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 ...
- 1,837
-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 + ...
- 1,137
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 $n\geq 10$.
...
- 11
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 ...
- 73
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 ...
- 1,711
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 ...
- 1,699
-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 ...
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
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-...
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),...
- 4,135
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 ...
- 4,135
1
vote
0
answers
123
views
PDFs for Olympiad preparation
Could someone please recommend me some pdf files containing theory for topics that come up often in maths olympiads? I'm currently working through one about inequalities, and I'm really enjoying it. I ...
- 41
4
votes
1
answer
187
views
question relating to the Euler's totient function
I just cam across a question in number theory which relates to Euler's totient function. The question is the following:
We have a positive integer $n>1$. Find the sum of all numbers $x$, such that $...
user814992
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
3
votes
0
answers
126
views
After $2013$ such transformations, how many number $2013$ are there on the line if the given numbers are $1$ and $1000$?
Several natural numbers are given on a line, we perform a transformation as follow:for every pair of consecutive integers on the line, write sum of those two numbers in the middle of them. After $2013$...
- 4,135
0
votes
2
answers
277
views
Find the smallest natural n with the property [closed]
Find the smallest natural $n$ with the property that it enters any $n$ distinct numbers from the set $\{1, 2, 3,\dots , 999\}$ we can find four distinct numbers $a, b, c, d$ such that $a + 2b + 3c = d$...
- 513
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 &...
- 2,314
4
votes
1
answer
166
views
Prove that the number of beautiful positive integers in the set $\{ 2^{20},\; 2^{20}+1,\; 2^{20}+2, \; ..., \; 2^{21}-1 \}$ is divisible by 17
Definition. Let a positive integer $n$ be written in binary numeral system. We shall say that a some digit of the $n$ is interesting if this digit is not equal to the adjacent digit to the right of it ...
- 952
0
votes
1
answer
182
views
Prove that the overlap of some two of these surfaces has an area greater than or equal to $\frac{1}{9}$.
The union of nine planar surfaces, each of area equal to 1, has a total area equal to 5. Prove that the overlap of some two of these surfaces has an area greater than or equal to $\frac{1}{9}$.
It is ...
1
vote
3
answers
196
views
number of ways to choose subsets from 11 boys and 12 girls where number of girls in the subset is one more than boys
Disclaimer: This is from AIME 2020 that has ended yesterday. https://www.maa.org/math-competitions/about-amc/events-calendar
A club has 11 boys, 12 girls. We need to choose a subset of kids from them,...
- 755
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
0
votes
0
answers
76
views
Combinatorics with Bashy
We call a set of positive integer good, if the greatest common divisor of all of the elements in this set is $1$.
$a_n$ is the number of good subsets of $\{1,2,...,n\}$. Find all integer $n \ge 2019$, ...
- 4,266
1
vote
2
answers
126
views
Square Chessboard Problem [duplicate]
Show that there is a $6$ x $4$ board whose squares are all black or white, where no rectangle has the four vertex squares of the same color. Also show that on each $7$
x $4$ board whose squares are ...
- 1,591
1
vote
2
answers
94
views
Number Theory Problem: Zonal Informatics Olympiad Help?!
In Gutenberg’s printing press, each line of text is assembled by placing individual metal letters in a rack, applying ink to the letters and then pressing them onto paper.
Gutenberg needs to print N ...
- 113
2
votes
1
answer
262
views
Existence of a subset such the product of its elements is a perfect square
Suppose $S \subset \{1,2,3,\ldots, 200\}$ such $|S|=50 $. Prove there exist a non empty subset of $S$ such that the product of its elements is a perfect square.
I have a solution, but I want to know ...
3
votes
2
answers
559
views
How many pairs of natural numbers can be formed whose LCM will be $49000$?
I'm stuck on this question.
First, I factorize $49000$.
$49000 = 2^3 \cdot 5^3 \cdot 7^2$
I've just assumed the two numbers : $$P=(2^a \cdot 5^b \cdot 7^c) \text{ and } Q=(2^x \cdot 5^y \cdot 7^z)...
7
votes
5
answers
276
views
On existence of positive integer solution of $\binom{x+y}{2}=ax+by$
How can I prove this?
Prove that for any two positive integers $a,b$ there are two positive integers $x,y$ satisfying the following equation:
$$\binom{x+y}{2}=ax+by$$
My idea was that $\binom{x+...
- 376
5
votes
0
answers
2k
views
Good books to learn olympiad geometry,number theory, combinatorics and more
I want to start learning olympiad mathematics more seriously, and I would like to have advice on some good books or pdfs to learn with.
I have background but not a big background. For example I know ...
- 2,510
4
votes
1
answer
550
views
Game: two stacks of coins and reducing by divisor of number of coins on the stack
Player $A$ and $B$ play a game: There are two stacks of coins, first stack has $a=12$ coins, second - has $b=8$ coins. Each player choose a stack and remove $d$ coins, if choose first stack $d|a$, ...
- 914
2
votes
1
answer
4k
views
Number of positive integral solutions of $a+b+c+d+e=20$ such that $a<b<c<d<e$ and $(a,b,c,d,e)$ is distinct
This is from a previous question paper for an entrance exam I am preparing for.
https://www.allen.ac.in/apps/exam-2014/jee-advanced-2014/pdf/JEE-Main-Advanced-P-I-Maths-Paper-with-solution.pdf (Link ...
- 53
1
vote
1
answer
265
views
How many numbers less than $100$ can be expressed as a sum of distinct factorials?
How many numbers less than $100$ can be expressed as a sum of distinct factorials?
Example:
a) $4 = 2! + 2!$
b) $3 = 2! + 1!$
- 535
1
vote
2
answers
936
views
How many triples satisfy $ab + bc + ca = 2 + abc $
$a^2 + b^2 + c^2 - \frac{a^3 + b^3 + c^3 - 3abc}{a+b+c} = 2 + abc$
How many triples $(a,b,c)$ satisfies the statement? Here $a,b,c > 1$.
It is easy to simplify the statement to
$$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
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