Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.

Find the smallest natural number $n$ which has the following properties: a) Its decimal representation has a 6 as the last digit. b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

[u]Set 7[/u] [b]p19.[/b] The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square. [b]p20.[/b] Tetrahedron $ABCD$ has equilateral triangle base $ABC$ and apex $D$ such that the altitude from $D$ to $ABC$ intersects the midpoint of $\overline{BC}$. Let $M$ be the midpoint of $\overline{AC}$. If the measure of $\angle DBA$ is $67^o$, find the measure of $\angle MDC$ in degrees. [b]p21.[/b] Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square. PS. You should use hide for answers.

Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$? [i]Proposed by Jesse Zhang[/i]

Let $M$ is a four digit positive interger. Write $M$ backwards and get a new number $N$.(e.g $M=1234$ then $N=4321$) Let $C$ is the sum of every digit of $M$. If $M,N,C$ satisfies (i)$d=\gcd(M-C,N-C)$ and $d<10$ (ii)$\dfrac{M-C}{d}=\lfloor\dfrac{N}{2}+1\rfloor$ (1)Find $d$. (2)If there are "m(s)" $M$ satisfies (i) and (ii), and the largest $M$=$M_{max}$. Find $(m,M_{max})$

Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

For each positive integer $n$ we consider the sequence of $2004$ integers$$\left [n+\sqrt{n}\right ],\left [n+1+\sqrt{n+1}\right ],\left [n+2+\sqrt{n+2}\right ],\ldots ,\left [n+2003+\sqrt{n+2003}\right ]$$Determine the smallest integer $n$ such that the $2004$ numbers in the sequence are $2004$ consecutive integers. Clarification: The brackets indicate the integer part.

Let $$P(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n$$ where $a_0,\ldots,a_n$ are integers. Show that if $P$ takes the value $2020$ for four distinct integral values of $x$, then $P$ cannot take the value $2001$ for any integral value of $x$. [i]Proposed by Ángel Plaza[/i]

Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.

Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.

Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$.

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Let $p \ge 2$ be a prime number and $\frac{a_p}{b_p}= 1 +\frac12+ .. +\frac{1}{p^2 -1}$, where $a_p$ and $b_p$ are two relatively prime positive integers. Compute gcd $(p, b_p)$.

For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers [i] good [/i] if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good. ([i]In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.[/i]) [i] Lajos Hajdu and Andras Sarkozy, Hungary [/i]

Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1, 2, . . . , ab$, putting the numbers $1, 2, . . . , b$ in the first row, $b + 1, b + 2, . . . , 2b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $ij$ in the cell in row $i$ and column $j$. (Examples are shown for a $3 \times 4$ table below.) [img]https://cdn.artofproblemsolving.com/attachments/6/8/a0855d790069ecd2cd709fbc5e70f21f1fa423.png[/img] Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is $1200$. Compute $a + b$.

Find all prime numbers $ p$ and $ q$ such that $ p$ divides $ q\plus{}6$ and $ q$ divides $ p\plus{}6$.

Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying \[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]

Let $p$ be a prime, $n$ be a positive integer, and let $\mathbb{Z}_{p^n}$ denote the set of congruence classes modulo $p^n.$ Determine the number of functions $f: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n}$ satisfying the condition \[ f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n} \] for all $a,b \in \mathbb{Z}_{p^n}.$

Let $a_0, a_1,...$ be a sequence of non-negative integers and $b_0, b_1,... $ be a sequence of non-negative integers defined by the following rule: $b_i=gcd(a_i, a_{i+1})$ for every $i=>0$ Is it possible every positive integer to occur exactly once in the sequence $b_0, b_1,... $