Ten ants are on the real line. At time t=0,th | vedclass

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(C) Let the position of the $k$-th ant at time $t$ be $x_k(t)$.Given $x_k(0) = k^2$ and $x_k(1) = (11-k)^2$.Since the speed is uniform,the velocity $v

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$17$

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(C) Let the position of the $k$-th ant at time $t$ be $x_k(t)$.Given $x_k(0) = k^2$ and $x_k(1) = (11-k)^2$.Since the speed is uniform,the velocity $v_k = x_k(1) - x_k(0) = (11-k)^2 - k^2 = 121 - 22k + k^2 - k^2 = 121 - 22k$.Thus,$x_k(t) = k^2 + (121 - 22k)t$.Two ants $i$ and $j$ (where $i $i^2 + (121 - 22i)t = j^2 + (121 - 22j)t$$i^2 - j^2 = (121 - 22j - 121 + 22i)t$$(i-j)(i+j) = 22(i-j)t$Since $i 
eq j$,we can divide by $(i-j)$:$t = \frac{i+j}{22}$.Since $1 \le i Thus,$t \in \{\frac{3}{22}, \frac{4}{22}, \dots, \frac{19}{22}\}$.The number of distinct values is $19 - 3 + 1 = 17$.

Ten ants are on the real line. At time $t=0$,the $k$-th ant starts at the point $k^2$ and,travelling at a 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

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