Find the sum of all the $5$-digit integers which are not multiples of $11$ and whose digits are $1, 3, 4, 7, 9$.

Define $(a_n)$ a sequence, where $a_1= 12, a_2= 24$ and for $n\geq 3$, we have: $$a_n= a_{n-2}+14$$ a) Is $2023$ in the sequence? b) Show that there are no perfect squares in the sequence.

Let $a, b$ and $c$ be positive integers satisfying the equation $(a, b) + [a, b]=2021^c$. If $|a-b|$ is a prime number, prove that the number $(a+b)^2+4$ is composite. [i]Proposed by Serbia[/i]

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$ [i]Proposed by Romeo Meštrović, Montenegro[/i]

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,\cdots$ be an infinity sequence of positive integers such that $a_1=2021$ and $$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$$ for all positive integers $n$. Prove that for any integer $n\ge 2$, $a_n$ is the product of at least $2n$ (not necessarily distinct) primes.

Let $a$ be a prime number and $n > 2$ an integer. Find all integer solutions of the equation $x^n +ay^n = a^2z^n$ .

Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.

Find the remainder after division of $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ by $7$.

$(\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 11$? $(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?

There exist $4$ positive integers $a,b,c,d$ such that $abcd \neq 1$ and each pair of them have a GCD of $1$. Two functions $f,g : \mathbb{N} \rightarrow \{0,1\}$ are multiplicative functions such that for each positive integer $n$ we have : $$f(an+b)=g(cn+d)$$ Prove that at least one of the followings hold. $i)$ for each positive integer $n$ we have $f(an+b)=g(cn+d)=0$ $ii)$ There exists a positive integer $k$ such that for all $n$ where $(n,k)=1$ we have $g(n)=f(n)=1$ (Function $f$ is multiplicative if for any natural numbers $a,b$ we have $f(ab)=f(a)f(b)$) Proposed by [i]Navid Safaii[/i]

Let there be $A=1^{a_1}2^{a_2}\dots100^{a_100}$ and $B=1^{b_1}2^{b_2}\dots100^{b_100}$ , where $a_i , b_i \in N$ , $a_i + b_i = 101 - i$ , ($i= 1,2,\dots,100$). Find the last 1124 digits of $P = A * B$.

Let $n{}$ and $m$ be positive integers, $n>m>1$. Let $n{}$ divided by $m$ have partial quotient $q$ and remainder $r$ (so that $n = qm + r$, where $r\in\{0,1,...,m-1\}$). Let $n-1$ divided by $m$ have partial quotient $q^{'}$ and remainder $r^{'}$. a) It appears that $q+q^{'} =r +r^{'} = 99$. Find all possible values of $n{}$. b) Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.

Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$. [i]Proposed by Viktor Simjanoski[/i]

For a positive integer $n$ we define $f (n) = \max X_1^{X_2^{...^{X_k}}}$ where the maximum is taken over all possible decompositions of natural numbers $n = X_1X_2...X_k$. Determine $f(n)$.

Let $p$ be a sufficiently large prime. Show that the number of distinct residues taken by the set $$\{1 + \frac12 + ... + \frac{1}{n}: n = 1, 2,..., p - 1\}$$ modulo $p$ has at least $\sqrt[4]{p}$ elements. (Carlo Sanna)

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

Find all pairs $(m, n)$ of positive integers for which $4 (mn +1)$ is divisible by $(m + n)^2$.

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.

For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?

For positive integers $a,n$, define $F_n(a)=q+r$, where $a=qn+r$ ($q,r$ are nonnegative integers, $0\leq q<n$). Find the largest integer $A$, there are positive integers $n_1,n_2,n_3,n_4,n_5,n_6$, for all positive integer $a\leq A$, $F_{n_6}(F_{n_5}(F_{n_4}(F_{n_3}(F_{n_2}(F_{n_1}(a))))))=1$.