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min integer from brackets in (29:28:27:26:... :17:16)/(15:14:13:12: ...:3:2)

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1999 p1

February 20, 2020
minFractionInteger Polynomialalgebra

Problem Statement

We are allowed to put several brackets in the expression 29:28:27:26:...:17:1615:14:13:12:...:3:2\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2} always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained.
Remark: in this problem, (29:28):27:...:16(15:14):13:...:2,\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2}, is valid position of parenthesis, on the other hand (29:28):27:...:1615:(14:13):...:2\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2} is forbidden.
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