2
votes
2answers
37 views

An arctan problem including a diophantine equation

This is a follow-up question to An equation of the form A + B + C = ABC . I totally messed up with making the equation from the question specification . Actually the question was $$ ...
0
votes
1answer
84 views

An equation of the form A + B + C = ABC

So I was on a SPOJ spree until I came across this question . The question says $$\tan(\frac{1}{A}) = \tan(\frac{1}{B}) + \tan(\frac{1}{C})$$ where we have to find the $\min(B+C)$ for a fix $A$ where ...
-2
votes
3answers
160 views

Equation $a^{n}+b^{n}=2008$ has no integers solutions. [closed]

Prove that the equation $a^{n}+b^{n}=2008$ has no solutions for $a,b,n\in\mathbb{Z}, n\geq2.$
13
votes
4answers
432 views

prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$

show that this equation $$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$ have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe ...
2
votes
0answers
56 views

An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
1
vote
2answers
2k views

Diophantine Equation (2014 AMC 12A)

There are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and $$5x^2 + kx + 12 = 0 $$ has at least one integer solution for $x$. What is $N$? (My idea was to consider the equation ...
3
votes
2answers
279 views

Finding all positive integers $x,y,z$ that satisfy $3^x - 5^y = z^2$

Find all positive integers $x,y,z$ that satisfy: $$3^x - 5^y = z^2.$$ I think that $(x,y,z)= (2,1,2)$ will be the only solution. But how to prove that?
1
vote
1answer
80 views

What kind of methods there are to solve a Diophantine equation from IMO longlist?

Namely, in IMO longlist 1987 were given the equation $3z^2=2x^3+385x^2+256x-58195$ and asked to find its integer points. How can I find those? I tried to substitute $z=12k,x=6t$ to get ...
0
votes
0answers
48 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
0
votes
1answer
56 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
10
votes
3answers
253 views

Math contest proof problem fractions

Could someone help me with this? Let $x, y, z$ be positive integers with greatest common divisor $1$. If $\frac 1 x +\frac 1 y=\frac 1 z$, then show that $\sqrt{x + y}$ is an integer.
7
votes
2answers
133 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
308 views

Find all integer solutions to $x^2+4=y^3$. [duplicate]

Find all integer solutions to $x^2+4=y^3$. Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
7
votes
1answer
190 views

(USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$

Find the integer solutions: $$a·b^5+3=x^3,a^5·b+3=y^3$$ This is the first problem of today's USAJMO (has finished),I only find a trival result that $x\equiv y \pmod6$ and $abxy≠0 \pmod 3$. Thanks in ...
5
votes
2answers
353 views

Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$

Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$. The contest is just ended, so you may freely answer. (I did not attend the contest: it is an Italian contest for schools and ...
19
votes
3answers
962 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.
4
votes
1answer
202 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
296 views

Modification of 5th question from BMO'81

First of all I will introduce original problem (Question 5 from British Mathematical Olympiad). You can find complete list of BMO'81 there BMO'81. Find, with proof, the smallest possible value ...
1
vote
1answer
105 views

Proof- set uniqueness

Moderator Note: This question is from a contest which ended 22 October 2012. Suppose that for $1\leq y\leq x$, and $x\geq 3$, $$\Gamma_{x,y}=\left\{\left\lfloor\frac{2^x-1}{2^{y-1}}n - 2^{x-y} ...
4
votes
1answer
385 views

How to find all rational numbers satisfy this equation?

Find all rational number $a,b,c$ satisfy: $$a+b+c=abc$$ I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
1
vote
2answers
127 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 ...