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

Step-by-step solution

Velocity of any ant U=(11-k)²-k²=121-22 k Now at any time distance travelled by any ant will be S = S ₀+ ut Where S ₀ is the initial position Now two ants will be at same position If S_ i =S_ i array l k _ i ²-22 k _ i t +121 t = k _ j ²-22 k _ j t +121 t t = k _ j ²- k _ i ² 22 ( k _ j - k _ i ) ; t = k _ j + k _ i 22 ( as k _ i k _ j ) array Now for i =1 Values of t will be 3 22 , 4 22 , 5 22 , . 11 22 (9 values) i=2 values of t will be 4 22 , 5 22 , 11 22 , 12 22 We can see there is only 1 distinct value Similar