A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

The product of Albrecht’s three favorite numbers is $2022$, and if we add one to each number, their product will be $1514$. What is the sum of their squares, if we know their sum is $0$?

Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\[2^x\cdot3^y-5^z\cdot7^w=1.\]

We call a divisor $d$ of a positive integer $n$ [i]special[/i] if $d + 1$ is also a divisor of $n$. Prove: at most half the positive divisors of a positive integer can be special. Determine all positive integers for which exactly half the positive divisors are special.

Jacob and Laban take turns playing a game. Each of them starts with the list of square numbers $1, 4, 9, \dots, 2021^2$, and there is a whiteboard in front of them with the number $0$ on it. Jacob chooses a number $x^2$ from his list, removes it from his list, and replaces the number $W$ on the whiteboard with $W + x^2$. Laban then does the same with a number from his list, and the repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by $4$ after a player has taken his turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible. What is the greatest number $K$ such that Jacob can guarantee to get at least $K$ sheep by the end of the game, no matter how Laban plays?

Describe all the ways to color each natural number as one of three colors so that the following condition is satisfied: if the numbers $a$, $b$ and $c$ (not necessarily different) satisfy the condition $2000(a + b) = c$, then they either all the same color or three different colors

Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.

Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?

Determine all pairs $(x, y)$ of positive integers for which the equation \[x + y + xy = 2006\] holds.

Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.

Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]R4.16 / P1.4[/b] Adam and Becky are building a house. Becky works twice as fast as Adam does, and they both work at constant speeds for the same amount of time each day. They plan to finish building in $6$ days. However, after $2$ days, their friend Charlie also helps with building the house. Because of this, they finish building in just $5$ days. What fraction of the house did Adam build? [b]R4.17[/b] A bag with $10$ items contains both pencils and pens. Kanye randomly chooses two items from the bag, with replacement. Suppose the probability that he chooses $1$ pen and $1$ pencil is $\frac{21}{50}$ . What are all possible values for the number of pens in the bag? [b]R4.18 / P2.8[/b] In cyclic quadrilateral $ABCD$, $\angle ABD = 40^o$, and $\angle DAC = 40^o$. Compute the measure of $\angle ADC$ in degrees. (In cyclic quadrilaterals, opposite angles sum up to $180^o$.) [b]R4.19 / P2.6[/b] There is a strange random number generator which always returns a positive integer between $1$ and $7500$, inclusive. Half of the time, it returns a uniformly random positive integer multiple of $25$, and the other half of the time, it returns a uniformly random positive integer that isn’t a multiple of $25$. What is the probability that a number returned from the generator is a multiple of $30$? [b]R4.20 / P2.7[/b] Julia is shopping for clothes. She finds $T$ different tops and $S$ different skirts that she likes, where $T \ge S > 0$. Julia can either get one top and one skirt, just one top, or just one skirt. If there are $50$ ways in which she can make her choice, what is $T - S$? [u]Set 5[/u] [b]R5.21[/b] A $5 \times 5 \times 5$ cube’s surface is completely painted blue. The cube is then completely split into $ 1 \times 1 \times 1$ cubes. What is the average number of blue faces on each $ 1 \times 1 \times 1$ cube? [b]R5.22 / P2.10[/b] Find the number of values of $n$ such that a regular $n$-gon has interior angles with integer degree measures. [b]R5.23[/b] $4$ positive integers form an geometric sequence. The sum of the $4$ numbers is $255$, and the average of the second and the fourth number is $102$. What is the smallest number in the sequence? [b]R5.24[/b] Let $S$ be the set of all positive integers which have three digits when written in base $2016$ and two digits when written in base $2017$. Find the size of $S$. [b]R5.25 / P3.12[/b] In square $ABCD$ with side length $13$, point $E$ lies on segment $CD$. Segment $AE$ divides $ABCD$ into triangle $ADE$ and quadrilateral $ABCE$. If the ratio of the area of $ADE$ to the area of $ABCE$ is $4 : 11$, what is the ratio of the perimeter of $ADE$ to the perimeter of $ABCE$? [u]Set 6[/u] [b]R6.26 / P6.25[/b] Submit a decimal n to the nearest thousandth between $0$ and $200$. Your score will be $\min (12, S)$, where $S$ is the non-negative difference between $n$ and the largest number less than or equal to $n$ chosen by another team (if you choose the smallest number, $S = n$). For example, 1.414 is an acceptable answer, while $\sqrt2$ and $1.4142$ are not. [b]R6.27 / P6.27[/b] Guang is going hard on his YNA project. From $1:00$ AM Saturday to $1:00$ AM Sunday, the probability that he is not finished with his project $x$ hours after $1:00$ AM on Saturday is $\frac{1}{x+1}$ . If Guang does not finish by 1:00 AM on Sunday, he will stop procrastinating and finish the project immediately. Find the expected number of minutes $A$ it will take for him to finish his project. An estimate of $E$ will earn $12 \cdot 2^{-|E-A|/60}$ points. [b]R6.28 / P6.28[/b] All the diagonals of a regular $100$-gon (a regular polygon with $100$ sides) are drawn. Let $A$ be the number of distinct intersection points between all the diagonals. Find $A$. An estimate of $E$ will earn $12 \cdot \left(16 \log_{10}\left(\max \left(\frac{E}{A},\frac{A}{E}\right)\right)+ 1\right)^{-\frac12}$ or $0$ points if this expression is undefined. [b]R6.29 / P6.29 [/b]Find the smallest positive integer $A$ such that the following is true: if every integer $1, 2, ..., A$ is colored either red or blue, then no matter how they are colored, there are always 6 integers among them forming an increasing arithmetic progression that are all colored the same color. An estimate of $E$ will earn $12 min \left(\frac{E}{A},\frac{A}{E}\right)$ points or $0$ points if this expression is undefined. [b]R6.30 / P6.30[/b] For all integers $n \ge 2$, let $f(n)$ denote the smallest prime factor of $n$. Find $A =\sum^{10^6}_{n=2}f(n)$. In other words, take the smallest prime factor of every integer from $2$ to $10^6$ and sum them all up to get $A$. You may find the following values helpful: there are $78498$ primes below $10^6$, $9592$ primes below $10^5$, $1229$ primes below $10^4$, and $168$ primes below $10^3$. An estimate of $E$ will earn $\max \left(0, 12-4 \log_{10}(max \left(\frac{E}{A},\frac{A}{E}\right)\right)$ or $0$ points if this expression is undefined. PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.

Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

For primes $a, b,c$ that satisfy the following, calculate $abc$. $\bullet$ $b + 8$ is a multiple of $a$, $\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$ $\bullet$ $b + c = a^2 - 1$.

If $p$ and $q$ are odd integers, prove that the equation $$x^2 + 2px + 2q = 0$$ has no rational roots.

Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1 $. Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions: [b]I.[/b] $m\leq x\leq n$ for all $x\in S$; [b]II.[/b] the product of all elements in $S$ is the square of an integer.

Find all $(m, n)$ positive integer pairs such that both $\frac{3n^2}{m}$ and $\sqrt{n^2+m}$ are integers.

Let $a$ and $b$ be positive integers. Show that if $a^3+b^3$ is the square of an integer, then $a + b$ is not a product of two different prime numbers.

Find all positive integers $x, y, z$ such that $2^x + 2^y + 2^z = 2336$.

Let us call a positive integer [i]pedestrian[/i] if all its decimal digits are equal to 0 or 1. Suppose that the product of some two pedestrian integers also is pedestrian. Is it necessary in this case that the sum of digits of the product equals the product of the sums of digits of the factors? [i]Viktor Kleptsyn, Konstantin Knop[/i]

$(x_1, y_1,z_1)$ and $(x_2, y_2, z_2)$ are triples of real numbers such that for every pair of integers $(m,n)$ at least one of $x_{1m} + y_{1n} + z_1$, $x_{2m} + y_{2n} + z_2$ is an even integer. Prove that one of the triples consists of three integers.

Solve in nonnegative integers the following equation : $$21^x+4^y=z^2$$

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that \[\sum_{t \mid n} d(t)^3\]is divisible by $35$?