All Questions
57
questions
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
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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 $d|a$, ...
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