Found problems: 988
Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$
Find the numerical value of $a + b + c$.
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$
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Find all the integers $x,y$ satisfying equation $x^2+x=y^4+y^3+y^2+y$.
Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$
Solve in integers the following equation:
\[y=2x^2+5xy+3y^2\]
Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$
Solve into $N$: $$a^2 = 2^b +c^4$$
Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation
\[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\]
where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.
Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.
Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$
Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.
Let $a, b, c$, and $d$ be four distinct integers.
Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.
Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.
Determine all integer solutions of the system \[2uv-xy=16,\] \[xv-yu=12.\]
Let $n$ be an odd positive integer. Prove that if the equation $\frac1x+\frac1y=\frac4n$ has a solution in positive integers $x,y$, then $n$ has at least one divisor of the form $4k-1$, $k\in\mathbb N$.
Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.
Determine all triples $(a, b, c)$ of positive integers such that
$$a! +b! = 2^{c!} .$$
Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.
Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.
Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.
Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$.
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