Math Question is Russian I think and I am looking for its contest paper. 1/2 * 3/4 * 5/6 ... 99/100 < 1/10

Question

Help! I found the French Math Olympiad question which is the same question. But I am looking for the Russian version of the question. I know it exists because I saw it a long time ago and I remember it was the first question of 5 questions. However, I can't say if it's written in Russian, but I'm pretty sure it is.

They're both good answers. Thank you for responding. My answer was
1/2 * 3/4 * 5/6 * 7/8 < 1/8^(1/2)
1*1/2*2 * 3*3/4*4 * 5*5/6*6 * 7*7/8*8 < 1/8
multiply both sides by 8
11/22 33/44 55/66 77/8 < 1
3/22 35/44 57/66 7/8 all of these are under 1
it works because 3*5/4*4 = (4-1)(4+1)/(4)(4)
x^2-1/x^2 all terms < 1

It's good to see both your solutions. I thought my solution was the only one. I was proud of myself, darnit. Oh well.

Answer 1

I don't know the wording but if you're looking for the solution:

1/2 * [3/4 * 5/6 ... 99/100] < 1/10
=> [3/4 * 5/6 ... 99/100] < 1/5

3/4 * 5/6 * 7/8 * 9/10 = (3*5*7*9)/(4*6*8*10)
= (2^4-1)(2^6-1)/2^7(2^4-1)
= (2^6-1)/2^7

For a general or inductive result:
1/2 * 3/4 * 5/6 ... (2n-1)/2n
= [1*3*5* ... (2n-1)]/[2*1*2*2*2*3...2*n]
= [1*3*5* ... (2n-1)]/2ⁿ[1*2*3...2ⁿ]

Answer 2

Also neat is to show that the product is equal to the probability of tossing a fair coin 100 times and getting exactly fifty heads and fifty tails.

It is the multiplication by 63/64 that drops you below 0.1.

(2n - 1) / 2n = (1 - 1/(2n))
ln ((2n - 1) / 2n) = ln(1 - 1/(2n))

ln(1 - x) = -(x + (1/2)x^2 + (1/3)x^3 + ...)

If P is the product then ln(P) = sum of ln(1 - 1/(2n)) for n = 1, 2, ..., 50.

All terms are negative, so gather enough terms to drop the sum below - ln 10. That seems too complicated an approach for solving this, though.

Google will let you limit your search to a particular language. Use Google translate to translate math olympiad to Russian then search for the result.

Answer 3

The gist of the communicate obtainable (sorry, there are too many pages to grant each and all of the links) is that wands help to concentration the magic resident in the witch or wizard, yet wandless magic remains a threat, extremely for terribly proficient human beings like Snape and Dumbledore. some examples ... Apparition Assuming one's Animagus variety (or Metamorphpagus) Accio (the Summoning allure) Elves can do magic with out making use of wands Lumos