input.txt

1555 1630
1600 1635
1600 1640
1610 1640
1625 1720
1635 1720
1645 1740
1650 1720
1710 1730
1715 1810
1720 1740
1725 1810

main.cpp

#include <iostream>
#include <fstream>
#include <vector>
#include <string>

using namespace std;

void split (string &txt, vector<int> &out, char ch);
void search_solution (vector <vector <int>> data, int maximum, int x, vector <int> &push);
int search_max (vector <int> target);

int main()
{
    vector <int> push;
    vector <vector <int>> data;
    string line;
    ifstream input;
    input.open("input.txt",ios::in);
    while (getline(input,line)) {
        vector <int> line_vector;
        split (line, line_vector, ' ');
        data.push_back(line_vector);
    }
    /* DEBUG PRINT
    for (unsigned int i = 0; i < data.size(); i++) {
        cout << i+1 << ". elem: " << data[i][0] << ">>" << data[i][1] << " length: " << data[i][2] << endl;
    }
    */
    int max_main = 0;
    search_solution(data, max_main, 0, push);
    cout << search_max(push) << endl;
    input.close();
    return 0;
}

void split (string &txt, vector<int> &out, char ch) {
    int i = 0;
    while (true) {
        if (txt[i] == ' ') {
            string begin_str = txt.substr(0,i);
            string end_str = txt.substr(i+1,txt.size()-i);
            int begin_int = stoi(begin_str.substr(0,2))*60 + stoi(begin_str.substr(2,2));
            int end_int = stoi(end_str.substr(0,2))*60 + stoi(end_str.substr(2,2));
            out.push_back(begin_int);
            out.push_back(end_int);
            out.push_back(end_int-begin_int);
            break;
        }
        i++;
    }
}

void search_solution (vector <vector <int>> data, int maximum, int x, vector <int> &push) {
    for (unsigned int i = x+1; i < data.size(); i++) {
        if (data[x][1] <= data[i][0]) {
            search_solution(data, maximum+1, i, push);
        }
    }
    push.push_back(maximum+1);

}

int search_max (vector <int> target) {
    int maximum = 0;
    for (unsigned int i = 0; i < target.size(); i++) {
        if (target[i] > maximum) {
            maximum = target[i];
        }
    }
    return maximum;
}

Great solution! You are awesome! :)