All Questions
57
questions
4
votes
1
answer
125
views
can one select 102 17-element subsets of a 102-element set so that the intersection of any two of the subsets has at most 3 elements
Can one select 102 17-element subsets of a 102-element set so that the intersection of any two of the subsets has at most 3 elements?
I'm not sure how to approach this problem. I think it might be ...
- 1,837
-1
votes
2
answers
124
views
solve in positive integers $\frac{1}a + \frac{1}b + \frac{1}c = \frac{4}5$ [duplicate]
Solve in positive integers $\frac{1}a + \frac{1}b + \frac{1}c = \frac{4}5$ (i.e. find all triples $(a,b,c)$ of positive integers satisfying the equation).
The expression is equivalent to $5(bc + ac + ...
- 1,137
1
vote
2
answers
446
views
Prove that there is a perfect cube between n and 3n for any integer n≥10
I was solving one of the Number theory problems from Mathematical Olympiad Challenges,
And the problem goes like :
Prove that there is a perfect cube between $n$ and $3n$ for any integer $n\geq 10$.
...
- 11
0
votes
0
answers
90
views
Number of 1-runs
A binary string is a word containing only $0$s and $1$s. In a binary string, a
1-run is a non-extendable substring containing only $1$s. Given a positive
integer n, let B(n) be the number of $1$-runs ...
0
votes
1
answer
121
views
Prove that you can get all combinations of coins in 4n - 1 moves
I was participating in a high-school math olympiad qualification contest and this was one of the problems I didn't manage to solve. The solutions will be posted in a month or so, but I'm very eager to ...
- 73
4
votes
1
answer
122
views
The largest possible number of inversions in a sequence of positive integers whose sum is $2014$
In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the ...
- 1,699
3
votes
5
answers
817
views
How many whole numbers between $100$ and $800$ contain the digit $2$?
I had a very strange doubt in this question while I was solving it. Now in order to solve first I calculated the three digit numbers which won't have $2$ at all in them and the number of such three ...
- 1,699
-1
votes
2
answers
131
views
There are $n$ persons present at a meeting. Every two persons are either friends of each other or strangers to each other. BMO round $2$ , $1972$
There are $n$ persons present at a meeting. Every two persons are either friends of each other or strangers to each other.
No to friends have a friend in common. Every to strangers
have two and only ...
12
votes
2
answers
506
views
Integers less than $7000$ achievable by starting with $x=0$ and applying $x\to\lceil x^2/2\rceil$, $x\to\lfloor x/3\rfloor$, $x\to9x+2$
Problem
Robert is playing a game with numbers. If he has the number $x$, then in the next move, he can do one of the following:
Replace $x$ by $\lceil{\frac{x^2}{2}}\rceil$
Replace $x$ by $\lfloor{\...
- 707
1
vote
1
answer
122
views
Number of solutions number theory problem
I am wondering how many nonnegative solutions the following Diophantine equation has: $$x_1+x_2+x_3+\dots+x_n=r$$ if $x_1 \leq x_2 \leq x_3 \leq \dots \leq x_n$
I know if a sequence can be non-...
1
vote
1
answer
113
views
Prove that $S$ has the same property $P_k$ of $majority$ for all positive integers $k$.
Let $n$ be a positive integer and let $S \subseteq \{0, 1\}^n$ be a set of binary strings of length $n$. Given an odd number $x_1, \dots, x_{2k + 1} \in S$ of binary strings (not necessarily distinct),...
- 4,125
1
vote
1
answer
94
views
Partition the numbers into disjoint pairs , and the replace each pair with it's non negative difference .
The numbers $1,2, \cdots, 2^n$ , $n>2$ is a natural number are written on a board . The following procedure is performed n times: partition the numbers into disjoint pairs , and the replace each ...
- 4,125
1
vote
0
answers
122
views
PDFs for Olympiad preparation
Could someone please recommend me some pdf files containing theory for topics that come up often in maths olympiads? I'm currently working through one about inequalities, and I'm really enjoying it. I ...
- 41
4
votes
1
answer
185
views
question relating to the Euler's totient function
I just cam across a question in number theory which relates to Euler's totient function. The question is the following:
We have a positive integer $n>1$. Find the sum of all numbers $x$, such that $...
user814992
11
votes
1
answer
356
views
Counting the number of decimals that satisfy a condition
This is supposedly a problem from a $\textbf{Chinese Math Olympiad team selection test}$ but it looks like an interesting combination of combinoatorics and number theory.
$\textbf{Problem:}$
We have ...
- 484