be a positive integer that is not divisible by 
.
be a positive integer. Prove that none of the numbers of the form 
is a square number.
and has exactly 
be real numbers such that 
Find the maximum value of the expression 
different bus lines in a city, each stopping at exactly 
stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city.
Y by cubres
An interesting problem:
There are
events 
that are each continuous and last on a certain time interval. Each event has a weight 
However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:
For each
consider 
the sum of 
over all 
such that 
and are not compatible.
1. At each step, delete the event that has the maximal
If there are multiple such events, delete the event with the minimal weight.
2. Update all
3. Repeat until all are 
There are
events 
that are each continuous and last on a certain time interval. Each event has a weight 
However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:For each
consider 
the sum of 
over all 
such that 
and are not compatible.1. At each step, delete the event that has the maximal
If there are multiple such events, delete the event with the minimal weight.2. Update all
3. Repeat until all
are 
This post has been edited 1 time. Last edited by Maximilian113, Apr 13, 2025, 12:56 AM
Reason: clarity
Reason: clarity
