All Questions
57
questions
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
23
votes
3
answers
739
views
Is it possible to cover $\{1,2,...,100\}$ with $20$ geometric progressions?
Recall that a sequence $A=(a_n)_{n\ge 1}$ of real numbers is said to be a geometric progression whenever $\dfrac{a_{n+1}}{a_n}$ is constant for each $n\ge 1$. Then, replacing $20$ with $12$, the ...
- 15.5k
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
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
14
votes
2
answers
209
views
How many numbers can we select from $\{1,2,...2016\}$ such that sum of any four of them cannot be divided by $11$
How many numbers can we select from $\{1,2, \ldots, 2016\}$ such that sum of any four of them cannot be divided by $11$
It's not hard to come up with some combinations, but the question is how to ...
- 755
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
11
votes
1
answer
357
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
7
votes
5
answers
276
views
On existence of positive integer solution of $\binom{x+y}{2}=ax+by$
How can I prove this?
Prove that for any two positive integers $a,b$ there are two positive integers $x,y$ satisfying the following equation:
$$\binom{x+y}{2}=ax+by$$
My idea was that $\binom{x+...
- 376
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
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
6
votes
4
answers
4k
views
Books for maths olympiad
I want to prepare for the maths olympiad and I was wondering if you can recommend me some books about combinatorics, number theory and geometry at a beginner and intermediate level. I would appreciate ...
5
votes
3
answers
111
views
$x_1 + x_2 + x_3 \le 50$ solutions
The book shows the answer as attached.
Their equation,
$$x_1 + x_2 + x_3 + y = 50 \implies x_1 + x_2 + x_3 = 50 - y$$
How is that the same as solving,
$$x_1 + x_2 + x_3 \le 50$$
???
- 1,736
5
votes
1
answer
757
views
Find all whole number solutions of the following equation
While training for a math olympiad(university level) I stumbled upon the following problem. Find all $n, k \in \mathbb{N}$ such that
$${ n \choose 0 } + {n \choose 1}+{n \choose 2} + {n \choose 3} = ...
- 4,188
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