Found problems: 988
If three integers $p, q$ and $r$ apply that $$p + q^2 = r ^2.$$Show that $6$ adds up to $pqr$ .
Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied.
(Walther Janous)
Suppose that $p$ is an odd prime such that $2p+1$ is also prime. Show that the equation $x^{p}+2y^{p}+5z^{p}=0$ has no solutions in integers other than $(0,0,0)$.
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
Determine all triplets of real numbers $(x, y, z)$ that satisfy the equation $4xyz = x^4 + y^4 + z^4 + 1$.
Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$
Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that
$$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$
Find all couples $(x,y)$ over the positive integers such that:
$7^x+x^4+47=y^2$
Find all integer pair $(m,n)$ such that $7^m=5^n+24$.
$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$
Find the smallest natural number $n$ for which there exists a natural number $x$ such that
$$(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.$$
Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$.
(a) Show that there are infinitely many good numbers.
(b) Show that if $n$ is a good number, then $7 \nmid n$.
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$
holds.
[i](Walther Janous)[/i]
Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.
Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.
Given a prime number $p$, find all integer solutions of $p(x+y) = xy$.
For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that
(i) $a+b+c+d=n$
(ii) $a \ge b \ge d$
(iii) $a \ge c \ge d$
Prove that for all $n$, $p(n) = q(n)$.
Find all triples $(x,y,z)$ of positive integers such that
$$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$
Prove that the equation $ (x\plus{}y)^n\equal{}x^m\plus{}y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.
For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$