Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.

Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.

Let $n$ be an integer larger than $1$ and let $S$ be the set of $n$-element subsets of the set $\{1,2,\ldots,2n\}$. Determine \[\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )\] where $[x,y]$ is the least common multiple of the integers $x$, $y$.

Compute the sum of all positive integers $n$ for which there exists a real number $x$ satisfying $$\left(x +\frac{n}{x} \right)^n= 2^{20}.$$

Find the largest positive integer $k$ such that we can find a set $A \subseteq \{1,2, \ldots, 100 \}$ with $k$ elements such that, for any $a,b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.

Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $$n_{k+1} = n_k + P(n_k)$$ Can it be unbounded for some $n_1$?

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

How many three-digit positive integers $\underline{a}$ $\underline{b}$ $\underline{c}$ are there whose nonzero digits $a$, $b$, and $c$ satisfy $$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$ (The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$) $\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$

Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors?

Solve the equation \[ x^3+2y^3-4x-5y+z^2=2012, \] in the set of integers.

Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.

A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

Let $p=3\cdot 10^{10}+1$ be a prime and let $p_n$ denote the probability that $p\mid (k^k-1)$ for a random $k$ chosen uniformly from $\{1,2,\cdots,n\}$. Given that $p_n\cdot p$ converges to a value $L$ as $n$ goes to infinity, what is $L$? [i]Proposed by Vijay Srinivasan[/i]

Does there exist a natural number $N$ which is a power of$2$, such that one can permute its decimal digits to obtain a different power of $2$?

How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

How many positive integers $n$ between $1$ and $2024$ (both included) are there such that $\lfloor{\sqrt{n}}\rfloor$ divides $n$? (For $x \in \mathbb{R}, \lfloor{n}\rfloor$ denotes the greatest integer less than or equal to $x$.) $(A) 44$ $(B) 132$ $(C) 1012$ $(D) 2024$

Suppose that $d(n)$ is the number of positive divisors of natural number $n$. Prove that there is a natural number $n$ such that $$ \forall i\in \mathbb{N} , i \le 1402: \frac{d(n)}{d(n \pm i)} >1401 $$ [i]Proposed by Navid Safaei and Mohammadamin Sharifi [/i]

A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

[b]p1.[/b] A permutation on $\{1, 2,..., n\}$ is an ordered arrangement of the numbers. For example, $32154$ is a permutation of $\{1, 2, 3, 4, 5\}$. Does there exist a permutation $a_1a_2... a_n$ of $\{1, 2,..., n\}$ such that $i+a_i$ is a perfect square for every $1 \le i \le n$ when a) $n = 6$ ? b) $n = 13$ ? c) $n = 86$ ? Justify your answers. [b]p2.[/b] Circle $C$ and circle $D$ are tangent at point $P$. Line $L$ is tangent to $C$ at point $Q$ and to $D$ at point $R$ where $Q$ and $R$ are distinct from $P$. Circle $E$ is tangent to $C, D$, and $L$, and lies inside triangle $PQR$. $C$ and $D$ both have radius $8$. Find the radius of $E$, and justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/f/b/4b98367ea64e965369345247fead3456d3d18a.png[/img] [b]p3.[/b] (a) Prove that $\sin 3x = 4 \cos^2 x \sin x - \sin x$ for all real $x$. (b) Prove that $$(4 \cos^2 9^o - 1)(4 \cos^2 27^o - 1)(4 cos^2 81^o - 1)(4 cos^2 243^o - 1)$$ is an integer. [b]p4.[/b] Consider a $3\times 3\times 3$ stack of small cubes making up a large cube (as with the small cubes in a Rubik's cube). An ant crawls on the surface of the large cube to go from one corner of the large cube to the opposite corner. The ant walks only along the edges of the small cubes and covers exactly nine of these edges. How many different paths can the ant take to reach its goal? [b]p5.[/b] Let $m$ and $n$ be positive integers, and consider the rectangular array of points $(i, j)$ with $1 \le i \le m$, $1 \le j \le n$. For what pairs m; n of positive integers does there exist a polygon for which the $mn$ points $(i, j)$ are its vertices, such that each edge is either horizontal or vertical? The figure below depicts such a polygon with $m = 10$, $n = 22$. Thus $10$, $22$ is one such pair. [img]https://cdn.artofproblemsolving.com/attachments/4/5/c76c0fe197a8d1ebef543df8e39114fe9d2078.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For positive integers $n$, let $f_2(n)$ denote the number of divisors of $n$ which are perfect squares, and $f_3(n)$ denotes the number of positive divisors which are perfect cubes. Prove that for each positive integer $k$ there exists a positive integer $n$ for which $\frac{f_2(n)}{f_3(n)}=k$.

Find the minimum $k>2$ for which there are $k$ consecutive integers such that the sum of their squares is a square.

A pair $(m, n)$ of positive integers is called “[i]linked[/i]” if $m$ divides $3n + 1$ and $n$ divides $3m + 1$. If $a, b, c$ are distinct positive integers such that $(a, b)$ and $( b, c)$ are linked pairs, prove that the number $1$ belongs to the set $\{a, b, c\}$

Let $k_0$ be a unit semi-circle with diameter $AB$. Assume that $k_1$ is a circle of radius $r_1=\frac12$ that is tangent to both $k_0$ and $AB$. The circle $k_{n+1}$ of radius $r_{n+1}$ touches $k_n,k_0$, and $AB$. Prove that: (a) For each $n\in\{2,3,\ldots\}$ it holds that $\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4$. (b) $\frac1{r_n}$ is either a square of an even integer, or twice a square of an odd integer.

Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally tangent to $\Gamma_1$ at $X$ and externally tangent to $\Gamma_2$ at $Y$. Line $PQ$ meets $\Gamma_2$ again at $R$, line $XY$ meets $\Gamma_2$ again at $Z$, and lines $PZ$ and $XR$ meet at $M$. As $\omega$ varies, the locus of point $M$ encloses a region of area $\tfrac{p}{q} \pi$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$. [i]Proposed by Michael Kural[/i]