MathDB

11.4

Part of VMEO IV 2015

Problems(1)

2015 students standing in rows, not all tell the truth

Source: 2015 VMEO IV Juniors 11.4 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

9/15/2021
Students in a school are arranged in an order that when you count from left to right, there will be nn students in the first row, n−1n-1 students in the second row, n−2n - 2 students in the third row,... until there is one student in the nnth row. All the students face to the first row. For example, here is an arrangement for n=5n = 5, where each ∗* represents one student: ∗* ∗∗* * ∗∗∗* * * ∗∗∗∗* * * * ∗∗∗∗∗* * * * * ( first row)
Each student will pick one of two following statement (except the student standing at the beginning of the row): i) The guy before me is telling the truth, while the guy standing next to him on the left is lying. ii) The guy before me is lying, while the guy standing next to him on the left is telling the truth.
For n=2015n = 2015, find the maximum number of students telling the truth. (A student is lying if what he said is not true. Otherwise, he is telling the truth.)
combinatorics