Find all positive integers $n$ for which there exists a positive integer $k$ such that the decimal representation of $n^k$ starts and ends with the same digit.

Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$.

Find the minimum value of $m + n$, where $m$ and $n$ are positive integers satisfying: $2023 \vert m + 2025n$ $2025 \vert m + 2023n$ [i]Proposed by Prudencio Guerrero Fernández [/i]

A prime $p>3$ is given. Let $K$ be the number of such permutations $(a_1, a_2, \ldots, a_p)$ of $\{ 1, 2, \ldots, p\}$ such that $$a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1$$ is divisible by $p$. Prove $K+p$ is divisible by $p^2$.

All the two-digit numbers from $19$ to $80$ are written in a line without spaces. Is the obtained number $192021....7980$ divisible by $1980$?

3. Show that there do not exist positive integers $a$, $b$, $c$ and $d$, pairwise co-prime, such that $ab+cd$, $ac+bd$ and $ad+bc$ are odd divisors of the number $(a+b-c-d)(a-b+c-d)(a-b-c+d)$.

[b]p1.[/b] Two trains, $A$ and $B$, travel between cities $P$ and $Q$. On one occasion $A$ started from $P$ and $B$ from $Q$ at the same time and when they met $A$ had travelled $120$ miles more than $B$. It took $A$ four $(4)$ hours to complete the trip to $Q$ and B nine $(9)$ hours to reach $P$. Assuming each train travels at a constant speed, what is the distance from $P$ to $Q$? [b]p2.[/b] If $a$ and $b$ are integers, $b$ odd, prove that $x^2 + 2ax + 2b = 0$ has no rational roots. [b]p3.[/b] A diameter segment of a set of points in a plane is a segment joining two points of the set which is at least as long as any other segment joining two points of the set. Prove that any two diameter segments of a set of points in the plane must have a point in common. [b]p4.[/b] Find all positive integers $n$ for which $\frac{n(n^2 + n + 1) (n^2 + 2n + 2)}{2n + 1}$ is an integer. Prove that the set you exhibit is complete. [b]p5.[/b] $A, B, C, D$ are four points on a semicircle with diameter $AB = 1$. If the distances $\overline{AC}$, $\overline{BC}$, $\overline{AD}$, $\overline{BD}$ are all rational numbers, prove that $\overline{CD}$ is also rational. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(F_{n})_{n\geq1}$ the Fibonacci sequence. Find all $n \in \mathbb{N}$ such that for every $k=0,1,...,F_{n}$ \[ {F_{n}\choose k} \equiv (-1)^{k} \ (mod \ F_{n}+1) \]

Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots+n^3$ is divided by $n+5$, the remainder is $17.$

Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ if integers. Banana may make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, upon which Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to figure out Anana's $n$-tuple for certain?

Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

Prove that if $n$ is an odd natural number, then $1^n +2^n +\cdots +n^n$ is divisible by $n^2$.

A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.) Given this information, find all possible values for the number of elements of $S$.

Find all triples $(p,q,r)$ of pairwise distinct primes such that \[p\mid q+r, q\mid r+2p, r\mid p+3q.\]

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

How many 4-digit numbers have exactly $9$ divisors from the set $\{1,2,3,4,5,6,7,8,9,10\}$? [i]Proposed by Ethan Gu[/i]

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

Let $n$ be a positive integer. Show that there exist a polynomial $P(x)$ with integer coefficient that satisfy the following [list] [*]Degree of $P(x)$ is at most $2^n - n -1$ [*]$|P(k)| = (k-1)!(2^n-k)!$ for each $k \in \{1,2,3,\dots,2^n\}$ [/list]

Find all positive integers $a$ and $b$ such that $ ab+1 \mid a^2-1$

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

Let $S(n)$ denote the sum of digits of a natural number $n$. Find all $n$ for which $n+S(n)=2004$.

Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$.

Let $a$ and $m$ be positive integers and $p$ be an odd prime number such that $p^m\mid a-1$ and $p^{m+1}\nmid a-1$. Prove that (a) $p^{m+n}\mid a^{p^n}-1$ for all $n\in\mathbb N$, and (a) $p^{m+n+1}\nmid a^{p^n}-1$ for all $n\in\mathbb N$.