2 unknowns, 1 equation ... (BMO 2010)

[wesleydeng]

The average of (1, 2, ... n) is (n+1)/2. Easy to calculate that remove 1 it increases by1/2 and remove n it decreases by1/2. So the original average is between 40.25 and 41.25, which means it is either 40.5 or 41. Therefore n=80 or 81. n=80 does not work because of divisibility. n=81 and 61 is the only answer.

[goyaldev]

Beautiful… will use this in mock interviews😂😂

[YunruiHu]

brilliant use of upper and lower bounds! never thought about using that to figure out n.

[OliverGoodman-todd]

Hi J pi, I was wondering if you could explain why you decide to find bounds for particular questions. I see it's quite an important technique in olympiads but I can never spot when to do it or why it may be useful. Thanks!

[ayushrudra]

oh thats actually smart - i was just gonna solve the diophantine equation and hope it wasn't crazy annoying as opposed to solving the inequality

[derwolf]

You can avoid testing case n=80 by noticing that 3(n-1)/4 must be an integer ==> 4|(n-1) ==> n=81.

[deadkiller]

Sir, can you help me on a math problem?
33998.74 = 30000(1 + 0.042/n)^3n
I tried using the Lambert W function but it didn't work, and I didn't want to approximate using Newton-Raphson method.

[archangecamilien]

I forgot the meaning of (40)3/4, lol, such bad notation, but they used to use it back in the day, in grade school and such, lol...I'm guessing 40 + 3/4...anyway, lol, a brute force way of doing this is simply to substitute n=2, then n=3, etc, for n, and solve for x, etc...should give a different answere each time...hell, lol, why not just solve for x?...It should be a function of n, etc...the thing I'm not 100% sure about is what that silly (very bad) notation meant, lol, I think that's what it meant...