All Questions
11
questions
1
vote
1
answer
283
views
Showing there exists a sequence that majorizes another
The exact quantity of gas needed for a car to complete a single loop around a track is distrubuted among $n$ containers placed along the track. Show that there exists a point from which the car can ...
- 1,418
1
vote
1
answer
2k
views
Making all row sums and column sums non-negative by a sequence of moves
Real numbers are written on an $m\times n$ board. At each step, you are allowed to change the sign of every number of a row or of a column. Prove that by a sequence of such steps, you can always make ...
- 3,914
6
votes
5
answers
2k
views
Least wasteful use of stamps to achieve a given postage
You have sheets of $42$-cent stamps and
$29$-cent stamps, but you need at least
$\$3.20$ to mail a package. What is the
least amount you can make with the $42$-
and $29$-cent stamps that is ...
- 36.6k
2
votes
1
answer
139
views
HMMT 2014 #9, how many times has Lucky performed the procedure when there are 20 tails-up coins?
There is a heads up coin on every integer of the number line. Lucky is initially standing on the zero
point of the number line facing in the positive direction. Lucky performs the following procedure: ...
- 2,031
2
votes
1
answer
105
views
determine the least number of rounds that needs to be played so that every child is satisfied
A group of N children, who are numbered 1, 2, . . . , N, want to play hide and seek. In a single round of hide and seek, there will one seeker, and N −1 hiders. Children like to hide and not seek and ...
- 49
15
votes
3
answers
946
views
Partition 100 people, 4 from each country into 4 groups with conditions
This is a problem from the $2005$ All-Russian Olympiad. Problem is as follows:
$100$ people from $25$ countries, four from each country, sit in a circle.
Prove that one may partition them onto $4$ ...
- 381
9
votes
3
answers
535
views
What is the largest possible number of moves that can be taken to color the whole grid?
Consider a $10\times 10$ grid. On every move, we color $4$ unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the $4$ squares is ...
- 90.7k
7
votes
2
answers
3k
views
Given $2n$ points in the plane, prove we can connect them with $n$ nonintersecting segments
Given $2n$ points in the plane such that no three points lie on one line. Prove that it is possible to draw $n$ segments such that each segment connects a pair of these points and no two segments ...
- 1,418
4
votes
2
answers
1k
views
Algorithm to uniquely determine a number using two adjacent digits
(Russia)
Arutyun and Amayak perform a magic trick as follows. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak covers two adjacent digits by a black disc. Then Arutyun ...
- 3,914
1
vote
1
answer
1k
views
Sum over all contiguous partitions of an array
Consider an array $A$ of length $n$. We can split $A$ into contiguous segments called pieces and store them as another array $B$ . For example , if $A = [1,2,3]$, we have the following arrays of ...
- 13
1
vote
1
answer
87
views
An optimization problem to find the consecutive day subset with maximum value - ZIO $2006$, P$1$
Hello everybody! The above problem is a combinatorics problem I got wrong. :( This is ZIO $2006$, P$1$.
For the first part, I got as answer: $3$ which is wrong. What I did to try my value which I ...
- 779