Found problems: 988
Show that the equation $y^{2}=x^{3}+2a^{3}-3b^2$ has no solution in integers if $ab \neq 0$, $a \not\equiv 1 \; \pmod{3}$, $3$ does not divide $b$, $a$ is odd if $b$ is even, and $p=t^2 +27u^2$ has a solution in integers $t,u$ if $p \vert a$ and $p \equiv 1 \; \pmod{3}$.
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation
$$ax+by+cz=n.$$
Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$,
$$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$
Find all pairs of integers $(m, n)$ such that $$m+n = 3(mn+10).$$
Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $
($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)
Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?
One morning each member of Angela's family drank an $ 8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 7$
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy
\[(ab + 1)(bc + 1)(ca + 1) = 84.\]
Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.
Find all positive integers $x, y, z$ such that $2^x + 21^y = z^2$
Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$
Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$.
Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying
$$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$
Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.
Determine all three integers $(x, y, z)$ that are solutions of the system
$$x + y -z = 6$$
$$x^3 + y^3 -z^3 = 414$$
Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satis ed by at least two pairs of integers $(x, y)$ with $1 < x \le y$.
Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$
where $d$ is the smallest divisor of $n$ which is greater than $1$.
Find all positive integers $n$ for which the equation \[a+b+c+d=n \sqrt{abcd}\] has a solution in positive integers.
Prove that, if $a$ and $b$ are given integers, the system of equatìons
$$x + y + 2z + 2t = a$$
$$2x - 2y + z- t = b$$
has a solution in integers $x, y,z,t$.
Consider the equation
\[(3x^3+xy^2)(x^2y+3y^3)=(x-y)^7\]
(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.
(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.
Let $x,y$ be the positive integers such that $3x^2 +x = 4y^2 + y$.
Prove that $x - y$ is a perfect (square).
Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books?