Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

Let $S$ be a set of nonnegative integers such that [list] [*] there exist two elements $a$ and $b$ in $S$ such that $a,b>1$ and $\gcd(a,b)=1$; and [*] for any (not necessarily distinct) element $x$ and nonzero element $y$ in $S$, both $xy$ and the remainder when $x$ is divided by $y$ are in $S$. [/list] Prove that $S$ contains every nonnegative integer. [i]Jacob Paltrowitz[/i]

Let p be a prime and $a_1 , a_2 , ..., a_k$ pairwise incongruent modulo p . Prove that $[\sqrt {k-1}]$ of the elements can be selected from $a_i$'s such that adding any numbers different from the selected ones will never give a number divisible by p .

If $d$ is a positive integer such that $d \mid 5+2022^{2022}$, show that $d=2x^2+2xy+3y^2$ for some $x, y \in \mathbb{Z}$ iff $d \equiv 3,7 \pmod {20}$.

Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.

Determine all triples of positive integers $(x, y, z)$ that satisfy $$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.$$

A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

Let $f(n) = n + \lfloor \sqrt{n} \rfloor$. Prove that for every positive integer $m$, the integer sequence $m, f(m), f(f(m)), \dots$ contains at least one square of an integer.

(a) $n$ is a natural number. $d_1,\dots,d_n,r_1,\dots ,r_n$ are natural numbers such that for each $i,j$ that $1\leq i < j \leq n$ we have $(d_i,d_j)=1$ and $d_i\geq 2$. Prove that there exist an $x$ such that (i) $1 \leq x \leq 3^n$ (ii)For each $1 \leq i \leq n$ \[x \overset{d_i}{\not{\equiv}} r_i\] (b) For each $\epsilon >0$ prove that there exists natural $N$ such that for each $n>N$ and each $d_1,\dots,d_n,r_1,\dots ,r_n$ satisfying the conditions above there exists an $x$ satisfying (ii) such that $1\leq x \leq (2+\epsilon )^n$. Time allowed for this exam was 75 minutes.

Does there exist an infinite sequence $a_1,a_2,\dotsc$ of real numbers which is bounded, not periodic, and satisfies the recursion $a_{n+1}=a_na_{n-1}+1$?

Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions: $\bullet$ $r$ is a positive integer with exactly $8$ positive divisors. $\bullet$ $p$ and $q$ are prime numbers.

Find all pairs $(m, n)$ of positive integers satsifying $m^6+5n^2=m+n^3$.

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

Find all positive integers $n\geqslant 2$ such that $d_{i+1}/d_i$ is an integer for all $1\leqslant i < k$, where $1=d_1<d_2<\dots <d_k=n$ are all the positive divisors of $n$. [i]Proposed by Pavel Ciurea[/i]

Determine the smallest positive integer $\ N $ such that there exists 6 distinct integers $\ a_1, a_2, a_3, a_4, a_5, a_6 > 0 $ satisfying: (i) $\ N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 $ (ii) $\ N - a_i$ is a perfect square for $\ i = 1,2,3,4,5,6 $.

Find a sequence of positive integers $f(n)$ ($n \in \mathbb{N}$) such that: (i) $f(n) \leq n^8$ for any $n \geq 2$; (ii) for any distinct $a_1, \cdots, a_k, n$, $f(n) \neq f(a_1) + \cdots+ f(a_k)$.

[b]p1.[/b] Find the smallest whole number $n \ge 2$ such that the product $(2^2 - 1)(3^2 - 1) ... (n^2 - 1)$ is the square of a whole number. [b]p2.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p3.[/b] Three cars are racing: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p4.[/b] There are $11$ big boxes. Each one is either empty or contains $8$ medium-sized boxes inside. Each medium box is either empty or contains $8$ small boxes inside. All small boxes are empty. Among all the boxes, there are a total of $102$ empty boxes. How many boxes are there altogether? [b]p5.[/b] Ann, Mary, Pete, and finally Vlad eat ice cream from a tub, in order, one after another. Each eats at a constant rate, each at his or her own rate. Each eats for exactly the period of time that it would take the three remaining people, eating together, to consume half of the tub. After Vlad eats his portion there is no more ice cream in the tube. How many times faster would it take them to consume the tub if they all ate together? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.

Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$

A sequence of numbers $x_{1},x_{2},x_{3},\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50}=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer

Prove that we can give a color to each of the numbers $1,2,3,...,2013$ with seven distinct colors (all colors are necessarily used) such that for any distinct numbers $a,b,c$ of the same color, then $2014\nmid abc$ and the remainder when $abc$ is divided by $2014$ is of the same color as $a,b,c$.

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.

Assume that $a$ and $b$ are natural numbers with $a \ge b$ so that $ \sqrt{a+\sqrt{a^2-b^2}}$ is a natural number. Show that $a$ and $b$ have the same parity.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.