Math Olympiad question, only 5 % of the students solved correctly! What about you?
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- Опубліковано 8 вер 2024
- A fantastic equation which looks so easy but very hard to solve❗ I gave the students this problem as an exercise but no one solved absolutely correctly! They all got the same answer. However, 95% of them didn't know how to prove! Watch the video and learn the method.
256^x = 1/x
256 = (1/x)^(1/x)
4^4 = (1/x)^(1/x) [I'm not going to use the Lambert function.]
4 = 1/x
x = 1/4
256^x=1/x. Raise both sides to 1/x power to get 256=(1/x)^(1/x). 256=4^4=(1/x)^(1/x) so 4=1/x and x=1/4. But how can we prove that there is only one solution?
Create two functions, one for LHS and one for RHS. f(x)=256^x and g(x)=1/x. f(x) has a range of y>0 so there can only be intersections when y>0. When y>0, g(x) strictly decreases and f(x) strictly increases. Therefore there can only be one intersection: x=1/4.
X= 1/4. I worked it out in my head in a few seconds.
I'm surprised that Math Olympiads seemingly use multiples of 2 so much in their problems. For us who grew up with computers and some hobbyist assembler programming in the 1980s, this is trivial to solve by mental calculation in a minute. We know by heart what 2^n is, up to 65536 (A double byte 2^16 was the maximum used for adressing memory and such). So we immediately see 256^x as (2^8)^x and that 2^(8/4)=1/(1/4). If the base were 3 or 5, one would actually have to calculate.
And 2^20=1048576 which is 1M
Exactly, as some one who has worked with machine code in the 1970s at University and work, with binary bit patches to assembled programs, I saw x = 1/4 without consciously thinking. 256 represented a byte, used to represent an ASCII character.
@@michaeledwards2251 I'm glad that I'm a highschooler but could solve this in less than a minute in my mind 😃 Academic Curriculum in India is tough 🙂
Why complicate. Take log to the base 2 on both sides and you will get the result with proof. Of course by trial 4 can be found in a
few seconds.
There's a much simpler way to prove 1/4 is the only answer. When 0
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here's a much simpler way to prove 1/4 is the only answer. x>0: the expression on the left monotonically increases, and the expression on the right monotonically decreases. Only one intersection is possible.
@@meirlev9613... It is easier to see that it happens in (0,1] interval because 0^0 has an undefined limit and 256^x gets smaller than 256 while 1/x gets smaller in (0, 1]. Basically what you are saying that f(x) = A^x and g(x) = 1/x = x^(-1) two functions cross in (0,1] interval seen by graphing where A = 256 is used in f(x).
Sketch the two graphs and you see there is only one solution
Possible to make an overkill solution by using Lambert W-function.
Not necessary here but solves the general case when it's not a multiple of 2 and easy to do in the head.
(256^x)' >0 and (1/x)'0, so there is at most one positive solution
256^x>0 and 1/x
В левой части - возрастающая функция, в правой - убывающая. Следовательно, уравнение имеет не более одного корня, а значение х=1/4 легко подбирается.
X=1/4.
If I'm a bit more free, I'll watch the video.
Спасибо.
rewrite 256^x as (2^8)^x, then 2^8x. Now just replace 8x with u and x with u/8. now you get 2^u = 8/u. u = 2, x = 2/8 = 1/3
x=W(ln(256))/ln(256)
x is 1/4 by inspection before watching the video...
Very Good
Too funny. I'm a genius so 5% is enough to me, x=√0,0625.
Ma'am what is your reference idea to solve this🙏🙏🙏
I mean what is your approach for to every question 🙏🙏🙏
x=1/4
256^x==x=1/x=1x=1 Stanford university ACADEMIC general Doctor
(256^X)/X=1 Правило N^0=1 и на 0 делить нельзя (смотря кому!!).
Значит X стремится к нулю!! Lim X->0 😅🤣😂
Merci
Х=1/16
X=0.25
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x = 0,25.
瞪眼法，x=1/4
Ummmm you can do that in your head just by using common sense.
X=1/t. T isn't 4 4:26
No, t=4, and so x=1/4.