Practice With Solutions

2300

Solve in integers the equation $x^2 + y^2 - 1 = 4xy$

2301

Solve in $\textit{rational}$ numbers the equation $x^2 - dy^2 = 1$ where $d$ is an integer.

2302

Let $x$ be a positive real number, and $\lfloor{x}\rfloor$ be the largest integer that not exceeding $x$. Prove that there exist infinity number of positive integers, $n$, such that $\lfloor{\sqrt{2}}\ n\rfloor$ is a perfect square.

2303

Show that there are infinitely many integers $n$ such that $2n + 1$ and $3n + 1$ are perfect squares, and that such $n$ must be multiples of $40$.

2304

Show that the equation $x^2 + y^3 = z^4$ has infinitely many integer solutions.

2306

Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$

for some natural number $k < n$.

2307

Prove that if $m=2+2\sqrt{28n^2 +1}$ is an integer for some $n\in\mathbb{N}$, then $m$ is a perfect square.

2308

Prove that if the difference of two consecutive cubes is $n^2$, $n\in\mathbb{N}$, then $(2n-1)$ is a square.

2309

If $n$ is an integer such that the values of $(3n+1)$ and $(4n+1)$ are both squares, prove that $n$ is a multiple of $56$.

2310

Let $p$ be a prime. Prove that the equation $x^2-py^2 = -1$ has integral solution if and only if $p=2$ or $p\equiv 1\pmod{4}$.

2312

If $p$ is a prime of the form $4k+3$, show that exactly one of the equations $x^2-py^2=\pm 2$ has an integral solution.

2315

Solve in non-negative integers the equation $$x^3 + 2y^3 = 4z^3$$

2316

Solve in nonnegative integers the equation $$2^x -1 = xy$$

2317

Solve in integers the equation $x^2+y^2+z^2-2xyz=0$

2318

For any given positive integer $n > 2$, show that there exists a right triangle with all sides' lengths are integers and one side's length equals $n$.

2321

Show that $x^4 + y^4 = z^2$ is not solvable in positive integers.

2322

Find all primes $p$ for which there exist positive integers $x$, $y$, and $n$ such that $$p^n = x^3+y^3$$

2323

Let $a_1, a_2, \cdots, a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \cdots = a_{2n+1}$.

2325

Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.

2331

Show that there exists an infinite sequence of positive integers $a_1, a_2, \cdots$ such that $$S_n=a_1^2 + a_2^2 + \cdots + a_n^2$$ is square for any positive integer $n$.

2334

Let $a$, $b$, $c$, $d$, and $e$ be five positive integers. If $ab+c=3115$, $c^2+d^2=e^2$, both $a$ and $c$ are prime numbers, $b$ is even and has $11$ divisors. Find these five numbers

2336

Solve in positive integers the equation $$m^2 - n^2 - 3n = 5$$

2337

Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.

2338

Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]

2343

Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied: $ (1)$ $ n$ is not a perfect square; $ (2)$ $ a^{3}$ divides $ n^{2}$.

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