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 times at which at least two ants are at the same location is

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

Answer 1

The correct option is C $17$

Velocity of any ant $u=(11-k{)}^2-k^2=121-22k$
Distance travelled by any ant is given by
$s=s_0+ut$
Two ants are at same position if,
$s_i=s_j\Rightarrow (k_i{)}^2+t(121-22k_i)=(k_j{)}^2+t(121-22k_j)\Rightarrow t=\frac{{k_i}^2-{k_j}^2}{22(k_i-k_j)}=\frac{k_i+k_j}{22}$ [$\because k_i\ne k_j$]

When $i=1$
The value of $t$ will be $\frac{3}{22},\frac{4}{22},\frac{5}{22},............,\frac{11}{22}\left(9\text{values}\right)$

When $i=2$
The value of $t$ will be $\frac{4}{22},\frac{5}{22},\frac{6}{22},............,\frac{12}{22}\left(9\text{values}\right)$
but only one distinct value.
Similarly for $i=3,4,5,6,7,8,9$ we get only $1$ distinct value each.
Total distinct values $=17$