Olympiad Question! NO CALCULATOR! | Solve this Math Challenge
Vložit
- čas přidán 18. 03. 2022
- Learn how to solve this algebraic if-then problem. Step-by-step tutorial by PreMath.com
#OlympiadMathematics #OlympiadPreparation #CollegeEntranceExam - Jak na to + styl
Wonderful. Sir you have a way dealing with numbers. Too good! You make it look so simple. Even when the problem is so tough. Awesome 👏👏
Thank you for your feedback! Cheers!
You are awesome Bernard. Keep it up 😀
Love and prayers from the USA!
Sir you are much awesome i started recently watching your video and I do it eadily
I found it slightly easier to use a = 45677. It turned out to be the answer, which was somewhat coincidental, but taking the value exactly between the two cube roots reduces the expansion of x into 6a^2+2 as the a^3 and a^1 terms cancel, which is fewer terms to use and the coefficients are smaller.
Excellent approach!
Thank you for your feedback! Cheers!
You are awesome Jay. Keep it up 😀
Sir, You are doing great👍🏻👍🏻👍🏻👍👍👍👍.
LOVE FROM INDIA🇮🇳🇮🇳🇮🇳🇮🇳
Yes this is what i did mentally. I started with x=(b+2)³-b³ but then i realised the better cancellation with x=(a+1)³-(a-1)³
My math question
czcams.com/video/CWw3mypD_ko/video.html
czcams.com/video/Z67zK4B6cfU/video.html😊😊
Noodling around a little, I came up with this.
Those two numbers being cubed, differ by 2. So let's set n = their average:
n = 45677.
Because then, what I'm about to do will make the odd terms in the expansion, cancel.
Then, since (n ± 1)³ = n³ ± 3n² + 3n ± 1, we get
x = (n+1)³ - (n-1)³ = 6n² + 2
And now we can see why the question was set up the way it was. Because
√⅙(x-2) = n = 45677
Let's see how PreMath works this...
Fred
Before watching:
Let us consider the general expression (a+1)^3-(a-1)^3. We can use Newton's binomium to expand and solve these:
(a+1)^3 = a^3+3a^2+3a+1
(a-1)^3 = a^3-3a^2+3a-1
--------‐------------------------------- -
(a+1)^3-(a-1)^3=6a^2+2
Now:
45678^3-45676^3
= (45677+1)^3-(45677-1)^3
= 6*45677^2+2
SQRT((6*45677^2+2-2)/6)
= SQRT(6*45677^2/6)
= SQRT(45677^2)
= +/- 45677
After watching: Or that. But mine is simpler.
My method too
Excellent Ben!
Glad to hear that!
Thank you for sharing! Cheers!
You are awesome. Keep it up 😀
Excellent Mark!
Glad to hear that!
Thank you for sharing! Cheers!
You are awesome. Keep it up 😀
czcams.com/video/Z67zK4B6cfU/video.html😊😊
Your answer is mistaken. The range of square root function in positive, so your answer of -45677 is erroneous. Also you substitute numbers way too early. The method presented is much simpler and more instructive than your approach.
You will improve by avoiding unnecessary arithmetic at early stages. Simplify algebraically before doing arithmetic.
Also there is difference between solution to equation x^2 = 9 {+3, -3} and the sqrt(9)=3. Yours is a common mistake. Hope this helps.
تمرين جميل. وشرح واضح مرتب . شكرا لكم وبارك فيكم وعليكم والله يحفظكم ويرعاكم ويحميكم وينصركم جميعا . تحياتنا لكم من غزة فلسطين .
i found slightly easier to use difference of two cubes and use 45677 as a substitute leading to
x+1=45678
x-1=45676
2((2x)^2-(x^2-1)
then sub to √(x-2)/6
√(6x^2/6)
Excellent!
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Pileo. Keep it up 😀
Lovely question and beautifully done. Thank you
Brilliant sir! thank you for sharing
Great video in so many ways again! Love your channel!
Glad you enjoy it!
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Tobias. Keep it up 😀
Love and prayers from the USA!
Great stuff. You're so patient.
Excellent!
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Tony. Keep it up 😀
Clear and precise. Excellent.
very well done, steps are easy to follow and understand, great job
Wonderful calculations
you explained very well.
Thank you
Wow i'm a genius I'm so happy i solved it 💪🏼✨✨✨
Excellent Nesrine!
Glad to hear that!
Thank you for sharing! Cheers!
You are awesome. Keep it up 😀
Love and prayers from the USA!
amazing!
Great !👏
Whoah, to designate 45676 as "a" was an outstanding move! I wouldn't come up with this. Thank you for video 😅
Excellent explanation
Today's question very pretty !!!
Very nice sir thanks for the question
Ans :) square root of 45679
Amazing Professor.👍👍👍
Glad you think so!
Thank you for sharing! Cheers!
You are awesome Ramani. Keep it up 😀
I solved it similarly to how you did it. I even used the variable a. I used a = 45677 so that I cubed (a-1) and (a+1). Everything canceled and simplified.
Excellent Bill!
Glad to hear that!
Thank you for sharing! Cheers!
You are awesome. Keep it up 😀
V nicely solved
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Niru. Keep it up 😀
Marvelous. Thanks a lot
Glad you liked it!
You are very welcome.
So nice of you, dear
Thank you for your feedback! Cheers!
You are awesome. Keep it up. 👍
Love and prayers from the USA! 😀
Superb🤩
Brilliant as usual
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Venkat. Keep it up 😀
But why i cant use simplified multiplying expression that is (a+2)^3 - a^3 = x^3-y^3 = (x-y)(x^2+xy+y^2)?
Easiest question solved without pen
Terrific method------- really does simplify the calculation
Excellent John!
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
So delicious a problem thank you sir
Beautiful
Very good solution thanks 👍👍🌹😊🇮🇳🇮🇳
You are very welcome.
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Kafil. Keep it up 😀
Love and prayers from the USA!
Put 45677 = a ,as such 45678 = a + 1 & 45676 = a -1 , now cube them & get substract x = 6 a^2 + 2 , put the value under radical sign ✓ 6a^2 +2 - 2/6 = ✓a^2 = a = 45677
Very helpful👍
Thanks for sharing😊😊
Stay connected
You are very welcome.
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
Nicely explained.thank u sir for this type of problems it gives a good mind work to math lovers.
You are very welcome.
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Sanjoy. Keep it up 😀
My math question
czcams.com/video/CWw3mypD_ko/video.html
Very nice!
Thanks for the visit
Glad to hear that! Cheers!
You are awesome Ram. Keep it up 😀
nice question
If I have Mo for that, I'll watch the video to see how you did.
You state in this and many of your videos that the problems are Olympiad. Will you indicate which Olympiad? For example IMO 2011, SMO 2018 problem number, etc. Thanks for what you do.
Whooow sir , again nicee ,, 😃
Thank you for your feedback! Cheers!
You are awesome Rifat. Keep it up 😀
@@PreMath thanks 😊
Amazing 🤩
Excellent!
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Hasan. Keep it up 😀
45677 is the answer
Great Bhuwan
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
I solved in a similar way but used the equation a³ - (a-2)³. (a=45678)
It reduced to 6a²-12a+8
Then from sqrt[(a-1)²] = a-1 = 45677
I considered u=45677; by substituting, I went down to sqrt(6u^2/6) or simply sqrt(u^2) which is u!
I solved using a=45678, so (a-2)=45676. The procedure is the same, obviously, we simplify to (x-2)/6 = (a-1)^2.
Good
Brilliant thnx a lot
Most welcome Pranav 😊
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
you have to consider the positive and negative root bro
45677 is the answer.
It can be easily solved by taking 45676 as "a".😁
Excellent!
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
Out of all the variables you could have chosen, you chose x which has already been assigned to a specific value. Brilliant!
@@Namchha1 my bad.i didn't saw it properly 😅
@@flute...4394 It's all good. What matters is that you do the math correctly which you did.
Thank you
You are very welcome.
Thank you for your feedback! Cheers!
You are awesome Mahalakshmi. Keep it up 😀
Great video however when taking the square root of a number we should always consider t he absolute value and then observe that it must be positive in this case therefore the absolute value sign can be dropped. Sqrt(a+1)^2) is not always a+1. But can also be -a-1. Generally speaking.
czcams.com/video/Z67zK4B6cfU/video.html😊😊
I had a hunch, so I substituted _much_ smaller numbers in my head to see if my hunch had merit. It did. Just went from there. 😆
Good point. If you just took last digits, you'd have
x = 8³ - 6³ = 512 - 216 = 296
√[(x-2)/6] = √[(296 - 2)/6] = √[294/6] = √49 = 7
and even that example is highly suggestive. Might lead to doing the algebra and verifying the general result.
Fred
I didn't use the same method but I found the same result.
Many different methods possible!
Excellent!
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
Thank
You are very welcome.
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
Love and prayers from the USA!
Simple √[(x-2)/6]=45677.
👍👍👍👍。Awesome.
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Chin. Keep it up 😀
🙏🏼 sir,I have solved without calculators , answer sharing 45677
Excellent Kalyan!
Glad to hear that!
Thank you for sharing! Cheers!
You are awesome. Keep it up 😀
This is definetly too easy to be an olympiad question but its a nice question
Nice🌹🌹🌹🙏🙏
Glad to hear that!
Thank you for your feedback! Cheers!
You are awesome Engineer. Keep it up 😀
45677 not as well.
I do it easily
Yes, but sqrt (a+1)^2 = |a+1|. So, WE have two solutions.
problem asked to evaluate sqrt((x-2)/6), so WE have one solution since range of sqrt is nonnegative
@@MyOneFiftiethOfADollar one can say it afterwards but technically there are two solutions. After analysing the problem, only the positive one is accepted.
@@albancal2002 Technically you are mistaken and it is quite a common mistake. At beginning of problem, sqrt((x-2)/6) = ?
What you are claiming is equivalent to believing sqrt(9)= 3 or -3 which is certainly false. Functions are single valued.
Now x^2=9 has 3 and -3 as solutions which may be part of your misunderstanding.
The gentleman who did the video did not get two answers either. Hope this helps.
@@MyOneFiftiethOfADollar Wrong ! How do you write the number 9 as a product of two identical numbers ? 3*3=9 but (-3)*(-3)=9 also, right ? Now let's say that we need to calculate the sqrt(9): what do we do? We rewrite it as follows : sqrt((+/-3)^2), right ? The sqrt and the square eliminate each-other; what does we get? +/-3 which correponds to |3|.
As for our example, one must say that there are two solutions for a but since we are asked to find the real value of a function, only the positive one will accepted. Hope this had helped. Period.
@@albancal2002 Hate to break it to you, but as a mathematician, I can verify that _My One Fiftieth Of A Dollar_ is right on this one.
When x ≥ 0, √(x²) = |x| = x
When x ≤ 0, √(x²) = |x| = -x
[Note that when x = 0, both are true!]
So √[(±3)²] = 3, not ±3.
And since the question asks for the square root of a number, the root is positive.
Now, in mathematics, we sometimes say that a number has two square roots. This is not strictly true, but is meant to acknowledge that there are two numbers whose square is any given value (except 0, of course). And when extending the square-root function into the complex field, a "cut line" must be chosen in its definition, in order to make it a function, which by definition *must* be single-valued.
Fred
it's not 1
asnwer=2 isit 🤣🤣🤣🤣🤣
45677
45677?
Super Mustafiz
Thank you for your feedback! Cheers!
You are awesome. Keep it up 😀
cant it be -45677 ??
No because the squareroot of any positive real number is a positive real number.
@@bulzangeorge1054 but the square root of x should be + or - underroot x , isnt it ??
@@imranakhtar4145 no. It's always positive
But when you have sqrt(x^2)=k when x and k are real numbers and k is positive then you will have |x|=k which means x=k or x=-k.
@@imranakhtar4145 the definition of the squareroot is sqrt(x^2)=|x| (the absolute value of x).
without calculator it's not possible cause technique is rout X-2 / 6 = its full root x-2 / 2.449 or root X-2 / 6 = 45681 not Same So. it's 45681