All Questions
57
questions
2
votes
1
answer
4k
views
Number of positive integral solutions of $a+b+c+d+e=20$ such that $a<b<c<d<e$ and $(a,b,c,d,e)$ is distinct
This is from a previous question paper for an entrance exam I am preparing for.
https://www.allen.ac.in/apps/exam-2014/jee-advanced-2014/pdf/JEE-Main-Advanced-P-I-Maths-Paper-with-solution.pdf (Link ...
- 53
1
vote
1
answer
265
views
How many numbers less than $100$ can be expressed as a sum of distinct factorials?
How many numbers less than $100$ can be expressed as a sum of distinct factorials?
Example:
a) $4 = 2! + 2!$
b) $3 = 2! + 1!$
- 535
1
vote
2
answers
936
views
How many triples satisfy $ab + bc + ca = 2 + abc $
$a^2 + b^2 + c^2 - \frac{a^3 + b^3 + c^3 - 3abc}{a+b+c} = 2 + abc$
How many triples $(a,b,c)$ satisfies the statement? Here $a,b,c > 1$.
It is easy to simplify the statement to
$$ab + bc + ca =...
- 3,146
1
vote
1
answer
65
views
Maximal Consecutive Integer Sequence
I'm doing up solutions to some junior Olympiad problems and am somewhat stumped by one of the questions:
Can you find a sequence of 14 consecutive positive integers such that each is divisible by ...
- 2,158
6
votes
4
answers
4k
views
Books for maths olympiad
I want to prepare for the maths olympiad and I was wondering if you can recommend me some books about combinatorics, number theory and geometry at a beginner and intermediate level. I would appreciate ...
5
votes
1
answer
757
views
Find all whole number solutions of the following equation
While training for a math olympiad(university level) I stumbled upon the following problem. Find all $n, k \in \mathbb{N}$ such that
$${ n \choose 0 } + {n \choose 1}+{n \choose 2} + {n \choose 3} = ...
- 4,188
-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 ...
- 2,374
2
votes
1
answer
178
views
Divisibility of a summation
Let $n , l, k, p$ be positive integers, and $1\leq p\leq n$. Let $B(n, l, k, p)$ be the cardinality of the following set
\begin{eqnarray}
\{(a_1, a_2, \cdots, a_n)\in\mathbb{Z}^{\oplus n}|\ \ 0\leq ...
- 699
3
votes
1
answer
84
views
Putnam: Show that $a(n)=b(n+2)$
Let $a(n)$ be the number of representations of positive integer $n$ as a sum of 1's and 2's taking order into account.
$$ \text{Example $n=4$: } (1+1+1+1), (1+2+1),(1+1+2),(2+1+1),(2+2)\implies a(4)=...
- 195
1
vote
2
answers
402
views
Number of divisors $d$ of $n^2$ so that $d\nmid n$ and $d>n$
I just wanted to share this nutshell with you guys, it is a little harder in this particular case of the problem:
Find the number of divisors $d$ of $a^2=(2^{31}3^{17})^2$ so that $d>a$. What is ...
- 106k
3
votes
3
answers
1k
views
Probability that the eventually a six on a dice will appear.
Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime ...
- 11.2k
23
votes
3
answers
739
views
Is it possible to cover $\{1,2,...,100\}$ with $20$ geometric progressions?
Recall that a sequence $A=(a_n)_{n\ge 1}$ of real numbers is said to be a geometric progression whenever $\dfrac{a_{n+1}}{a_n}$ is constant for each $n\ge 1$. Then, replacing $20$ with $12$, the ...
- 15.5k
0
votes
1
answer
121
views
PIE Problem with divisors
Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}$.
Let $n(A)$ be the number of positive integers that divide $10^{10}$ let $n(B)$ be the number of ...
- 1,736
5
votes
3
answers
111
views
$x_1 + x_2 + x_3 \le 50$ solutions
The book shows the answer as attached.
Their equation,
$$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
3
votes
2
answers
95
views
A grandmother is giving out apples to her grandchildren.
A grandmother has 7 grandchildren, and 14 apples to give. How many ways can she give apples to her grandchildren so that each grandchild gets aT LEAST one? (but she has to get rid of hers).
This ...
- 1,736