Problems

Problem 1

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

Problem 2

For a given integer $$d$$, solve $$x^2-dy^2=1$$ in the set of rational numbers.

Problem 3

Let $$(x,y)=(a,b)$$, $$a,b\in\mathbb{N}$$ be the smallest integer solution of $$x^2-dy^2=1$$. Consider the sequence defined by $$y_0=0$$, $$y_1=b$$, $$y_{n+1}=2ay_n-y_{n-1}$$ for $$n\geq1$$. Show that $$ay_n^2+1$$ is a square for each $$n$$. Show that if $$ay^2+1$$ is a square for some $$y\in\mathbb{N}$$, then $$y=y_n$$ for some $$n$$.

Problem 4

Prove that $$5x^2+4$$ or $$5x^2-4$$ is a perfect square if and only if $$x$$ is a term in the Fibonacci sequence.

Problem 5

Find all $$n\in\mathbb{N}$$ such that $$\displaystyle\binom n{k-1}= 2\binom nk+\binom n{k+1}$$ for some natural number $$k< n$$.

Problem 6

Let $$a\in\mathbb{N}$$ and $$d=a^2-1$$. If $$x,y$$ are integers and the absolute value of $$m=x^2-dy^2$$ is less than $$2a+2$$, prove that $$|m|$$ is a perfect square.

Problem 7

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.

Problem 8

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

Problem 9

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

Problem 10

Prove that the equation $$x^2-dy^2=-1$$ is solvable in integers if and only if so is $$x^2-dy^2=-4$$.

Problem 11

Let $$p$$ be a prime. Prove that the equation $$x^2-py^2=-1$$ has integral solutions if and only if $$p=2$$ or $$p\equiv 1$$ (mod 4).

Problem 12

If $$p$$ is a prime of the form $$4k+3$$, show that exactly one of the equations $$x^2-py^2=\pm2$$ has an integral solution.

Problem 13

Prove that $$3^n-2$$ is a square only for $$n=1$$ and $$n=3$$.

Problem 14

Prove that if $$\displaystyle \frac{x^2+1}{y^2}+4$$ is a perfect square, then this square equals 9.

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