Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \] where $m\geq n\geq p \geq 1$ are integer numbers. [i]Ioan Bogdan[/i]

Find, with proof, all pairs $(x, y)$ of integers satisfying the equation $3x^2+ 4 = 2y^3$.

The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$.

Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

Natural numbers $a$ and $b$ are relatively prime. Prove that the greatest common divisor of $(a+b)$ and $(a^2+b^2)$ is either $1$ or $2$.

[b]rolling cube[/b] $a$,$b$ and $c$ are natural numbers. we have a $(2a+1)\times (2b+1)\times (2c+1)$ cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.) prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane. time allowed for this question was 1 hour.

For natural $n$, define $f_n=[2^n\sqrt{69}]+[2^n\sqrt{96}]$ Prove that there are infinite even integers and infinite odd integers that appear in number $f_1,f_2,\dots$. [i](tatari/nightmare)[/i]

Let $ p_1,p_2,\ldots,p_k $ be the distinct prime divisors of $ n $ and let $ a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k} $ for $ n\geq 2 $. Show that for every positive integer $ N\geq 2 $ the following inequality holds: $ \sum_{k=2}^{N} a_2a_3 \cdots a_k <1 $ [i]Laurentiu Panaitopol[/i]

Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer. a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice. b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?

Let $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-[i]complete [/i] if all elements of $S$ are divisors of $n$, and if $d_1$ and $d_2$ are any elements of $S$, then $n| d_1$ and gcd $(d_1, d_2)$ are in $S$. How many $2310$-complete sets are there?

$[x,y]-[x,z]=y-z$ and $x \neq y \neq z \neq x$ Prove, that $x|y,x|z$

The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following: [list](i) $1\le a_1<a_2<\cdots < a_n\le 50$ (ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that \[mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n. \] [/list]Prove that $n\le 16$ and determine the number of $n$-tuples $(a_1,a_2,\ldots,a_n$) satisfying these conditions for $n=16$.

The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions: [list] [*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$ [*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$ [/list] Prove that $a_n=n$ for all positive integers $n$. [i]Proposed by José Alejandro Reyes González[/i]

The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Prove that for any fixed integer $a$ equation $$(m!+a)^2=n!+a^2$$ has finitely many solutions in positive integers $m,n$

Let $ a_n\equal{}1\plus{}\dfrac1{2^2}\plus{}\dfrac1{3^2}\plus{}\cdots\plus{}\dfrac1{n^2}$ Find $ \lim_{n\to\infty}a_n$

How many integers $1\le x \le 2021$ make the value of the expression $$\frac{2x^3 - 6x^2 - 3x -20}{5(x - 4)}$$ an integer?

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

Prove that if in the product of four consequtive natural numbers we add $1$, we get a perfect square.

A set of distinct positive integers is called [i]singular [/i] if, for each of its elements, after crossing out that element, the remaining ones can be grouped into two sets with no common elements such that the sum of the elements in the two groups is the same. Find the smallest positive integer $n>1$ such that there exists a singular set $A$ with $n$ items.

Find the sum of all prime numbers $p$ such that $p$ divides $$(p^2+p+20)^{p^2+p+2}+4(p^2+p+22)^{p^2-p+4}.$$

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.

[b]p1.[/b] A canoeist is paddling upstream in a river when she passes a log floating downstream,, She continues upstream for awhile, paddling at a constant rate. She then turns around and goes downstream and paddles twice as fast. She catches up to the same log two hours after she passed it. How long did she paddle upstream? [b]p2.[/b] Let $g(x) =1-\frac{1}{x}$ and define $g_1(x) = g(x)$ and $g_{n+1}(x) = g(g_n(x))$ for $n = 1,2,3, ...$. Evaluate $g_3(3)$ and $g_{1982}(l982)$. [b]p3.[/b] Let $Q$ denote quadrilateral $ABCD$ where diagonals $AC$ and $BD$ intersect. If each diagonal bisects the area of $Q$ prove that $Q$ must be a parallelogram. [b]p4.[/b] Given that: $a_1, a_2, ..., a_7$ and $b_1, b_2, ..., b_7$ are two arrangements of the same seven integers, prove that the product $(a_1-b_1)(a_2-b_2)...(a_7-b_7)$ is always even. [b]p5.[/b] In analyzing the pecking order in a finite flock of chickens we observe that for any two chickens exactly one pecks the other. We decide to call chicken $K$ a king provided that for any other chicken $X, K$ necks $X$ or $K$ pecks a third chicken $Y$ who in turn pecks $X$. Prove that every such flock of chickens has at least one king. Must the king be unique? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].