All Questions
57
questions
3
votes
1
answer
190
views
Remainder of a combination
Problem from a contest:
What is the remainder when $\binom{169}{13}$ is divided by $13^5$?
I thought that Wolstenholme's/Babbage's would help, but not entirely sure how.
- 617
1
vote
1
answer
63
views
Find $n$ where 15756 is the $n$th member of a set
It's a question from BNMO. It still haunts me a lot. I want to find an answer to this question.
Any number of the different powers of $5: 1,5,25,125$ etc is added one at a time to
generate the ...
- 1,583
1
vote
1
answer
342
views
Count ways to distribute candies
N students sit in a line, and each of them must be given at least one candy. Teacher wants to distribute the candies in such a way that the product of the number of candies any two adjacent students ...
- 272
3
votes
2
answers
1k
views
partitions and their generating functions and Partitions of n
A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
- 217
5
votes
1
answer
1k
views
Sequence where the sum of digits of all numbers is 7
BdMO 2014
We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
- 3,914
7
votes
1
answer
347
views
South Africa National Olympiad 2000 (Tile 4xn rectangle using 2x1 tiles)
Let $A_n$ be the number of ways to tile a $4×n$ rectangle using $2×1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3.
My attempt: Define basic shapes A, B and C, ...
- 495
7
votes
0
answers
180
views
Pairwise sums are equal
The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
- 2,325
20
votes
1
answer
694
views
Sum involving binomial coefficient satisfies congruence (A contest question)
Let $p$ be an odd prime, and denote $$f(x)=\sum_{k=0}^{p-1}\binom{2k}{k}^2x^k.$$
Prove that for every $x\in \mathbf Z$,$$(-1)^\frac{p-1}2f(x)\equiv f\left(\frac{1}{16}-x\right)\pmod{p^2}.$$
This is a ...
- 15.9k
7
votes
2
answers
3k
views
Exploring Properties of Pascal's Triangle $\pmod 2$
Moderator Note: This question is from a contest which ended 1 Dec 2012.
Consider Pascal's Triangle taken $\pmod 2$:
For simplicity, we will call a finite string of 0's and 1's proper if it occurs in ...
- 123
4
votes
1
answer
893
views
Card Shuffling [SPOJ]
The original question is posted on SPOJ, and included below:
Here is an algorithm for shuffling N cards:
1) The cards are divided into K equal piles, where K is a factor of N.
2) The ...
- 183
26
votes
4
answers
2k
views
Find the number of pairs of positive integers $(a, b)$ such that $a!+b! = a^b$
How many pairs of positive integers $(a, b)$ such that $a!+b! = a^b$?
A straight forward brute-force reveals that $(2,2)$ and $(2,3)$ are solutions and this seems to be the only possible solutions, I ...
- 22.5k
17
votes
1
answer
565
views
How many $N$ of the form $2^n$ are there such that no digit is a power of $2$? [duplicate]
How many $N$ of the form $2^n,\text{ with } n \in \mathbb{N}$ are there such that no digit is a power of $2$?
For this one the answer given is the $2^{16}$, but how could we prove that that this is ...
- 22.5k