Ten ants on a number line (KVPY SX problem)

Hi! I have some trouble solving this problem for KVPY SB/SX 2016. I could find out the number of crosses that happen between the ants which is 45 but couldnt find the number of distinct times in which atleast 2ants are at same position, which as per the key is 17. I request you to help me with the solution.

kvpy #math #combinatorics #puzzle

PS: Sorry for the very small question image. I took a screnshot from the question paper and dont know how to resize it

#Combinatorics

Easy Math Editor

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Comments

This Link might help you.

Rishabh Deep Singh - 3 years, 7 months ago

@Calvin Lin, @Brian Charlesworth,@Steven Chase, @Swapnil Das, @Deeparaj Bhat,@Agnishom Chattopadhyay,@Prakhar Bindal,@Rishabh Cool,@Rishabh Deep Singh, @Ronak Agarwal,@Azhaghu Roopesh M
Please help

Jaya Krishna - 3 years, 7 months ago

What have you tried? Where did you get stuck?

Calvin Lin Staff - 3 years, 7 months ago

What I did: The 1st ant goes from 1 to 100. The 10th from 100 to 1. 2nd from 4 to 81.... and so on. Each ant travels a disoacement of 11(11-2k), the 1st crosses 9 ants, 2nd crosses 8 ants, 3rd crosses 7ants and so on... summing which gives 45 cross8ngs. But the problem is, these do not happen all at distinct times . Im completely lost what to do with that

Jaya Krishna - 3 years, 7 months ago

@Jaya Krishna – Great, so you discovered that there are 45 "crossings", and now the uncertainty is if there are 3 (or more) ants that are at the same spot at the same time.

Can you solve for the distance and time that ant $i$ and $j$ meet? This will help us determine how distinct the (distance,time) is.

Calvin Lin Staff - 3 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$