number `x,y,z` such that : `cos x+cosy+cosz=0` and `cos3x+cos3y+cos3z=0`

It is given three positive real number `x,y,z` such that : `cos x+cosy+cosz=0` and `cos3x+cos3y+cos3z=0`.

Prove that: `cos 2x.cos 2y.cos 2z`≤ 0

Solution

Prove that : `cos 2x.cos 2y.cos 2z`≤ 0

Following formula :

`\cos3a=4\cos^3a-3\cos a` then we can see that:

Finding the sum of following problem: `S_n=1.1!+2.2!+3.3!+.....+n.n!` which `(n!)=1.2.3....n`

Finding the sum of following problem:
`S_n=1.1!+2.2!+3.3!+.....+n.n!` which `(n!)=1.2.3....n`

Solution

In order to solve kind of these problem, you need to start with the general term of the sequence:

Exactly, the general term of our problem here is `k.k!`

We can rewrite such that: `k.k! = (k+1-1)k! =(k+1)k!-k! =(k+1)!-k!`

From that we can replace the value of k following our main problem:

It is given two positive real numbers `x,y` which are satisfied that `4x+3y=11`. Find the maximum value of the following function: `f(x,y)=(x+6)(y+7)(3x+2y)`

It is given two positive real numbers `x,y` which are satisfied that `4x+3y=11`.

Find the maximum value of the following function:

`f(x,y)=(x+6)(y+7)(3x+2y)`

Solution

Find the maximum value of the following function:

It is give positive number `x,y,z` which satisfied that `x^2+y^2=z^2`. Prove that: `xyz` is divided by `x+y+z`.

Solution

From the following hypothesis: `x^2+y^2=z^2` we can rewrite such as: `x^2+y^2+2xy=z^2+2xy`

`(x+y)^2=z^2+2xy`

`2xy=(x+y+z)(x+y-z)`

We can see that: `(x+y+z)` or `(x+y-z)` is divided by 2 which `(x+y-z)` gives remain same to `(x+y+z)` then `(x+y-z)` is dived by 2 exactly.
Let, `x+y-z=2k` then `xy=k(x+y+z)`then `xyz` is dived by `x+y+z`.

Solution by: Thin Sokkean

Wednesday, November 25, 2020

Best Resources For Preparing IMO

In fact, If you want to join any competition, you need to train again and again with kind of those thing. Mathematics as well, You need to take time, spend time, work hard, keep patience. But, the most important things are resources.

You need to find the sample tests, sample form of the test so that you can get more understand and also more effective than someone who has no the resources.

Here are some of the resources that I used to find when I prepared for IMO:

Art of Problem Solving : an online forum for mathematical enthusiasts. It has an extensive contest section where you can always find problems to challenge yourself. In particular, the best for preparation for the IMO are past IMO and IMO shortlist problems, but also some other national or international Olympiads (EGMO, MEMO, USA, Canada, Russia …)
Yufei Zhao : handouts are also very good.
Evan Chen's Website : with his awesome handouts.
Euclid Geometry For IMO : from Evan Chen has recently been published. This book provides a very solid background in geometry. It assumes no previous knowledge
14 Number Theory Problems : You can find It in Amazon.
102 Combination Problem

I believe the most efficient way to prepare for the IMO is to do as many problems as you can. This needn’t be limited to the problems from one resource so don’t feel obliged to go through one book systematically. (Also, all mentioned books are available on the internet, but I am not sure if legally.)

Moldova Mathematical Olympiad | 2000 Q2

This problem is being picked up from the Moldova Mathematical Olympiad in 2002 (Question 02) . In this following page, you will see how to find the root of non-linear equation system with more power of variables.

Compare these number : 50^99 and 99!

In other to compare two of number, we have several methods to prove . In here, I will use the division operation to see which number is bigger.

If A is bigger than B, we can write in the fraction A/B is bigger than 1.

Friday, November 13, 2020

New Zealand Mathematical Olympiad 2019 Question 5

Noticed that: All IMO Problems are not too difficult for you to solve, It is just because you either rarely study or never face with kind of those problems. For my opinion, you should research more about those problems to make sure that you have ability to join to contest

2015-2016 Slovakian Math Olympiad

Slovakian Math Olympiad | 2015-2016

1985 International Mathematics Olympiad

This is the problem that I picked up from 1985 International Mathematics Olympiad Long List Problems. I will make the solution for you by my Microsoft Word and convert it to images for you to see my solution.

Noticed that: All IMO Problems are not too difficult for you to solve, It is just because you either rarely study or never face with kind of those problems. For my opinion, you should research more about those problems to make sure that you have ability to jion to contest.

Solution By Thin Sokkean

Sunday, October 18, 2020

What Is The Remainder When 2022^2020 Is Divided By 20

What Is The Remainder When 2022^2020 Is Divided By 20?

This problem is a bit harder If you do not know about basic Theory Numbers as well as Modulo's Theory.

Why those theories are important?

Remember that: All dividing problem which is finding the remainder, you must to understand clear the division of natural number. It starts from 2 to 11 or more than that. Exponential formula is the important once for you to use as the power. After that, modulo's theory as I have uploaded already either in this site or my YouTube Channel.

Here is the solution for you:

Monday, October 5, 2020

Why 1=2

Why 1=2

In Mathematics operation and logical 1 is equal to 1, can not equal to 2. But, the question here is 1=2? How do you think about this statements?
It is the common problem that you guy should think about its mistake. For me, I can not believe it. I will find the proof for you which is the statement above is truth.

Let's 100=100

Click Here To Find More Math Problem

As you see in the solution, and the conclusion 1 is equal to 2.
Let's give any idea about this problem and find the mistake of the solution.
Thank you!

Friday, September 18, 2020

Vietnamese's Mathematics Problem

Vietnamese's Mathematics Problem

These problems I have researched in many Vietnam Math Books to share in my blog. But, I hope these problems can help you to fulfill your shortages of Math. If you want to be good at Math, the only thing you must do is Do More Math Problems Every Day. Push yourself, motivate yourself to be crazy with study, especially Math.

Let's try to solve these problems:
I will release the link for you to download the book so that you can find the solutions by yourself.

Click Here To Download Book

Click Here To Download This Book

Click Here To Download

I hope my efforts can be a small factor of your studies.

Thank you!

Enjoy your study

Wednesday, September 16, 2020

Find The Last Digit Number Of 2023 The Power of 2023

Find The Last Digit Number Of 2023 The Power of 2023

There are many way to find the last digit number of complicated Math Problems. But, In here I want to shower the best solution to find the last digit number of every single Mathematics amount by just using Dividing Operation.

If you want to be out-standing with these type of problem, you should understand clearly about Modulo Theory.

Solution:

Here Is The Video That I have Made For You.

Click Here To Watch Video

Sunday, August 30, 2020

Remainder Theory

The Remainder Theorem is useful for evaluating polynomials at a given value of x, though it might not seem so, at least at first blush. This is because the tool is presented as a theorem with a proof, and you probably don't feel ready for proofs at this stage in your studies. Fortunately, you do not "have" to understand the proof of the Theorem, you just need to understand how to use the Theorem.

The Remainder Theorem starts with an unnamed polynomial p(x), where "p(x)" just means "some polynomial p whose variable is x". Then the Theorem talks about dividing that polynomial by some linear factor x – a, where a is just some number.

Math Out-Standing Student Problem For Grade 12

Math Out-Standing Student Problem For Grade 12

In 2020 of Examination Mathematics For Out-Standing Students in Camboida was cancelled cause of Covid-19, but, Before cancelling, this problem was the most popular problem which occurred mostly in every province.

You can try to solve by yourself or you can research more about two variables function. There are ton of problems on the internet, but here I just picked some of them to share with all of you.

You can watch solution which was solved by me, where you can check the tips, method, the theory that I used to solve It.

Thank you very much!

Monday, August 24, 2020

Perfect Problems

Common Problems

These are the problems of common daily problems that you all should try to solve everyday.
I have research these problems for all you, specially our Cambodian students to challenge with other countries in the local and around the world.

Day 01

Examinations Of Math Out Standing Student For Grade 12 In 2020

Examinations Of Math Out Standing Student For Grade 12 In 2020

These are some tests of Math For Out Standing Student in 2020 which I picked up from the internet. You can see list of these problems are in my book, and also some problems I already solved in my channel. You can see that: Limit, sum of sequence, find the general term of sequence, find the derivative of function, find the sum of function, geometry, function equation, trigonometry function, polynomial function, one variable function, two variables function, find the maximum and minimum of function, trigonometry equation, and so on. These type of problems are the common problems for Math Out Standing For Grade 12 in Cambodia.

Here are some of these tests:

Sihanouk 2020

Phom Penh 2020

You can find solution by yourself

Khmer Math Out-Standing Student For Grade 12

Khmer Math Out-Standing Student For Grade 12

These are problems, I picked up from many examinations, many books both Khmer and Foreign, and so on. All you guy can copy or download this problems from my blog to try by yourself to solve them. I guarantee that you will understand and get close to succeed of your examination.

My recommendation to you, you should have basic of Mathematics with theory, operation, and also being better in the normal class as well before you try to solve theses problems.

You also can find the solutions in my YouTube Channel: Thin Sokkean
Then, you go to the playlist of " Math Out Standing Student" while you can find other problems and solutions as well.

I hope that these can help you to be an out standing student .
Have a good study !

These are some common problems that I think you should know If you want to join the testing.