Question

Ten ants are on the real line. At time $t=0$, the $k^{th}$ ant starts at the point $k^2$ and travelling at uniform speed, reaches the point $(11-k{)}^2$ at time $t=1$. The number of distinct time at which at least two ants are at the same location is?

• A. $45$
• B. $11$
• C. $17$
• D. $9$

Answer 1

Answer: (C)

$17$

Velocity of $k^{th}$ ant $={(11-k)}^2-k^2=121-22k$

Position of $k^{th}$ ant at time $t:x_k=k^2+t(121-22k)$

Let $p^{th}$ ant and $k^{th}$ ant ($k$ and $p$ are distinct) be at the same position at time $t$.

$\Rightarrow k^2+t(121-22k)=p^2+t(121-22p)$

$\Rightarrow (k^2-p^2)-22t(k-p)=0$

$\Rightarrow k+p=22t$

$k$ and $p$ both range from $1$ to $10$ and are distinct, so $(k+p)$ attains each value in the set $\{3,4,\mathrm{5...18},19\}$.

Hence, there are 17 distinct times at which at least two ants are at the same location.