Found problems: 916
What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]
Solve in integers the system of equations: $$x^2-y^2=z$$
$$3xy+(x-y)z=z^2$$
Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$
Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.
How many integer solutions exist that satisfy this equation?
$$x+4y-343\sqrt{x}-686\sqrt{y}+4\sqrt{xy}+2022=0$$.
Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.
Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$
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]
Determine all positive integer solutions $(x, y, z, t)$ of the equation \[(x+y)(y+z)(z+x)=xyzt\] for which $\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1$.
Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.
Find all solutions in integers of $x^{3}+2y^{3}=4z^{3}$.
What is the smallest perfect square that ends in $9009$?
Solve the equation in integer numbers $$y^3-x^3=91$$
Show that the equation
$$a^2b=2017(a+b)$$
has no solutions for positive integers $a$ and $b$.
[i]Proposed by Oriol Solé[/i]
Let $x, y$ be integers and $p$ be a prime for which
\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.
If $p$ is a prime positive integer and $x,y$ are positive integers,
find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1)
If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).
Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\]
Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.
Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.
$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.