Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.

[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$. [b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer. [b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy. [b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers. [b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$. [b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$ where $m|496$ means $496$ is divisible by $m$. [b]p7.[/b] What is the value of $${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$ [b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.

Given $m, n$ such that $m > n^{n-1}$ and the number $m+1$, $m+2$,$ ...$, $m+n$ are composite. Prove that there exist distinct primes $p_1, p_2, ..., p_n$ such that $m + k$ is divisible by $p_k$ for each $k = 1, 2, ...$

Let $n \geq 2$ be a positive integer. For a positive integer $a$, let $Q_a(x)=x^n+ax$. Let $p$ be a prime and let $S_a=\{b | 0 \leq b \leq p-1, \exists c \in \mathbb {Z}, Q_a(c) \equiv b \pmod p \}$. Show that $\frac{1}{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.

Define the sequence $\{a_n\}_{n\geq 1}$ as $a_n=n-1$, $n\leq 2$ and $a_n=$ remainder left by $a_{n-1}+a_{n-2}$ when divided by $3$ $\forall n\geq 2$. Then $\sum_{i=2018}^{2025}a_i=$? (A)$6$ (B)$7$ (C)$8$ (D)$9$

(a) Find a pair of integers (x,y) such that $15x^2 +y^2 = 2^{2000}$ (b) Does there exist a pair of integers $(x,y)$ such that $15x^2 + y^2 = 2^{2000}$ and $x$ is odd?

Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that $$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$ where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$

p1. Let $A = \{(x, y)|3x + 5y\ge 15, x + y^2\le 25, x\ge 0, x, y$ integer numbers $\}$. Find all pairs of $(x, zx)\in A$ provided that $z$ is non-zero integer. p2. A shop owner wants to be able to weigh various kinds of weight objects (in natural numbers) with only $4$ different weights. (For example, if he has weights $ 1$, $2$, $5$ and $10$. He can weighing $ 1$ kg, $2$ kg, $3$ kg $(1 + 2)$, $44$ kg $(5 - 1)$, $5$ kg, $6$ kg, $7$ kg, $ 8$ kg, $9$ kg $(10 - 1)$, $10$ kg, $11$ kg, $12$ kg, $13$ kg $(10 + 1 + 2)$, $14$ kg $(10 + 5 -1)$, $15$ kg, $16$ kg, $17$ kg and $18$ kg). If he wants to be able to weigh all the weight from $ 1$ kg to $40$ kg, determine the four weights that he must have. Explain that your answer is correct. p3. Given the following table. [img]https://cdn.artofproblemsolving.com/attachments/d/8/4622407a72656efe77ccaf02cf353ef1bcfa28.png[/img] Table $4\times 4$ ​​is a combination of four smaller table sections of size $2\times 2$. This table will be filled with four consecutive integers such that: $\bullet$ The horizontal sum of the numbers in each row is $10$ . $\bullet$ The vertical sum of the numbers in each column is $10$ $\bullet$ The sum of the four numbers in each part of $2\times 2$ which is delimited by the line thickness is also equal to $10$. Determine how many arrangements are possible. p4. A sequence of real numbers is defined as following: $U_n=ar^{n-1}$, if $n = 4m -3$ or $n = 4m - 2$ $U_n=- ar^{n-1}$, if $n = 4m - 1$ or $n = 4m$, where $a > 0$, $r > 0$, and $m$ is a positive integer. Prove that the sum of all the $ 1$st to $2009$th terms is $\frac{a(1+r-r^{2009}+r^{2010})}{1+r^2}$ 5. Cube $ABCD.EFGH$ is cut into four parts by two planes. The first plane is parallel to side $ABCD$ and passes through the midpoint of edge $BF$. The sceond plane passes through the midpoints $AB$, $AD$, $GH$, and $FG$. Determine the ratio of the volumes of the smallest part to the largest part.

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Prove that for every positive integer $d > 1$ and $m$ the sequence $a_n=2^{2^n}+d$ contains two terms $a_k$ and $a_l$ ($k \neq l$) such that their greatest common divisor is greater than $m$. [i]Proposed by T. Hakobyan[/i]

A sequence $a_0, a_1, \dots , a_n, \dots$ of positive integers is constructed as follows: [list] [*] If the last digit of $a_n$ is less than or equal to $5$, then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits, the process stops.) [*] Otherwise, $a_{n+1}= 9a_n$. [/list] Can one choose $a_0$ so that this sequence is infinite?

Find all positive integers $k$ for which there exists a function $f: \mathbb{N}\to\mathbb{Z}$ satisfying $f(1997)=1998$ and $f(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b$.

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]

Prove that there are infinitely many different triangles in coordinate plane satisfying: 1) their vertices are lattice points 2) their side lengths are consecutive integers [b]Remark[/b]: Triangles that can be obtained by rotation or translation or shifting the coordinate system are considered as equal triangles

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]

Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

Find all positive integers $m$ with the next property: If $d$ is a positive integer less or equal to $m$ and it isn't coprime to $m$ , then there exist positive integers $a_{1}, a_{2}$,. . ., $a_{2019}$ (where all of them are coprimes to $m$) such that $m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}$ is a perfect power.

Find all ordered triples $(a,b,c)$ of positive integers such that the value of the expression \[\left (b-\frac{1}{a}\right )\left (c-\frac{1}{b}\right )\left (a-\frac{1}{c}\right )\] is an integer.

Can the positive integers be partitioned into $12$ subsets such that for each positive integer $k$, the numbers $k, 2k,\ldots,12k$ belong to different subsets?

Let $n$ be an integer greater than $1$ such that $n$ could be represented as a sum of the cubes of two rational numbers, prove that $n$ is also the sum of the cubes of two non-negative rational numbers. Proposed by [i]Navid Safaei[/i]

Prove that there exists no innite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.