All Questions
57
questions
4
votes
1
answer
126
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 ...
- 9,567
5
votes
0
answers
2k
views
Good books to learn olympiad geometry,number theory, combinatorics and more
I want to start learning olympiad mathematics more seriously, and I would like to have advice on some good books or pdfs to learn with.
I have background but not a big background. For example I know ...
CommunityBot
- 1
-1
votes
2
answers
125
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 + ...
- 26.1k
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 $n\geq 10$.
...
- 3,187
-1
votes
1
answer
128
views
golden ratio of a fraction
This is a computational exercise, but I am looking to attempt on a calculation on a golden ratio. I am trying to compute that of the continued fraction for the golden ratio $(1+\sqrt{5})/2$, and I am ...
- 431k
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 ...
- 593
4
votes
1
answer
123
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 ...
- 106k
3
votes
5
answers
824
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 ...
- 7,388
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
votes
2
answers
132
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 ...
- 70.4k
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-...
- 2,322
1
vote
0
answers
346
views
Limit in Olympiad discursive question
Let $M,k$ be two positive integers. Define $X_{M,k}$ as the set of the numbers $p_1^{\alpha_1}\cdot p_2^{\alpha_2} \cdots p_r^{\alpha_r}$ where $p_i$ are prime numbers such that $M \leq p_1 < p_2 &...
- 138
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),...
- 3,598
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 ...
- 16.4k