All Questions
Tagged with algebraic-combinatorics contest-math
7
questions
9
votes
1
answer
542
views
Let $B\subset A = \{1,2,3,...,99,100\}$ and $|B|= 48$. Prove that exist $x,y\in B$, $x\ne y$ such that $11\mid x+y$.
Let $B\subset A = \{1,2,3,...,99,100\}$ and $|B|= 48$.
Prove that exist $x,y\in B$, $x\ne y$ such that $11\mid x+y$.
Proof: Let $P_0:= \{11,22,...,99\}$ and for $i=
1,2,...49$ and $11\nmid i$ make ...
- 90.7k
4
votes
1
answer
422
views
Given a positive integer $n$, some straight lines and lattice points such... Prove that the number of the lines is at least $n(n+3)$.
Given a positive integer $n$ and some straight lines in the plane
such that none of the lines passes through $(0,0)$, and such that every lattice point
$(a,b)$, where $ 0\leq a,b\leq n$ are integers ...
- 90.7k
8
votes
5
answers
268
views
Algebraic proof of $\sum_{k}\binom{n}{2k}\binom{2k}{k}2^{n-2k}=\binom{2n}n$ (Combinatoric proof is given)
I had a IMO training about double counting. Then, there is a problem which I hope there is a combinatoric proof. Here comes the problem:
For every positive integer $n$, let $f\left(n\right)$ be ...
- 4,043
6
votes
2
answers
457
views
There are $n$ different $3$-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= j$.
Determine all possible values of positive integer $n$, such that there are $n$ different $3$-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not=...
- 90.7k
22
votes
3
answers
735
views
A conference uses $4$ main languages. Prove that there is a language that at least $\dfrac{3}{5}$ of the delegates know.
A conference uses $4$ main languages. Any two delegates always have a common language that they both know. Prove that there is a language that at least $\dfrac{3}{5}$ of the delegates know.
Source: ...
- 90.7k
10
votes
1
answer
307
views
Does in plane exist $22$ points and $22$ such circles that each circle contains at least $7$ points and each point is on at least $7$ circles.
Does in plane exist $22$ points and $22$ such circles that each circle contains at least $7$ points and each point is on at least $7$ circles.
I have solved this one but now I can't remember how I ...
- 90.7k
1
vote
2
answers
823
views
What is the expected value of the product of the number of heads you get and the number of tails you get when you flip n coins? [closed]
Ex.
If you get $h$ heads and $n-h$ tails the product would be $n(n-h)$ I want to know the expected value of this product.
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