The 12th Indian National Mathematical Olympiad
1997年第12届印度奥林匹克数学竞赛 |
- ABCD is a parallelogram. A line through C does not pass through the interior of ABCD and meets the lines AB, AD at E, F respectively. Show that AC2 + CE·CF = AB·AE + AD·AF.
- Show that there do not exist positive integers m, n such that m/n + (n+1)/m = 4.
- a, b, c are distinct reals such that a + 1/b = b + 1/c = c + 1/a = t for some real t. Show that t = -abc.
- In a unit square, 100 segments are drawn from the center to the perimeter, dividing the square into 100 parts. If all parts have equal perimeter p, show that 1.4 < p < 1.5 .
- Find the number of 4 x 4 arrays with entries from {0, 1, 2, 3} such that the sum of each row is divisible by 4, and the sum of each column is divisible by 4.
- a, b are positive reals such that the cubic x3 - ax + b = 0 has all its roots real. α is the root with smallest absolute value. Show that b/a < α ≤ 3b/2a.
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