Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$.

Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.) $\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313$

Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that: i) In the set $P_n$ there are infinitely many squares. ii) In the set $P_n$ there are no squares.

[b]p13.[/b] Suppose $\overline{AB}$ is a radius of a circle. If a point $C$ is chosen uniformly at random inside the circle, what is the probability that triangle $ABC$ has an obtuse angle? [b]p14.[/b] Find the second smallest positive integer $c$ such that there exist positive integers $a$ and $b$ satisfying the following conditions: $\bullet$ $5a = b = \frac{c}{5} + 6$. $\bullet$ $a + b + c$ is a perfect square. [b]p15.[/b] A spotted lanternfly is at point $(0, 0, 0)$, and it wants to land on an unassuming CMU student at point $(2, 3, 4)$. It can move one unit at a time in either the $+x$, $+y$, or $+z$ directions. However, there is another student waiting at $(1, 2, 3)$ who will stomp on the lanternfly if it passes through that point. How many paths can the lanternfly take to reach its target without getting stomped? PS. You should use hide for answers.

Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$

Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy: $$p+q=r(p-q)^n$$ [i]Proposed by Şahin Emrah[/i]

Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.

Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.

Let $\displaystyle{p,q}$ be relatively prime positive integers. Prove that \[\displaystyle{ \sum_{k=0}^{pq-1} (-1)^{\left\lfloor \frac{k}{p}\right\rfloor + \left\lfloor \frac{k}{q}\right\rfloor} = \begin{cases} 0 & \textnormal{ if } pq \textnormal{ is even}\\ 1 & \textnormal{if } pq \textnormal{ odd}\end{cases}}\] [i]Proposed by Alexander Bolbot, State University, Novosibirsk.[/i]

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

Let $n$ be a positive integer, which gives remainder $4$ of dividing by $8$. Numbers $1 = k_1 < k_2 < ... < k_m = n$ are all positive diivisors of $n$. Show that if $i \in \{ 1, 2, ..., m - 1 \}$ isn't divisible by $3$, then $k_{i + 1} \le 2k_{i}$.

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.

A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$. Prove that every positive integer is dapper.

Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will [b]not[/b] happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$.

Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied: (i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$. (ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$. (iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. [i](Ivan Gochev, Hristo Minchev)[/i]

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!

Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation $$p^3 + pm + 2zm = m^2 + pz + z^2$$

Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

Prove: for all positive integers $m, n$ $\frac 1m + \frac 1{m+1} + \dotsb + \frac 1 {m+n} $ is not an integer.

Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that [list] [*] $f(x \cdot y) = f(x) + f(y)$ [*] $f(30) = 0$ [*] $f(x)=0$ always when the units digit of $x$ is $7$ [/list]