Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{(2k-1)^2+a_k^2}$, where $n\in\mathbb{N}$, $a_1, a_2, \cdots, a_n\in\mathbb{R}^+$, and $a_1+a_2+\cdots a_n = 17$. If there exists a unique $n$ such that $S_n$ is also an integer, find $n$.
Find the smallest $n$ such that $\frac{1}{n}(1^2 + 2^2 + \cdots + n^2)$ is a square of an integer.
Solve in integers $y^2=x^4 + x^3 + x^2 +x +1$.
Solve in integers $x^3 + (x+1)^3 + \cdots + (x+7)^3 = y ^3$
Solve in positive integers $y^2 = x^2 + x + 1$
Solve in integers $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} = \frac{3}{5}$
Prove there is no integer solutions to $x^2 = y^5 - 4$.
Find all positive integer solutions to: $x^2 + 3y^2 = 1998x$.
Prove that there exist infinite many triples of consecutive integers each of which is a sum of two squares. For example: $8 = 2^2 + 2^2$, $9 = 3^2 + 0^2$, and $10=3^1 + 1^2$
Find all triangles whose sides are consecutive integers and areas are also integers.
Find all positive integers $k$, $m$ such that $k < m$ and
$$1+ 2 +\cdots+ k = (k +1) + (k + 2) +\cdots+ m$$
Prove that there are infinitely many positive integers $n$ such that $(n^2+1)$ divides $n!$.
There are $100$ tigers, $100$ foxes, and $100$ monkeys in the animal kingdom. Tigers always tell the truth; Foxes always tell lies; and monkeys sometimes tell the truth but sometimes not. These $300$ animals are divided into $100$ groups, each of which has exactly two of the same kind and one of another kind. Now comes the Kong Fu Panda. He asks every animal: "is there a tiger in your group?" and receives $138$ "yes" answers. Then he asks everyone: "is there a fox in your group?" and receives $188$ positive answer this time. Find the number of monkeys who tell the truth both time.
Find all 3-digit integer $\overline{abc}$ that satisfy $\overline{abc} = (a + b + c)^3$.
Find all nonnegative integers $x$ and $y$ such that $x^3+y^3 = (x+y)^2$.
Find the number of paris $(a, b)$ of nonnegative integers that satisfy $6a+7b=1000$
Prove that the sum of $n$ consecutive perfect squares cannot be a perfect square for $n=3, 4, 5,$ and $6$.
Find all nonnegative integers $n$ such that there are integers $a$ and $b$ with the property:
$$n^2 = a + b \qquad\text{and}\qquad n^3 = a^2 + b^2$$
Find all pairs of positive integers $(n;m)$ satisfying $3n^2 + 3n + 7 = m^3$.
- Solve the following diophantine equation in natural numbers: $$y^2 = 1 + x + x^2 + x^3 + x^4$$
Find all pairs of positive integers $(a; b)$ such that $\frac{a}{b} + \frac{21b}{25a}$ is a positive integer.
Let $x; y; z$ be positive integers such that $(x; y; z) = 1$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. Prove that $x + y; x-z$ and $y-z$ are perfect squares.
Find all non-negative solutions to: $43^n-2^x3^y7^z = 1$.
Find all non-negative integers $n$ such that $2^{200}+2^{192}\cdot 15+2^n$ is a perfect square
Find all integers $a$, $b$, $c$ with $1 < a < b < c$ such that the number $(a-1)(b-1)(c-1)$ is a divisor of $(abc-1)$.