Let $1=d_1<d_2<....<d_k=n$ be all different divisors of positive integer n written in ascending order. Determine all n such that: \[d_6^{2} +d_7^{2} - 1=n\]

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties: (i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$; (ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite. Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic. [i]Proposed by Ang Jie Jun, Singapore[/i]

Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

Given two cubes $R$ and $S$ with integer sides of lengths $r$ and $s$ units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that $r=s$.

If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$ [i]Flavian Georgescu[/i]

Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.

Find all prime numbers $p, q, r$ satisfying the equation \[ p^4+2p+q^4+q^2=r^2+4q^3+1 \]

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

Let $S=\{x^2+2y^2\mid x,y\in\mathbb Z\}$. If $a$ is an integer with the property that $3a$ belongs to $S$, prove that then $a$ belongs to $S$ as well.

[b]p1. [/b] What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated. [img]https://cdn.artofproblemsolving.com/attachments/7/6/468fd9b81e9115a4a98e4cbf6dedf47ce8349e.png[/img] [b]p2.[/b] In triangle $\vartriangle ABC$, $\angle A = 30^o$. $D$ is a point on $AB$ such that $CD \perp AB$. $E$ is a point on $AC$ such that $BE \perp AC$. What is the value of $\frac{DE}{BC}$ ? [b]p3.[/b] Given that f(x) is a polynomial such that $2f(x) + f(1 - x) = x^2$. Find the sum of squares of the coefficients of $f(x)$. [b]p4. [/b] For each positive integer $n$, there exists a unique positive integer an such that $a^2_n \le n < (a_n + 1)^2$. Given that $n = 15m^2$ , where $m$ is a positive integer greater than $1$. Find the minimum possible value of $n - a^2_n$. [b]p5.[/b] What are the last two digits of $\lfloor (\sqrt5 + 2)^{2016}\rfloor$ ? Note $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p6.[/b] Let $f$ be a function that satisfies $f(2^a3^b)) = 3a+ 5b$. What is the largest value of f over all numbers of the form $n = 2^a3^b$ where $n \le 10000$ and $a, b$ are nonnegative integers. [b]p7.[/b] Find a multiple of $21$ such that it has six more divisors of the form $4m + 1$ than divisors of the form $4n + 3$ where m, n are integers. You can keep the number in its prime factorization form. [b]p8.[/b] Find $$\sum^{100}_{i=0} \lfloor i^{3/2} \rfloor +\sum^{1000}_{j=0} \lfloor j^{2/3} \rfloor$$ where $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p9. [/b] Let $A, B$ be two randomly chosen subsets of $\{1, 2, . . . 10\}$. What is the probability that one of the two subsets contains the other? [b]p10.[/b] We want to pick $5$-person teams from a total of $m$ people such that: 1. Any two teams must share exactly one member. 2. For every pair of people, there is a team in which they are teammates. How many teams are there? (Hint: $m$ is determined by these conditions). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.

The first four digits of a certain positive integer $n$ are $1137$. Prove that the digits of $n$ can be shuffled in such a way that the new number is divisible by 7.

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]

We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is [b]kawaii[/b] if \( T \) has the following properties: 1. \( T \) has at least two distinct elements; 2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \). For example, the subset \( T = \{31, 71, 2024\} \) is [b]kawaii[/b], but \( T = \{5, 15, 75\} \) is not [b]kawaii[/b] because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \). What is the largest possible number of elements that a [b]kawaii [/b]subset can have?

Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\ (a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\ (b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds. [i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]

Find the smallest positive integer $n$ such that $32^n = 167x + 2$ for some integer $x$

Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\] $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

How many ordered triples $(a, b, c)$ of integers $1 \le a, b, c \le 31$ are there such that the remainder of $ab+bc+ca$ divided by $31$ equals $8$?

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

Prove that if $ n $ is an integer, then $$ \frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30}$$ is also an integer.

What is the maximal number of elements we can choose form the set $\{1, 2, \ldots, 31\}$, such that the sum of any two of them is not a perfect square.

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?