For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

Determine all the pairs of positive integers $(a,b),$ such that $$14\varphi^2(a)-\varphi(ab)+22\varphi^2(b)=a^2+b^2,$$ where $\varphi(n)$ is Euler's totient function.

Let $a_1,a_2,...$ be an arithmetic sequence with the common difference between terms is positive. Assume there are $k$ terms of this sequence creates an geometric sequence with common ratio $d$. Prove that $n\ge 2^{k-1}$.

Given \(n, k \in \mathbb{N}\), prove that \((n-1)^2\) divides \(n^k - 1\) if and only if \(n-1 \mid k\).

We consider the proposition $ p(n)$: $ n^2\plus{}1$ divides $ n!$, for positive integers $ n$. Prove that there are infinite values of $ n$ for which $ p(n)$ is true, and infinite values of $ n$ for which $ p(n)$ is false.

For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.

Show that there is no positive integer $n$ such that $n + k^2$ is a perfect square for at least $n$ positive integer values of $k$.

$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that $\min{\{x,y,z\}} \le ab+bc+ca$.

Determine all pairs of positive integers $m, n,$ satisfying the equality $(2^{m}+1;2^n+1)=2^{(m;n)}+1$ , where $(a;b)$ is the greatest common divisor

$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?

Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.

A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

Find all ordered triples of positive integers $(a,b,c)$ where $$\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)=c+\frac{1}{c}.$$ [i]Proposed by vsamc[/i]

Find all pairs $(p,q)$ of prime numbers such that \[m^{3pq} \equiv m \pmod{3pq} \qquad \forall m \in \mathbb Z.\]

A positive integer $N$ is said to be $n-$ good if (i) $N$ has at least $n$ distinct prime divisors, and (ii) there exist distinct positive divisors $1, x_{2}, . . . , x_{n}$ whose sum is $N$ . Show that there exists an $n-$ good number for each $n\geq 6$.

Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.

Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

Joaquín and his brother Andrés go to class every day on the $62$ bus. Joaquín always pays for the tickets. Each ticket has a $5$-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely $62$. Andrés asks him if the sum of the digits of any of the tickets is $35$ and, knowing the answer, he can directly say the number of each ticket. What were those numbers?

Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$. [i](Walther Janous)[/i]

Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$. (BR Frenkin)

For positive integers $x,y$, find all pairs $(x,y)$ such that $x^2y + x$ is a multiple of $xy^2 + 7$.

Let $\pi$ be a permutation of the numbers from $1$ through $2012$. What is the maximum possible number of integers $n$ with $1 \le n \le 2011$ such that $\pi (n)$ divides $\pi (n + 1)$?

For positive integers $m$ and $n$ we define $T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right)$ (a) Prove that there are infinitely many pairs $(m,n)$ of positive integers for which $T(m,n) > 1$ and $T(n,m) > 1$. (b) Do there exist positive integers $m,n$ such that $T(m,n) = T(n,m) > 1$?

[u]Set 6[/u] [b]p16.[/b] Let $n! = n \times (n - 1) \times ... \times 2 \times 1$. Find the maximum positive integer value of $x$ such that the quotient $\frac{160!}{160^x}$ is an integer. [b]p17.[/b] Let $\vartriangle OAB$ be a triangle with $\angle OAB = 90^o$ . Draw points $C, D, E, F, G$ in its plane so that $$\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG,$$ and none of these triangles overlap. If points $O, A, G$ lie on the same line, then let $x$ be the sum of all possible values of $\frac{OG}{OA }$. Then, $x$ can be expressed in the form $m/n$ for relatively prime positive integers $m, n$. Compute $m + n$. [b]p18.[/b] Let $f(x)$ denote the least integer greater than or equal to $x^{\sqrt{x}}$. Compute $f(1)+f(2)+f(3)+f(4)$. [u]Set 7[/u] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \ge 0$. [b]p19.[/b] Find the least odd prime factor of $(F_3)^{20} + (F_4)^{20} + (F_5)^{20}$. [b]p20.[/b] Let $$S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+...$$ Compute $420S$. [b]p21.[/b] Consider the number $$Q = 0.000101020305080130210340550890144... ,$$ the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to $\frac{1}{Q}$. [u]Set 8[/u] [b]p22.[/b] In five dimensional hyperspace, consider a hypercube $C_0$ of side length $2$. Around it, circumscribe a hypersphere $S_0$, so all $32$ vertices of $C_0$ are on the surface of $S_0$. Around $S_0$, circumscribe a hypercube $C_1$, so that $S_0$ is tangent to all hyperfaces of $C_1$. Continue in this same fashion for $S_1$, $C_2$, $S_2$, and so on. Find the side length of $C_4$. [b]p23.[/b] Suppose $\vartriangle ABC$ satisfies $AC = 10\sqrt2$, $BC = 15$, $\angle C = 45^o$. Let $D, E, F$ be the feet of the altitudes in $\vartriangle ABC$, and let $U, V , W$ be the points where the incircle of $\vartriangle DEF$ is tangent to the sides of $\vartriangle DEF$. Find the area of $\vartriangle UVW$. [b]p24.[/b] A polynomial $P(x)$ is called spicy if all of its coefficients are nonnegative integers less than $9$. How many spicy polynomials satisfy $P(3) = 2019$? [i]The next set will consist of three estimation problems.[/i] [u]Set 9[/u] Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed $1,000,000$. [b]p25.[/b] Suppose a circle of radius $20192019$ has area $A$. Let s be the side length of a square with area $A$. Compute the greatest integer less than or equal to $s$. If $n$ is the correct answer, an estimate of $e$ gives $\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \}$ points. [b]p26.[/b] Given a $50 \times 50$ grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue? If $n$ is the correct answer, an estimate of $e$ gives $\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor$ points. [b]p27.[/b] The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is $\frac{\pi}{3\sqrt2} \approx 74.05\%$ of space (confirmed as recently as $2017!$), so we say that the packing density of spheres in three dimensions is about $0.74$. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than $10^8 \times d$. If $n$ is the correct answer, an estimate of e gives $\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\}$ points. PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.