Proof that if $4$ numbers (not necessarily distinct) are picked from $\{1, 2, \cdots, 20\}$, one can pick $3$ numbers among them and can label these $3$ as $a, b, c$ such that $ax \equiv b \;(\bmod\; c)$ has integral solutions.

A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary. John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well. What was Mary's Number?

Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions: (i) Every positive integer appears exactly once, (ii) For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$.

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

Let $n$ be a positive integer. A German set in an $n \times n$ square grid is a set of $n$ cells which contains exactly one cell in each row and column. Given a labelling of thecells with the integers from $1$ to $n^2$ using each integer exactly once, we say that an integer is a German product if it is the product of the labels of the cells in a German set. (a) Let $n=8$. Determine whether there exists a labelling of an $8 \times 8$ grid such that the following condition is fulfilled: The difference of any two German products is alwaysdivisible by $65$. (b) Let $n=10$. Determine whether there exists a labelling of a $10 \times 10$ grid such that the following condition is fulfilled: The difference of any two German products is always divisible by $101$.

Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.

Let $a$, $b$, $c$ be positive reals for which \begin{align*} (a+b)(a+c) &= bc + 2 \\ (b+c)(b+a) &= ca + 5 \\ (c+a)(c+b) &= ab + 9 \end{align*} If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]G.16[/b] A number $k$ is the product of exactly three distinct primes (in other words, it is of the form $pqr$, where $p, q, r$ are distinct primes). If the average of its factors is $66$, find $k$. [b]G.17[/b] Find the number of lattice points contained on or within the graph of $\frac{x^2}{3} +\frac{y^2}{2}= 12$. Lattice points are coordinate points $(x, y)$ where $x$ and $y$ are integers. [b]G.18 / C.23[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]G.19[/b] Cindy has a cone with height $15$ inches and diameter $16$ inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint? [b]G.20 / C.25[/b] An even positive integer $n$ has an odd factorization if the largest odd divisor of $n$ is also the smallest odd divisor of n greater than 1. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u] Set 5[/u] [b]G.21[/b] In the magical tree of numbers, $n$ is directly connected to $2n$ and $2n + 1$ for all nonnegative integers n. A frog on the magical tree of numbers can move from a number $n$ to a number connected to it in $1$ hop. What is the least number of hops that the frog can take to move from $1000$ to $2018$? [b]G.22[/b] Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for $10$ days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy $1024$ dollars, but then he learns the catch - after $10$ days, the amount of money he has must be a multiple of $11$ or he loses all his money. What is the largest amount of money Stan can have after the $10$ days are up? [b]G.23[/b] Let $\Gamma_1$ be a circle with diameter $2$ and center $O_1$ and let $\Gamma_2$ be a congruent circle centered at a point $O_2 \in \Gamma_1$. Suppose $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Let $\Omega$ be a circle centered at $A$ passing through $B$. Let $P$ be the intersection of $\Omega$ and $\Gamma_1$ other than $B$ and let $Q$ be the intersection of $\Omega$ and ray $\overrightarrow{AO_1}$. Define $R$ to be the intersection of $PQ$ with $\Gamma_1$. Compute the length of $O_2R$. [b]G.24[/b] $8$ people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible? [b]G.25[/b] Let $S$ be the set of points $(x, y)$ such that $y = x^3 - 5x$ and $x = y^3 - 5y$. There exist four points in $S$ that are the vertices of a rectangle. Find the area of this rectangle. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]

Find the remainder when $$\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor$$ is divided by $10^4$. [i]Proposed by Vijay Srinivasan[/i]

Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

Find all positive integers $n$ that are equal to the sum of digits of $n^2$.

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.

Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan.

Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$ [i]Flavian Georgescu[/i]

One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.

Find all composite odd positive integers, all divisors of which can be divided into pairs so that the sum of the numbers in each pair is a power of two, and each divisor belongs to exactly one such pair. [i]Proposed by Anton Trygub[/i]

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find $a+b$.

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

Find all pair of positive integers $(x, y)$ satisfying the equation \[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]

Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power: $$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$ [i]Proposed by Navid Safaei[/i]