Found problems: 288
Find all triplets of integers $(x,y,z)$ such that $$xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)=1$$
Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.
Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$.
Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.
Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$.
a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$.
b) Find the sum of the other components of all such pairs of numbers.
Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.
Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.
Brackets are to be inserted into the expression $10 \div 9 \div 8 \div 7 \div 6 \div 5 \div 4 \div 3 \div 2$ so that the resulting number is an integer.
(a) Determine the maximum value of this integer.
(b) Determine the minimum value of this integer.
The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.
[b][u]BdMO National Higher Secondary Problem 3[/u][/b]
Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$
Is $N$ finite or infinite?If $N$ is finite,what is its value?
Find all integers $n$ such that $\frac{n+4}{3n-2}$ is integer
The quadratic expression $ax^2+bx+c$ is the $4$-th power (of an integer) for any integer $x$. Prove that $a = b = 0$.
Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.
(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that
$$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$
(b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)
(a) Positive rational numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution?
(b) Positive real numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution?
[i](Andrew Brahms, USA)[/i]
If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer
Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.
For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute
$$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
Let $x_1,x_2,x_3,...,x_{102}$ be natural numbers such that $x_1<x_2<x_3<...<x_{102}<255$.
Prove that among the numbers $d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}$ there are at least $26$ equal.
Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$
Find all $(m, n)$ positive integer pairs such that both $\frac{3n^2}{m}$ and $\sqrt{n^2+m}$ are integers.
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.
The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?
Find all functions $f$ mapping non-empty finite sets of integers, to integers, such that
$$f(A+B)=f(A)+f(B)$$
for all non-empty sets of integers $A$ and $B$.
$A+B$ is defined as $\{a+b: a \in A, b \in B\}$.
How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2=m^2n^2$?
$\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$