Russia | Math Olympiad Question | You should know this trick!!

What a waste of time seeing this.

Easy: The 2nd one!!

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I ended up here on an insomniac night (very counterintuitive) and this is just the perfect voice I needed to calm down and have a good night. Thx!

How come 1/49 is equal to 1/6

My logic is comparing 50*50 and 49*50 where difference is 50, so even adding one more 49 to 49*50 still 50*50 is 1 more.

Use compound interest logic, you’ll do this much earlier. 50/49 is 1.02 approx raised to 50, will be much below 49, think of it as getting a 2% interest on your bank deposit. It will take maybe 25-30 years to double, and would max be 4 times in 50 years - hence 50/49 raised to 50 is def below 4, and when you divide by 1/49, it’s clearly below 1.

Try 3^3 & 2^4
4^4 & 3^5
5^5 & 4&6
At 5^5 and above, the second expression becomes larger and larger.
You can solve it graphically too

I completely understand

Higher power wins for all numbers > 5^5
49^51 is actually almost 20 times larger than 50^50.
(bruteforced it by excel)

alt+r,"cmd",enter,"py","50**50<49**51
Proof without a calculator.

Its izi 5 ⁵ is smaller than 4⁶

complete bullshit

answer it in 2 secs, but i dont know how to explain

Or u could just use the binomial theorm

Why not get everything to the power of 50 and compare the bases 50^50<(49^(1/50)*49)^50 since 50<52.97

asmr math

So basically to know to solve this you should already at the first moment suspect, that the numbers chosen (49 and 50 and 51) in the question were similar to each other with a reason. The one who created this question had to chose similar numbers to make the question solvable.

垃圾算法，一点都不严谨，证明过程有问题

I took the natural log of both sides and saw that it was 50 ln 50 vs 2450 ln 49. This showed a clear difference. The other method suggested is a more general approach that I didn't think of. Works better in the long term.

Beautiful ! Subbed.

Just compare logarithmic values.
51 ln 49 vs 50 ln 50
51 ln 49 vs 50 (ln 49 + ln 50/49)
ln 49 vs 50 ln (50/49)
ln 49 vs 50 ln (1+1/49)
Right hand side is smaller than it's first degree approximation since ln (1+x) is concave.
So right hand side is smaller than 50/49, which is way smaller than ln 49 = 2 ln 7 > 2 since 7 > e.
Left hand side is way bigger.
So 49^51 is bigger than 50^50.

You didn't prove here that (50/49)^49 < 3 (although that is true), What you can do is to prove that (1+1/n)^(n+1) is descending and therefore (50/49)^50 < 2^2 = 4. One more way is to find that x^(100-x) has negative derivative on [49, 50]. But of course the best is to take logarithm first.

How can we confirm that (1+1/n)^n is monotone increasing function?

Worst math explanation I've ever seen

I mean its 50/50 I would just take a guess and move on to the next question 😅 (all jokes aside great content and very interesting!)

you could argue that 50^x/49^(x+1) approaches 1 for x going to infinity. if u then calculate for x=1 you can see that thats where the fraction has its max value and is smaller then 1 - therefor for x=50 it will still be!

"...the letter e..." 😄. Thank you that was refreshing.

This is the exact same thing Math Window did.

If you are using an approximation for e, why not using an approximation for 1/49 = 0,0204 ???
Only math elegance 😂😂

I solved it differently, just made a new rule as well...
[n^n] is always < [(n-1)^(n+1)] given that n> (π+1).

I liked the elegance of this solution

You can just see it by looking at the numbers, that's how exponentials work Just like you can intuitively see that 1045 is larger than 983.

could that be extended to all numbers ?
this example : 50 over 50 versus 49 over 51
general example : n over n versus (n-1) over (n+1)

I tried 3² and 2³

Then 4³ and 3⁴

Then 5⁴ and 4⁵

I finally concluded 49⁵¹ is larger.

that was perfect

Well done. It's pretty clever how you've managed to prove it

This is not the correct answer. The 50 to the power of 50 is greater to the 49 to the power or 51. This is the answer.

В целом, размышления верные. Рассмешило только то, что в конце решения мадамка заменяет - вдруг - 1/49 на 1/6. Она совсем не понимает, что делает?..

I did not get the 1/6 part can someone explain ?

This is why after getting A's in Calculus I, II, III and Linear Algebra, I still never finished a degree in mathematics. As good as I am with logic, the above solution was just too abstract for me to get my head around.

I am an indian and i did binomial expansion and did it in like 10 seconds.Just equate (50/49)^50 to 49 first and 50/49 is 1.02 ,which can be written as (1+0.02).After that just multiply 50 to 0.02 so tha answer will come 1+(50×0.02)=2 which is less than 49. So 50^50 is way way smaller than 49^51.

I think u can compare also 3^3 - 2^4, and 4^4 - 3^5, 5^5 - 4^6 its( 27 - 16 ) ( 256 - 243 ) ( 3125 - 4096 ) and make conclusin that 50^50 is smaller then 49^51

Finding an unnecessarily complicated, inelegant and difficult solution is not a sign that one should be a mathematician.

If you compared 50^3 with 49^4 by calculator
Then easily you will conclude that 49^51 is greater than 50^50
QED

<3 * >1 * <1/6 = <1 you cant be serious

Все долго что-то считают, а ответ элементарный. Числа примерно одинаковые, а 50 умножается на 50 50 раз, а 49 - 51, то есть примерно одно и то же, только там, где 49, ещё в 49 раз больше.

50⁵⁰_49⁵¹ <=> ln50⁵⁰_ln49⁵¹ since these numbers are positive or belong to]0, infinite]
Equivalent to 50 ln50- 51 ln49
Equivalent to 50 ln50- 51[ ln(50-1)]
Equivalent to 50 ln50- 51[ ln50+ln1], we know ln1=0
New equation 50 ln50- 51 ln50
Which equals to
(50-51)ln50 < 0

Which means 50⁵⁰_49⁵¹<0
Which means 50⁵⁰ < 49⁵¹