Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.

In $R^d$ , all $n^d$ points of an n × n × ··· × n cube grid are contained in 2n - 3 hyperplanes. Prove that n ($n\geq3$) hyperplanes can be chosen from these so that they contain all points of the grid.

When the IMO is over and students want to relax, they all do the same thing: download movies from the internet. There is a positive number of rooms with internet routers at the hotel, and each student wants to download a positive number of bits. The load of a room is defined as the total number of bits to be downloaded from that room. Nobody likes slow internet, and in particular each student has a displeasure equal to the product of her number of bits and the load of her room. The misery of the group is defined as the sum of the students’ displeasures. Right after the contest, students gather in the hotel lobby to decide who goes to which room. After much discussion they reach a balanced configuration: one for which no student can decrease her displeasure by unilaterally moving to another room. The misery of the group is computed to be $M_1$, and right when they seemed satisfied, Gugu arrived with a serendipitous smile and proposed another configuration that achieved misery $M_2$. What is the maximum value of $M_1/M_2$ taken over all inputs to this problem? [i]Proposed by Victor Reis (proglote), Brazil.[/i]

Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.