We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

Are there integers $k$ and $m$ for which $$\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\, ?$$

Find all integers $n$ such that $n^4-8n+15$ is product of two consecutive integers

Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$ [i]Proposed by Oleksii Masalitin (Ukraine)[/i]

If there are integers $a,b,c$ such that $a^2+b^2+c^2-ab-bc-ca$ is divisible by a prime $p$ such that $\text{gcd}(p,\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$, then prove that there are integers $x,y,z$ such that $p=x^2+y^2+z^2-xy-yz-zx$.

Find all polynomials with integer coefficients $P$ such that for all positive integers $n$, the sequence $$0, P(0), P(P(0)), \cdots$$ is eventually constant modulo $n$. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$ $$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$ [i]Proposed by Navid Safaei[/i]

Let $p$ be a prime number and $a$ and $n$ positive nonzero integers. Prove that if $2^p + 3^p = a^n$ then $n=1$

Let $ p_i \geq 2$, $ i \equal{} 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let: \[ P \equal{} \{ x \equal{} \sum_{i \equal{} 1}^{n} x_i \prod_{j \equal{} 1, j \neq i}^{n} p_j \mid x_i \text{is a non \minus{} negative integer}, i \equal{} 1,2, \cdots n \} \] Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n \minus{} 2}{2} \cdot \prod_{i \equal{} 1}^{n} p_i$, and also find $ M$.

[b]p1.[/b] We say that six positive integers form a magic triangle if they are arranged in a triangular array as in the figure below in such a way that each number in the top two rows is equal to the sum of its two neighbors in the row directly below it. The triangle shown is magic because $4 = 1 + 3$, $5 = 3 + 2$, and $9 = 4 + 5$. $$9$$ $$4\,\,\,\,5$$ $$1\,\,\,\,3\,\,\,\,2$$ (a) Find a magic triangle such that the numbers at the three corners are $10$, $20$, and $2010$, with $2010$ at the top. (b) Find a magic triangle such that the numbers at the three corners are $20$, $201$, and $2010$, with $2010$ at the top, or prove that no such triangle exists. [b]p2.[/b] (a) The equalities $\frac12+\frac13+\frac16= 1$ and $\frac12+\frac13+\frac17+\frac{1}{42}= 1$ express $1$ as a sum of the reciprocals of three (respectively four) distinct positive integers. Find five positive integers $a < b < c <d < e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1.$$ (b) Prove that for any integer $m \ge 3$, there exist $m$ positive integers $d_1 < d_2 <... < d_m$ such that $$\frac{1}{d_1}+\frac{1}{d_2}+ ... +\frac{1}{d_m}= 1.$$ [b]p3.[/b] Suppose that $P(x) = a_nx^n +... + a_1x + a_0$ is a polynomial of degree n with real coefficients. Say that the real number $b$ is a balance point of $P$ if for every pair of real numbers $a$ and $c$ such that $b$ is the average of $a$ and $c$, we have that $P(b)$ is the average of $P(a)$ and $P(c)$. Assume that $P$ has two distinct balance points. Prove that $n$ is at most $1$, i.e., that $P$ is a linear function. [b]p4.[/b] A roller coaster at an amusement park has a train consisting of $30$ cars, each seating two people next to each other. $60$ math students want to take as many rides as they can, but are told that there are two rules that cannot be broken. First, all $60$ students must ride each time, and second, no two students are ever allowed to sit next to each other more than once. What is the maximal number of roller coaster rides that these students can take? Justify your answer. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that the lengths of all four sides and the two diagonals of $ABCD$ are rational numbers. If the two diagonals $AC$ and $BD$ intersect at a point $M$, prove that the length of $AM$ is also a rational number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n{}$ be a natural number. The pairwise distinct nonzero integers $a_1,a_2,\ldots,a_n$ have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by $a_1a_2\cdots a_n$ for any integer $k{}.$ Find the largest possible value of $a_n.$ [i]Proposed by F. Petrov and K. Sukhov[/i]

find all primes $p$, for which exist natural numbers, such that $p=m^2+n^2$ and $p|(m^3+n^3-4)$.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$

In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers? [i]Proposed by Morteza Saghafian [/i]

For any integer $n > 4$, prove that $2^n > n^2$.

Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.

Let us denote with $S(n)$ the sum of all the digits of $n$. a) Is there such an $n$ that $n+S(n)=1980$? b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$.

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Can we find positive integers $a$ and $b$ such that both $(a^2 + b)$ and $(b^2 + a)$ are perfect squares?

Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.