All Questions
Tagged with natural-numbers contest-math
15
questions
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45
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Consecutive multiplication of natural numbers problem [duplicate]
Prove that the product of any three consecutive natural numbers is not a perfect square. If there were four numbers, I know how to solve the problem, as I would somehow mention the number $ n(n+1)(n+2)...
user1340018
2
votes
2
answers
143
views
For any $f:\mathbb{Z_+}\to \mathbb{Z_+}$ there exist $f(a)\le f(a+b)\le f(a+2\cdot b)$
I am preparing for the USAJMO, however I got stuck on the following question. I would really appreciate any help/hint.
For any function ( not necessarily having Bijectivity ) $f:\mathbb{Z^+} \to \...
- 129
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0
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45
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Determine all ordered triples $(A, B, C)$ of elements from the set $\{1, 2, \ldots, 9\}$ for which the following holds
Determine all ordered triples $(A, B, C)$ of elements from the set $\{1, 2, \ldots, 9\}$ for which the following holds:
$ AAA + BBB + CCC = BAAC $
where $A$, $B$, and $C$ are used as digits in the ...
user1190379
5
votes
1
answer
229
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Proving $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}=2$ If $a,b,c,d \in \mathbb{N}$
Given pairwise distinct $a,b,c,d \in \mathbb{N}$, prove that $$E=2$$ if $E=\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$ is an integer.
My effort:
We have: $$\begin{aligned}
& E=\frac{...
- 13.2k
5
votes
1
answer
217
views
When does the square root of $f:\mathbb{N} \rightarrow \mathbb{N}$ exist? [duplicate]
Let $f:\mathbb{N} \rightarrow \mathbb{N}$. When does there exist a $g: \mathbb{N}\rightarrow \mathbb{N}$ such that $f(n)=g(g(n))$ for all $n$?. I don't think its always possible, for example $f(n)=n+1$...
- 366
12
votes
3
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2k
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The number of ways to represent a natural number as the sum of three different natural numbers
Prove that the number of ways to represent a natural number $n$ as the sum of three different natural numbers is equal to $$\left[\frac{n^2-6n+12}{12}\right].$$
It was in our meeting a year ago, but I ...
- 195k
3
votes
2
answers
637
views
Show that there do not exist any distinct natural numbers a,b,c,d such that
Show that there do not exist any distinct natural numbers a,b,c,d such that $a^3+b^3=c^3+d^3$ and $a+b=c+d$.
user754135
0
votes
1
answer
113
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Does a strictly increasing function satisfying this exist? [closed]
Is there a function $f: \mathbb{N}\rightarrow\mathbb{N}$ (strictly increasing) , such that the following is always true:
$$f(f(f(n))) = n + 2f(n)?$$
This is a homework problem from a contest math ...
- 353
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1
answer
1k
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How many $10-$digit numbers are divided by $11.111$ and all the digits are different?
The Problem:
How many $10-$digit numbers are divided by $11.111$ and all the digits are different?
A) $3250$
B) $3456$
C) $3624$
D) $3842$
E) $4020$
The Problematic point is, "all digits must ...
user548054
2
votes
2
answers
658
views
Making the sum of 5th power of integers, a perfect square.
Yesterday this question was posed in a contest. It contains pretty easy questions like asking range of $ab+bc+ca$ when $a^2+b^2+c^2=1$, etc.
But this question is something else. I haven't been able ...
- 1,514
7
votes
2
answers
1k
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Finding the last two digits of the given number.
Question:
Given $N=2^5+2^{{5}^{2}}+2^{{5}^{3}}+2^{{5}^{4}}... 2^{{5}^{2015}}$
Written in the usual decimal form, find the last two digits of the number $N$.
My attempt:
We know that every ...
- 1,514
1
vote
1
answer
68
views
Solving an equation for positive integral values of variables.
Question
Solve
$$y^3+3y^2+3y=x^3+5x^2-19x+20$$
for positive integers $x$ and $y$.
My approach:
I factorised the equation as
$(y-x)(x^2+y^2+3x+3y+3)=2[(x-1)(x-10)]$
and got two ...
- 1,514
1
vote
0
answers
97
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Two Perfect Squares--$(3n+1) \& (4n+1)$. [duplicate]
Assume $n$ is a Natural Number which satisfies the following 2 properties simultaneously:
$01$ . $(3n+1)$=$a$12 for some Natural Number $a$1.
$02$ . $(4n+1)$=$a$22 for some Natural Number $a$2.
...
- 413
6
votes
1
answer
1k
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Functional Equation $f(f(n))=3n$ [duplicate]
Came across this problem a little while ago but can't seem to get beyond a certain point.
Let $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(n+1)>f(n)$ and $$f(f(n))=3n$$ for all $n$. ...
- 3,690
-1
votes
3
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1k
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Find the smallest natural number that satisfy $13^N = 1 \pmod {2013}$
Moderator Note: This is a current contest question on Brilliant.org.
Find the smallest natural number that satisfy:
$$13^N = 1 \pmod {2013}$$
My idea is to use the Fermat's little theorem for ...
- 35.9k