For every positive integer $n$, define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$. Show that $f(n)$ is a perfect square if and only if $n=1$.

Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$, where $a,b,c,d,e$ are positive integers? Explain why.

Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?

Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Find all triples $(m,p,q)$ such that \begin{align*} 2^mp^2 +1=q^7, \end{align*} where $p$ and $q$ are ptimes and $m$ is a positive integer.

What is the sum of the first $31$ integers that can be written as a sum of distinct powers of $3$?

Determine all $n$ positive integers such that exists an $n\times n$ where we can write $n$ times each of the numbers from $1$ to $n$ (one number in each cell), such that the $n$ sums of numbers in each line leave $n$ distinct remainders in the division by $n$, and the $n$ sums of numbers in each column leave $n$ distinct remainders in the division by $n$.

An integer $n\ge2$ is called [i]resistant[/i], if it is coprime to the sum of all its divisors (including $1$ and $n$). Determine the maximum number of consecutive resistant numbers. For instance: * $n=5$ has sum of divisors $S=6$ and hence is resistant. * $n=6$ has sum of divisors $S=12$ and hence is not resistant. * $n=8$ has sum of divisors $S=15$ and hence is resistant. * $n=18$ has sum of divisors $S=39$ and hence is not resistant.

[u]Round 6[/u] [b]p16.[/b] Given that $x$ and $y$ are positive real numbers such that $x^3+y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a\sqrt{b}}{c}$ +d where $a$, $b$, $c$, and $d$ are positive integers such that $gcd(a,c) = 1$ and $b$ is square-free. Find $a +b +c +d$. [b]p17.[/b] In $\vartriangle DRK$ , $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the intersection of the altitudes of $\vartriangle DRK$. Find the value of $\lfloor DE +RE +KE \rfloor$. [b]p18.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find the expected number of jumps it takes Subaru to reach pad $7$. [u]Round 7[/u] This set has problems whose answers depend on one another. [b]p19.[/b] Let $B$ be the answer to Problem $20$ and let $C$ be the answer to Problem $21$. Given that $$f (x) = x^3-Bx-C = (x-r )(x-s)(x-t )$$ where $r$, $s$, and $t$ are complex numbers, find the value of $r^2+s^2+t^2$. [b]p20.[/b] Let $A$ be the answer to Problem $19$ and let $C$ be the answer to Problem $21$. Circles $\omega_1$ and $\omega_2$ meet at points $X$ and $Y$ . Let point $P \ne Y$ be the point on $\omega_1$ such that $PY$ is tangent to $\omega_2$, and let point $Q \ne Y$ be the point on $\omega_2$ such that $QY$ is tangent to $\omega_1$. Given that $PX = A$ and $QX =C$, find $XY$ . [b]p21.[/b] Let $A$ be the answer to Problem $19$ and let $B$ be the answer to Problem $20$. Given that the positive difference between the number of positive integer factors of $A^B$ and the number of positive integer factors of $B^A$ is $D$, and given that the answer to this problem is an odd prime, find $\frac{D}{B}-40$. [u]Round 8[/u] [b]p22.[/b] Let $v_p (n)$ for a prime $p$ and positive integer $n$ output the greatest nonnegative integer $x$ such that $p^x$ divides $n$. Find $$\sum^{50}_{i=1}\sum^{i}_{p=1} { v_p (i )+1 \choose 2},$$ where the inner summation only sums over primes $p$ between $1$ and $i$ . [b]p23.[/b] Let $a$, $b$, and $c$ be positive real solutions to the following equations. $$\frac{2b^2 +2c^2 -a^2}{4}= 25$$ $$\frac{2c^2 +2a^2 -b^2}{4}= 49$$ $$\frac{2a^2 +2b^2 -c^2}{4}= 64$$ The area of a triangle with side lengths $a$, $b$, and $c$ can be written as $\frac{x\sqrt{y}}{z}$ where $x$ and $z$ are relatively prime positive integers and $y$ is square-free. Find $x + y +z$. [b]p24.[/b] Alan, Jiji, Ina, Ryan, and Gavin want to meet up. However, none of them know when to go, so they each pick a random $1$ hour period from $5$ AM to $11$ AM to meet up at Alan’s house. Find the probability that there exists a time when all of them are at the house at one time. [b]Round 9 [/b] [b]p25.[/b] Let $n$ be the number of registered participantsin this $LMT$. Estimate the number of digits of $\left[ {n \choose 2} \right]$ in base $10$. If your answer is $A$ and the correct answer is $C$, then your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ [b]p26.[/b] Let $\gamma$ be theminimum value of $x^x$ over all real numbers $x$. Estimate $\lfloor 10000\gamma \rfloor$. If your answer is $A$ and the correct answer is $C$, then your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ [b]p27.[/b] Let $$E = \log_{13} 1+log_{13}2+log_{13}3+...+log_{13}513513.$$ Estimate $\lfloor E \rfloor$. If your answer is $A$ and the correct answer is $C$, your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167127p28823220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a positive integer $n$ and a prime $q$, prove that the number $n^q+(\frac{n-1}{2})^2$ can't be a power of $q$.

What is the smallest positive integer $n$ such that $2016n$ is a perfect cube?

Let $f_n(x)=\underbrace{xx\cdots x}_{n\ \text{times}}$ that is, $f_n(x)$ is a $n$ digit number with all digits $x$, where $x\in \{1,2,\cdots,9\}$. Then which of the following is $\Big(f_n(3)\Big)^2+f_n(2)$? [list=1] [*] $f_n(5)$ [*] $f_{2n}(9)$ [*] $f_{2n}(1)$ [*] None of these [/list]

Prove that there don't exist positive integer numbers $k$ and $n$ which satisfy equation $n^n+(n+1)^{n+1}+(n+2)^{n+2} = 2023^k$. [i]Proposed by Mykhailo Shtandenko[/i]

Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]

Let $n$ be a positive integer. Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$

Let $n$ be a positive square free integer, $S$ is a subset of $[n]:=\{1,2,\ldots ,n\}$ such that $|S|\ge n/2.$ Prove that there exists three elements $a,b,c\in S$ (can be same), satisfy $ab\equiv c\pmod n.$ [i]Created by Zhenhua Qu[/i]

$200$ natural numbers are written in a row. For any two adjacent numbers of the row, the right one is either $9$ times greater than the left one, $2$ times smaller than the left one. Can the sum of all these 200 numbers be equal to $24^{2022}$?

Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$

Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$ $\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72$

Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.

At the start of the day, the four numbers $(a_0,b_0,c_0,d_0)$ were written on the board. Every minute, Danny replaces the four numbers written on the board with new ones according to the following rule: if the numbers written on the board are $(a,b,c,d)$, then Danny first calculates the numbers \begin{align*} a'&=a+4b+16c+64d\\ b'&=b+4c+16d+64a\\ c'&=c+4d+16a+64b\\ d'&=d+4a+16b+64c \end{align*} and replaces the numbers $(a,b,c,d)$ with the numbers $(a'd',c'd',c'b',b'a')$. For which initial quadruples $(a_0,b_0,c_0,d_0)$, will Danny write at some point a quadruple of numbers all of which are divisible by $5780^{5780}$?

Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

Given a prime number $p\ge 3,$ and an integer $d \ge 1$. Prove that there exists an integer $n\ge 1,$ such that $\gcd(n,d) = 1,$ and the product \[P=\prod\limits_{1 \le i < j < p} {({i^{n + j}} - {j^{n + i}})} \text{ is not divisible by } p^n.\]