Call a natural number [i]simple[/i] if it is not divisible by any square of a prime number (in other words it is square-free). Prove that there are infinitely many positive integers $n$ such that both $n$ and $n+1$ are [i]simple[/i].

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2024$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2024$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

p1. Find all real numbers $x$ that satisfy the inequality $$\frac{x^2-3}{x^2-1}+ \frac{x^2 + 5}{x^2 + 3} \ge \frac{x^2-5}{x^2-3}+\frac{x^2 + 3}{x^2 + 1}$$ p2. It is known that $m$ is a four-digit natural number with the same units and thousands digits. If $m$ is a square of an integer, find all possible numbers $m$. p3. In the following figure, $\vartriangle ABP$ is an isosceles triangle, with $AB = BP$ and point $C$ on $BP$. Calculate the volume of the object obtained by rotating $ \vartriangle ABC$ around the line $AP$ [img]https://cdn.artofproblemsolving.com/attachments/c/a/65157e2d49d0d4f0f087f3732c75d96a49036d.png[/img] p4. A class farewell event is attended by $10$ male students and $ 12$ female students. Homeroom teacher from the class provides six prizes to randomly selected students. Gifts that provided are one school bag, two novels, and three calculators. If the total students The number of male students who received prizes was equal to the total number of female students who received prizes. How many possible arrangements are there of the student who gets the prize? p5. It is known that $S =\{1945, 1946, 1947, ..., 2016, 2017\}$. If $A = \{a, b, c, d, e\}$ a subset of $S$ where $a + b + c + d + e$ is divisible by $5$, find the number of possible $A$'s.

Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.

Let $n = 2^{p-1} (2^p - 1)$, and let $2^p- 1$ be a prime number. Prove that the sum of all (positive) divisors of $n$ (not including $n$ itself) is exactly $n$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $20 \div 2 - 0 \times 1 + 2 \times 5$? [b]p2.[/b] Today is Saturday, January $25$, $2020$. Exactly four hundred years from today, January $25$, $2420$, is again a Saturday. How many weekend days (Saturdays and Sundays) are in February, $2420$? (January has $31$ days and in year $2040$, February has $29$ days.) [b]p3.[/b] Given that there are four people sitting around a circular table, and two of them stand up, what is the probability that the two of them were originally sitting next to each other? [b]p4.[/b] What is the area of a triangle with side lengths $5$, $5$, and $6$? [b]p5.[/b] Six people go to OBA Noodles on Main Street. Each person has $1/2$ probability to order Duck Noodle Soup, $1/3$ probability to order OBA Ramen, and $1/6$ probability to order Kimchi Udon Soup. What is the probability that three people get Duck Noodle Soup, two people get OBA Ramen, and one person gets Kimchi Udon Soup? [b]p6.[/b] Among all positive integers $a$ and $b$ that satisfy $a^b = 64$, what is the minimum possible value of $a+b$? [b]p7.[/b] A positive integer $n$ is called trivial if its tens digit divides $n$. How many two-digit trivial numbers are there? [b]p8.[/b] Triangle $ABC$ has $AB = 5$, $BC = 13$, and $AC = 12$. Square $BCDE$ is constructed outside of the triangle. The perpendicular line from $A$ to side $DE$ cuts the square into two parts. What is the positive difference in their areas? [b]p9.[/b] In an increasing arithmetic sequence, the first, third, and ninth terms form an increasing geometric sequence (in that order). Given that the first term is $5$, find the sum of the first nine terms of the arithmetic sequence. [b]p10.[/b] Square $ABCD$ has side length $1$. Let points $C'$ and $D'$ be the reflections of points $C$ and $D$ over lines $AB$ and $BC$, respectively. Let P be the center of square $ABCD$. What is the area of the concave quadrilateral $PD'BC'$? [b]p11.[/b] How many four-digit palindromes are multiples of $7$? (A palindrome is a number which reads the same forwards and backwards.) [b]p12.[/b] Let $A$ and $B$ be positive integers such that the absolute value of the difference between the sum of the digits of $A$ and the sum of the digits of $(A + B)$ is $14$. What is the minimum possible value for $B$? [b]p13.[/b] Clark writes the following set of congruences: $x \equiv a$ (mod $6$), $x \equiv b$ (mod $10$), $x \equiv c$ (mod $15$), and he picks $a$, $b$, and $c$ to be three randomly chosen integers. What is the probability that a solution for $x$ exists? [b]p14.[/b] Vincent the bug is crawling on the real number line starting from $2020$. Each second, he may crawl from $x$ to $x - 1$, or teleport from $x$ to $\frac{x}{3}$ . What is the least number of seconds needed for Vincent to get to $0$? [b]p15.[/b] How many positive divisors of $2020$ do not also divide $1010$? [b]p16.[/b] A bishop is a piece in the game of chess that can move in any direction along a diagonal on which it stands. Two bishops attack each other if the two bishops lie on the same diagonal of a chessboard. Find the maximum number of bishops that can be placed on an $8\times 8$ chessboard such that no two bishops attack each other. [b]p17.[/b] Let $ABC$ be a right triangle with hypotenuse $20$ and perimeter $41$. What is the area of $ABC$? [b]p18.[/b] What is the remainder when $x^{19} + 2x^{18} + 3x^{17} +...+ 20$ is divided by $x^2 + 1$? [b]p19.[/b] Ben splits the integers from $1$ to $1000$ into $50$ groups of $20$ consecutive integers each, starting with $\{1, 2,...,20\}$. How many of these groups contain at least one perfect square? [b]p20.[/b] Trapezoid $ABCD$ with $AB$ parallel to $CD$ has $AB = 10$, $BC = 20$, $CD = 35$, and $AD = 15$. Let $AD$ and $BC$ intersect at $P$ and let $AC$ and $BD$ intersect at $Q$. Line $PQ$ intersects $AB$ at $R$. What is the length of $AR$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$. prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.

Does there exists a base in which the numbers of the form: \[ 10101, 101010101, 1010101010101,\cdots \] are all prime numbers?

Given an integer $m > 1$, we call the number $x{}$ dangerous if $x{}$ divides the number $y{}$, which is obtained by writing the digits of $x{}$ in base $m{}$ in reverse order, with $x\neq y$. Prove that if there exists a three-digit (in base $m$) dangerous number for a given $m$, then there exists a two-digit (in base $m$) dangerous number.

Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.

A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar reassignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Find all primes $p$ and $q$ such that $p$ divides $q^2-4$ and $q$ divides $p^2-1$.

Prove that the number $N = 10 ...050...01$ (1, 49 zeros, 5 , 99 zeros, 1) is a not cube of an integer.

Consider a right-angled triangle with integer-valued sides $a<b<c$ where $a,b,c$ are pairwise co-prime. Let $d=c-b$ . Suppose $d$ divides $a$ . Then [b](a)[/b] Prove that $d\leqslant 2$. [b](b)[/b] Find all such triangles (i.e. all possible triplets $a,b,c$) with perimeter less than $100$ .

Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$, at least one of the differences \[(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A\] is divisible by $n$.

Find all positive integer pairs $(a,b)$ for which the set of positive integers can be partitioned into sets $H_1$ and $H_2$ such that neither $a$ nor $b$ can be represented as the difference of two numbers in $H_i$ for $i=1,2$.

Determine all non negative integers $x$ and $y$ such that $6^x$ + $2^y$ + 2 is a perfect square.

Let $f:\mathbb N\to\mathbb R$ be given by $$f(n)=n^{\frac12\tau(n)}$$for $n\in\mathbb N=\{1,2,\ldots\}$ where $\tau(n)$ is the number of divisors of $n$. Show that $f$ is an injection.

Find all integers x,y,z with $2\leq x\leq y\leq z$ st $xy \equiv 1 $(mod z) $xz\equiv 1$(mod y) $yz \equiv 1$ (mod x)

Let $n\not\equiv 2\pmod{3}$. Is $\sqrt{\lfloor n+\tfrac {2n}{3}\rfloor+7},\forall n \in \mathbb {N}$, a natural number?

We consider all $7$-digit numbers that are obtained by swapping in all ways Possible digits of $1234567$. How many of them are divisible by $7$?

Determine all three-digit numbers $N$ such that the average of the six numbers that can be formed by permutation of its three digits is equal to $N$.

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

What is the last two digits of the number $(11^2 + 15^2 + 19^2 +
...

+ 2007^2)^
2$?