MathDB

18

Part of 2018 Online Math Open Problems

Problems(2)

2017-2018 Spring OMO Problem 18

Source:

4/3/2018
Suppose that a,b,ca,b,c are real numbers such that a<b<ca < b < c and a3−3a+1=b3−3b+1=c3−3c+1=0a^3-3a+1=b^3-3b+1=c^3-3c+1=0. Then 1a2+b+1b2+c+1c2+a\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a} can be written as pq\frac pq for relatively prime positive integers pp and qq. Find 100p+q100p+q.
Proposed by Michael Ren
2018-2019 Fall OMO Problem 18

Source:

11/7/2018
On Lineland there are 2018 bus stations numbered 1 through 2018 from left to right. A self-driving bus that can carry at most NN passengers starts from station 1 and drives all the way to station 2018, while making a stop at each bus station. Each passenger that gets on the bus at station ii will get off at station jj for some j>ij>i (the value of jj may vary over different passengers). Call any group of four distinct stations i1,i2,j1,j2i_1, i_2, j_1, j_2 with iu<jvi_u< j_v for all u,v∈{1,2}u,v\in \{1,2\} a good group. Suppose that in any good group i1,i2,j1,j2i_1, i_2, j_1, j_2, there is a passenger who boards at station i1i_1 and de-boards at station j1j_1, or there is a passenger who boards at station i2i_2 and de-boards at station j2j_2, or both scenarios occur. Compute the minimum possible value of NN.
Proposed by Yannick Yao
Online Math Open