Found problems: 988
For what pairs of integers $(x,y)$ does the inequality $x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}$ hold?
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.
Find all positive integers $a,b,c$ and prime $p$ satisfying that
\[ 2^a p^b=(p+2)^c+1.\]
Given a prime number $n \ge 5$. Prove that for any natural number $a \le \frac{n}{2} $, we can search for natural number $b \le \frac{n}{2}$ so the number of non-negative integer solutions $(x, y)$ of the equation $ax+by=n$ to be odd*.
Clarification:
* For example when $n = 7, a = 3$, we can choose$ b = 1$ so that there number of solutions og $3x + y = 7$ to be $3$ (odd), namely: $(0, 7), (1, 4), (2, 1)$
Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$
We say that a triple of integers $(x, y, z)$ is of [i]jenifer [/i] type if $x, y$, and $z$ are positive integers, with $y \ge 2$, and $$x^2 - 3y^2 = z^2 - 3.$$
a) Find a triple $(x, y, z)$ of the jenifer type with $x = 5$ and $x = 7$.
b) Show that for every $x \ge 5$ and odd there are at least two distinct triples $(x, y_1, z_1)$ and $(x, y_2, z_2)$ of jenifer type.
c) Find some triple $(x, y, z)$ of jenifer type with $x$ even.
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$
For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$.
Alexandr Yuran
Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .
Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]
Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$.
(1) $ $ $x\mid y^2-3, y\mid x^2-2$
(2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $
Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.
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.