Tagged Questions
9
votes
1answer
142 views
Show that $4mn-m-n$ can never be a square
Let $m$ and $n$ be positive integers. Show that $$4mn-m-n$$ can never be a square.
In my attempt I started by assuming for the sake of contradiction that $$4mn-m-n=k^2$$ for some $k \in ...
9
votes
0answers
154 views
A contest question
$p$ is an odd prime,denote $$f(x)=\sum_{k=0}^{p-1}\binom{2k}{k}^2x^k.$$
Prove that for every $x\in Z$,$$(-1)^\frac{p-1}2f(x)\equiv f(\frac{1}{16}-x)\pmod{p^2}.$$
This is a contest question,I do not ...
17
votes
1answer
149 views
To prove that $2^{3n}+2^n +1$ is not a perfect square.
Question: Prove that $2^{3n} + 2^n + 1$ cannot be a perfect square for any natural $n$.
I attempted this question and failed in two different ways.
1) I considered a polynomial $p(x) = x^3+ x + 1 - ...
10
votes
5answers
140 views
Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.
Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.($a^2+b^2$, is same $b^2+a^2$), $n \in \mathbb{N}.$
...
7
votes
2answers
107 views
For which integers x, y is $2^x + 3^y$ a square of a rational number?
For which integers x, y is $2^x + 3^y$ a square of a rational number?
(Of course $(x,y)=(0,1),(3,0)$ work)
3
votes
1answer
114 views
Find all integer solutions to $x^2+4=y^3$.
Find all integer solutions to $x^2+4=y^3$.
Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
5
votes
2answers
165 views
Does there always exist an odd number of elements?
Given a nonzero integer $k$, does there always exist a positive integer $n$ such that there are exactly an odd number of elements $i\in\{0,1,...,n-1\}$ with $\frac{2^n-1}4 < 2^ik \mod{2^n-1} < ...
10
votes
1answer
85 views
Proving existence of a square-free sequence
I found this problem and a solution sketch in a MathOverflow answer, and I thought it was nice enough to deserve more attention and a properly written solution.
Problem:
Prove that for each ...
9
votes
1answer
244 views
Prove that all prime divisors of $7a^2(a+1)-1$ are of the form $7k\pm1$
Question:
Let a be a positive integer. Prove that all prime divisors of $7a^2(a+1)-1$ are of the form $7k\pm1$
$a$ and $k \in \mathbb{N}$ .
16
votes
3answers
323 views
Finding all integer solutions of $5^x+7^y=2^z$
Find all integers $x,y,z$ such that $5^x+7^y=2^z$.
This one comes from an online contest that I arranged some years ago, and I can assure that a completely elementary solution exists.
5
votes
2answers
219 views
Divisibility - Math Olympiad
Show that for any positive integer $m$, there is an infinite number of pairs of integers $(x,y)$ satisfying the conditions:
i) $\gcd(x,y)=1 $;
ii) $y \mid x^2+m$;
iii) $x \mid y^2+m$.
2
votes
3answers
190 views
Integers with 15 divisors (from brilliant.org)
How many integers from 1 to 19999 have exactly 15 divisors?
Note: This is a past question from brilliant.org.
2
votes
0answers
71 views
How to make a string of numbers not to become a square number? [duplicate]
Moderator Note: this is a question from the Federal Mathematics Competition 2013.
Anja and Bernd are playing the following game: They take turns write down digits on the blackboard. Anja starts. ...
4
votes
1answer
82 views
Can the distance from the vertices of a square of integer width to an inscribed circle all be integer?
I'm looking for solutions to the following British Mathematical Olympiad question:
Suppose that $ABCD$ is a square and that $P$ is a point which is on the circle inscribed in the square. Determine ...
5
votes
1answer
66 views
Given $A$ and $B$, how many positive integers $N$ such that $N\times B$ has at least one divisior $D$ that lies in $N \lt D \le A$?
For two integers $A$ and $B$, how can we find the number of positive integers $N$ such that $N\times B$ has at least one divisior $D$ that lies in $N \lt D \le A$?
For example, if $A = 100$ and $B = ...
5
votes
1answer
163 views
$x^2+y^2=z^2(1+xy)$ prove $z=\min \{x;y;z\}$ (with $x,y,z \in \mathbb{Z^+}$)
$x,y,z \in \mathbb{Z^+}$ such that $x^2+y^2=z^2(1+xy)$. Prove $z=\min \{x;y;z\}$
$$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum $z>y \implies xy < x^2/z^2$, we have $xy \in Z ...
3
votes
2answers
112 views
LCM($1, 2, 3, …, n$)?
I want to find LCM($1, 2, 3, \ldots, n$) where $2 \le n \le 10^8$ .
LCM = Least Common Multiple
I'm trying to find a formula .
Please Help .
1
vote
2answers
65 views
Table clock and wall clock
I have two clocks - table clock and wall clock
A table clock gains 2 mins every 12 hours and a wall clock loses 1 min every 12 hours both are set at 12 noon on tuesday(date is not known ) we need to ...
6
votes
1answer
129 views
$x^4 + y^4 = z^2$
$x, y, z \in \mathbb{N}$,
$\gcd(x, y) = 1$
prove that $x^4 + y^4 = z^2$ has no solutions.
It is true even without $\gcd(x, y) = 1$, but it is easy to see that $\gcd(x, y)$ must be $1$
2
votes
1answer
146 views
Product of three consecutive positive integers is never a perfect power
I am trying to prove that the product of three consecutive positive integers is never a perfect power.Can anyone point to gaps in my proof and/or post an alternate solution?
Let the three positive ...
2
votes
1answer
91 views
The G.C.D of two numbers is $23$. The other two factors of the L.C.M are $13$ and $14$.
The G.C.D of two numbers is $23$. $13$ and $14$ are also factors of the L.C.M [out of some unknown number of factors]. What is the larger number?
The thing I am able to infer is that both numbers ...
0
votes
1answer
51 views
Polynomial that permutes residue classes
Prove that for any integers $d, e > 1$, the polynomial $f$ with integer coefficients permutes the residue classes modulo $p^d$ if and only if it permutes the residue classes modulo $p^e$ where $p$ ...
1
vote
1answer
235 views
Multiplication Table with a frame and picture of equal sum
Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")?
...
2
votes
1answer
106 views
Prove that all divisors of $\frac{p^p-1}{p-1}$ are of the form $pk+1$ where $p$ is prime and $k\in\mathbb{Z}$.
Prove that all divisors of
$$\frac{p^p-1}{p-1}$$
are of the form $pk+1$ where $p$ is prime and $k\in\mathbb{Z}$.
2
votes
2answers
61 views
For what values of the variable x does the following inequality hold:
$\ \frac{4x^2}{\Bigl(1-\sqrt{\ 1\ +2x}\Bigr)^2} <
2x+9$
... IMO-1960
3
votes
1answer
108 views
Find the minimum values of x, y, and z
If:
\begin{aligned}
\ {n} & = \ 2x ^2 \\
\ {n} & = \ 3y^3 \\
\ {n} & = \ 5z^5 \\
\end{aligned}
Find the minimum value for $x, y,$ and $z$.
Note: $n$ is the minimum value you can get ...
3
votes
0answers
242 views
Equation with divisors II
This is a link to my first question about this problem .
Upd$^{*}$: I've followed Matthew Conroy advice and found "amazing" numbers such as $2^6 \cdot p$, $3^6 \cdot p$.
Upd$^{**}$: If $n=p^6 ...
6
votes
1answer
228 views
Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.
The Olympiad-style question I was given was as follows:
A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
12
votes
2answers
480 views
Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
The question is:
Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$
I started off approaching this ...
1
vote
1answer
210 views
Help Understanding Solution? (Putnam and Beyond)
From Putnam and Beyond:
Proofs (Question 10:)
Let $n>1$ be an arbitrary real number and let $k$ be the number of
positive prime numbers less than or equal to $n$. Select $k + 1$ positive
...
6
votes
4answers
526 views
Seemingly invalid step in the proof of $\frac{a^2+b^2}{ab+1}$ is a perfect square?
Recall the famous IMO 1988 question 6:
Suppose that $\displaystyle\frac{a^2+b^2}{ab+1}=k\in\mathbb{N}$ for some $a,b\in\mathbb{N}$. Show that $k$ is a perfect square.
Solutions can be found:
...
4
votes
3answers
168 views
In how many ways we can choose three numbers the set of first $11$ natural numbers $(1,2,\cdots,11)$ so that their sum is a multiple of $3$?
In how many ways we can choose three numbers from first $11$ natural numbers $(1,2,\cdots,11)$ so that their sum is a multiple of $3$?
I tried using "stars and bars", but as this is about selection ...
9
votes
1answer
178 views
solution to $ 7^{a}+1 =3^{b}+5^{c} $ for natural $a$,$b$ and $c$
How do I solve $ 7^{a}+1 =3^{b}+5^{c} $ for natural $a$,$b$ and $c$?All I got after some modular arithmetic is that the $a$,$b$ and $c$ are all odd.The problem was posted on Art of Problem ...
1
vote
0answers
77 views
How many ways to fill the $N \times N$ board by nonnegative integers, such that sum of the numbers of each row and each column is $R$?
How many ways to fill the $4 \times 4$ board by nonnegative integers, such that sum of the numbers of each row and each column is $3$?
I wrote a brute-force and got $2008$ which seems to be the ...
11
votes
2answers
265 views
Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$
Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$.
I tried with expanding $n^{2003}+1$, but I got nothing pretty not useful. I also couldn't get any improvement, let ...
5
votes
3answers
268 views
Find remainder when dividing $9^{{10}^{{11}^{12}}}-5^{9^{10^{11}}}$ by $13$.
Find remainder when dividing
$$9^{{10}^{{11}^{12}}}-5^{9^{10^{11}}} \hspace{1cm} \text{by} \hspace{1.2cm} 13.$$
I tried transforming these who numbers separately to form $13k+n$ but failed.
10
votes
1answer
251 views
Prove that there can't be 985 divisors of $123456…19841985$
Numbers from 1 to 1985 are written one after another so they form a new number, $n=123456\ldots19841985$. Prove that there can't be 985 divisors of $n$.
This should be solved on paper, without ...
13
votes
2answers
854 views
Asking 2011 Putnam B6
I wish to ask today's Putnam problem B6:
Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ number of $\sum^{p-1}_{k=0} k! n^{k}$ is not divisble by $p$.
...
1
vote
2answers
123 views
Is there an easy way to determine when this fractional expression is an integer?
For $x,y\in \mathbb{Z}^+,$ when is the following expression an integer?
$$z=\frac{(1-x)-(1+x)y}{(1+x)+(1-x)y}$$
The associated Diophantine equation is symmetric in $x, y, z$, but I couldn't do ...
0
votes
1answer
80 views
Interscholastic Mathematic League Senior B Division #1
Let n be a positive integer less than 1000. If n^3 has 10 factors, compute the largest value of n.
2
votes
5answers
256 views
Finding the smallest positive integer a
Can we find the smallest positive integer $a$ such that $1971|50^n+a.23^n$ where n is odd?
Source:Problem Solving Strategies by Arthur Engel
0
votes
1answer
254 views
Find number of interesting numbers (China TST 2011)
A positive integer $n$ is known as an interesting number if $n$ satisfies
$$ \left\{\frac{n}{10^k}\right\} > \frac{n}{10^{10}} $$
for all $k=1, 2, \ldots, 9$, where $\{x\}=x - \lfloor x \rfloor$.
...
0
votes
1answer
225 views
How can I find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}})$?
How can one find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $$d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}}),$$ holds for all $k$-tuples $(x_1,x_2,\cdots,x_k)$ of ...
4
votes
1answer
224 views
CMO 2011 Sum of integers
The following problem was asked in the CMO 2011 and I'd be interested in finding various solutions for it. Here's the problem:
Fix a positive integer $d$, then for any integer $k$ there exists a ...
