There exist $5$ consecutive positive integers such that their sum is a square, and the sum of the middle three is a cube. What is the smallest one of these five numbers?
Let integers $a$, $b$ and $c$ satisfy $a + b + c = 0$, show that $\vert{a^3 + b^3 + c^3}\vert$ cannot be a prime number.
Solve the following equation in positive integers: $$x^2 +3x^2y^2 = 30y^2 + 517$$
Solve in positive integers the equation $x^3 - y^3 = xy + 61$.
Solve the equation in integers $(x^2-y^2)^2 = 1+16y$.
Show the equation $x^2 + y^2-8z^3 = 6$ has no integer solution.
Find all the integer solutions to the equation $xy - 10(x+ y)= 1$.
Solve in integers the equation $x^2 - xy +2x -3 y = 0$
Solve the equation in integers $x^2 +4xy + 5y^2 + 2x + 4y -7 =0$
Solve the equation in integers $x^2 - 2xy -3y^2 +3x-5y-6=0$
Show that the equation $x^4 + y^4 + z^4 = 2x^2y^2 + 2y^2 z^2 + 2z^2x^2 +24$ has no integer solution.
Let $n$ be a positive integer. If the equation $x + 2y + 2z = n$ has exactly $28$ positive integer solutions, find the value of $n$.
Let $x$, $y$, and $z$ be three positive integers, If
- $7x^2 - 3y^2 + 4z^2 = 8$
- $16x^2 - 7y^2 + 9z^2=-3$
Find the value of $x^2 + y^2 + z^2$
Solve the equation in integers $(x+y)^x = y^x + 1413$
Let $n$ be a positive integer, show that $11^{n+2} + 12^{2n+1}$ is a multiple of 133.
How many positive divisors does $20$ have?
How many integer solutions does the equation $(x+1)(y+1)=25$ have?
Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price p. $A$ day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has 12 plushies, Bob has 40, and Charlie has 52 but they all spent the same amount of money: \$42. How many plushies did Alice buy on the first day?
Real numbers $x, y, z$ satisfy the following equality: $$4(x + y + z) = x^2 + y^2 + z^2$$
Let $M$ be the maximum of $xy + yz + zx$, and let $m$ be the minimum of $xy + yz + zx$. Find $M + 10m$.
$x, y, z$ are positive real numbers that satisfy $x^3+2y^3+6z^3 = 1$. Let $k$ be the maximum possible value of $2x + y + 3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n + n$.
Find $\textit{any}$ quadruple of positive integers $(a, b, c, d)$ satisfying $a^3+b^4+c^5=d^{11}$ and $abc<10^5$.
How many positive integers, not exceeding $2019$, are relatively prime to $2019$?
Let $p$ be a prime number, computer $\varphi(p)$.
Let $p$ be a prime number and $n$ be a positive integer. Show that $\varphi(p^n)=p^n - p^{n-1}$ where $\varphi(n)$ is the Euler's totient function.
Show that if $a$ and $b$ are relatively prime, then $\varphi(a)\varphi(b)=\varphi(ab)$ where $\varphi(n)$ is Euler's totient function.