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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 $$... 1answer 84 views ### An equation of the form A + B + C = ABC So I was on a SPOJ spree until I came across this question . The question says$$\tan(\frac{1}{A}) = \tan(\frac{1}{B}) + \tan(\frac{1}{C})$$where we have to find the \min(B+C) for a fix A where ... 3answers 160 views ### Equation a^{n}+b^{n}=2008 has no integers solutions. [closed] Prove that the equation a^{n}+b^{n}=2008 has no solutions for a,b,n\in\mathbb{Z}, n\geq2. 4answers 432 views ### prove Diophantine equation has no solution \prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i) show that this equation$$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe ... 0answers 56 views ### An algorithm for solving linear diophantine equations? I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ... 2answers 2k views ### Diophantine Equation (2014 AMC 12A) There are exactly N distinct rational numbers k such that |k| < 200 and$$5x^2 + kx + 12 = 0 $$has at least one integer solution for x. What is N? (My idea was to consider the equation ... 2answers 279 views ### Finding all positive integers x,y,z that satisfy 3^x - 5^y = z^2 Find all positive integers x,y,z that satisfy:$$3^x - 5^y = z^2.$$I think that (x,y,z)= (2,1,2) will be the only solution. But how to prove that? 1answer 80 views ### What kind of methods there are to solve a Diophantine equation from IMO longlist? Namely, in IMO longlist 1987 were given the equation 3z^2=2x^3+385x^2+256x-58195 and asked to find its integer points. How can I find those? I tried to substitute z=12k,x=6t to get ... 0answers 48 views ### Find all rational solutions to x^3 - y^2 = 2. [duplicate] Find all rational solutions to x^3 - y^2 = 2. The only integers solutions are (3,\pm5): http://mathforum.org/library/drmath/view/51569.html 1answer 56 views ### Find all (a,b,c)\in\mathbb{Z}^3 such that b^2-4ac=-20, and -|a|<b\le|a|<|c|, or 0\le b\le|a|=|c|. Find all (a,b,c)\in\mathbb{Z}^3 such that b^2-4ac=-20, and either of the following is true: -|a|<b\le|a|<|c|, or 0\le b\le|a|=|c|. We see that if (a,b,c) is a solution, then so is ... 3answers 253 views ### Math contest proof problem fractions Could someone help me with this? Let x, y, z be positive integers with greatest common divisor 1. If \frac 1 x +\frac 1 y=\frac 1 z, then show that \sqrt{x + y} is an integer. 2answers 133 views ### For which integers x, y is 2^x + 3^y a square of a rational number? For which integers x, y is 2^x + 3^y a square of a rational number? (Of course (x,y)=(0,1),(3,0) work) 1answer 308 views ### Find all integer solutions to x^2+4=y^3. [duplicate] Find all integer solutions to x^2+4=y^3. Some obvious solutions are (x,y)=(\pm2,2). Are these the only ones? 1answer 190 views ### (USAJMO)Find the integer solutions:ab^5+3=x^3,a^5b+3=y^3 Find the integer solutions:$$a·b^5+3=x^3,a^5·b+3=y^3$$This is the first problem of today's USAJMO (has finished),I only find a trival result that x\equiv y \pmod6 and abxy≠0 \pmod 3. Thanks in ... 2answers 353 views ### Find all positive integers a, b, c such that 1/a + 1/b + 1/c = 4/3999 Find all positive integers a, b, c such that 1/a + 1/b + 1/c = 4/3999. The contest is just ended, so you may freely answer. (I did not attend the contest: it is an Italian contest for schools and ... 3answers 962 views ### Finding all integer solutions of 5^x+7^y=2^z Find all integers x,y,z such that 5^x+7^y=2^z. This one comes from an online contest that I arranged some years ago, and I can assure that a completely elementary solution exists. 1answer 202 views ### x^2+y^2=z^2(1+xy) prove z=\min \{x;y;z\} (with x,y,z \in \mathbb{Z^+}) x,y,z \in \mathbb{Z^+} such that x^2+y^2=z^2(1+xy). Prove z=\min \{x;y;z\}$$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum z>y \implies xy < x^2/z^2, we have xy \in Z ... 2answers 296 views ### Modification of 5th question from BMO'81 First of all I will introduce original problem (Question 5 from British Mathematical Olympiad). You can find complete list of BMO'81 there BMO'81. Find, with proof, the smallest possible value ... 1answer 105 views ### Proof- set uniqueness Moderator Note: This question is from a contest which ended 22 October 2012. Suppose that for 1\leq y\leq x, and x\geq 3,$$\Gamma_{x,y}=\left\{\left\lfloor\frac{2^x-1}{2^{y-1}}n - 2^{x-y} ...
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...
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 ...