Find the sum of all positive $30$-digit palindromes. The leading digit is not allowed to be $0$.

A sequence $\{x_n \}=x_0, x_1, x_2, \cdots $ satisfies $x_0=a(1\le a \le 2019, a \in \mathbb{R})$, and $$x_{n+1}=\begin{cases}1+1009x_n &\ (x_n \le 2) \\ 2021-x_n &\ (2<x_n \le 1010) \\ 3031-2x_n &\ (1010<x\le 1011) \\ 2020-x_n &\ (1011<x_n) \end{cases}$$ for each non-negative integer $n$. If there exist some integer $k>1$ such that $x_k=a$, call such minimum $k$ a [i] fundamental period[/i] of $\{x_n \}$. Find all integers which can be a fundamental period of some seqeunce; and for such minimal odd period $k(>1)$, find all values of $x_0=a$ such that the fundamental period of $\{x_n \}$ equals $k$.

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $ [asy] size(170); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0); draw(A--C--B--A--D--B--E--D); label("$A$",A,W); label("$B$",B,S); label("$C$",C,SW); label("$D$",D,NE); label("$E$",E,SE); [/asy] $ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 128\qquad\textbf{(C)}\ 153\qquad\textbf{(D)}\ 243\qquad\textbf{(E)}\ 256 $

The set $A{}$ of positive integers satisfies the following conditions: [list=1] [*]If a positive integer $n{}$ belongs to $A{}$, then $2n$ also belongs to $A{}$; [*]For any positive integer $n{}$ there exists an element of $A{}$ divisible by $n{}$; [*]There exist finite subsets of $A{}$ with arbitrarily large sums of reciprocals of elements. [/list]Prove that for any positive rational number $r{}$ there exists a finite subset $B\subset A$ such that \[\sum_{x\in B}\frac{1}{x}=r.\]

Let $ 1\le a_1<a_2<\cdots<a_m\le N$ be a sequence of integers such that the least common multiple of any two of its elements is not greater than $ N$. Show that $ m\le 2\left[\sqrt{N}\right]$, where $ \left[\sqrt{N}\right]$ denotes the greatest integer $ \le \sqrt{N}$

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Alisha has $6$ cupcakes and Tyrone has $10$ brownies. Tyrone gives some of his brownies to Alisha so that she has three times as many desserts as Tyrone. How many desserts did Tyrone give to Alisha? [b]p2.[/b] Bisky adds one to her favorite number. She then divides the result by $2$, and gets $56$. What is her favorite number? [b]p3.[/b] What is the maximum number of points at which a circle and a square can intersect? [b]p4.[/b] An integer $N$ leaves a remainder of 66 when divided by $120$. Find the remainder when $N$ is divided by $24$. [b]p5.[/b] $7$ people are chosen to run for student council. How many ways are there to pick $1$ president, $1$ vice president, and $1$ secretary? [b]p6.[/b] Anya, Beth, Chloe, and Dmitri are all close friends, and like to make group chats to talk. How many group chats can be made if Dmitri, the gossip, must always be in the group chat and Anya is never included in them? Group chats must have more than one person. [b]p7.[/b] There exists a telephone pole of height $24$ feet. From the top of this pole, there are two wires reaching the ground in opposite directions, with one wire $25$ feet, and the other wire 40 feet. What is the distance (in feet) between the places where the wires hit the ground? [b]p8.[/b] Tarik is dressing up for a job-interview. He can wear a chill, business, or casual outfit. If he wears a chill oufit, he must wear a t-shirt, shorts, and flip-flops. He has eight of the first, seven of the second, and three of the third. If he wears a business outfit, he must wear a blazer, a tie, and khakis; he has two of the first, six of the second, and five of the third; finally, he can also choose the casual style, for which he has three hoodies, nine jeans, and two pairs of sneakers. How many different combinations are there for his interview? [b]p9.[/b] If a non-degenerate triangle has sides $11$ and $13$, what is the sum of all possibilities for the third side length, given that the third side has integral length? [b]p10.[/b] An unknown disease is spreading fast. For every person who has the this illness, it is spread on to $3$ new people each day. If Mary is the only person with this illness at the start of Monday, how many people will have contracted the illness at the end of Thursday? [b]p11.[/b] Gob the giant takes a walk around the equator on Mars, completing one lap around Mars. If Gob’s head is $\frac{13}{\pi}$ meters above his feet, how much farther (in meters) did his head travel than his feet? [b]p12.[/b] $2022$ leaves a remainder of $2$, $6$, $9$, and $7$ when divided by $4$, $7$, $11$, and $13$ respectively. What is the next positive integer which has the same remainders to these divisors? [b]p13.[/b] In triangle $ABC$, $AB = 20$, $BC = 21$, and $AC = 29$. Let D be a point on $AC$ such that $\angle ABD = 45^o$. If the length of $AD$ can be represented as $\frac{a}{b}$ , what is $a + b$? [b]p14.[/b] Find the number of primes less than $100$ such that when $1$ is added to the prime, the resulting number has $3$ divisors. [b]p15.[/b] What is the coefficient of the term $a^4z^3$ in the expanded form of $(z - 2a)^7$? [b]p16.[/b] Let $\ell$ and $m$ be lines with slopes $-2$, $1$ respectively. Compute $|s_1 \cdot s_2|$ if $s_1$, $s_2$ represent the slopes of the two distinct angle bisectors of $\ell$ and $m$. [b]p17.[/b] R1D2, Lord Byron, and Ryon are creatures from various planets. They are collecting monkeys for King Avanish, who only understands octal (base $8$). R1D2 only understands binary (base $2$), Lord Byron only understands quarternary (base $4$), and Ryon only understands decimal (base $10$). R1D2 says he has $101010101$ monkeys and adds his monkey to the pile. Lord Byron says he has $3231$ monkeys and adds them to the pile. Ryon says he has $576$ monkeys and adds them to the pile. If King Avanish says he has $x$ monkeys, what is the value of $x$? [b]p18.[/b] A quadrilateral is defined by the origin, $(3, 0)$, $(0, 10)$, and the vertex of the graph of $y = x^2 -8x+22$. What is the area of this quadrilateral? [b]p19.[/b] There is a sphere-container, filled to the brim with fruit punch, of diameter $6$. The contents of this container are poured into a rectangular prism container, again filled to the brim, of dimensions $2\pi$ by $4$ by $3$. However, there is an excess amount in the original container. If all the excess drink is poured into conical containers with diameter $4$ and height $3$, how many containers will be used? [b]p20.[/b] Brian is shooting arrows at a target, made of concurrent circles of radius $1$, $2$, $3$, and $4$. He gets $10$ points for hitting the innermost circle, $8$ for hitting between the smallest and second smallest circles, $5$ for between the second and third smallest circles, $2$ points for between the third smallest and outermost circle, and no points for missing the target. Assume for each shot he takes, there is a $20\%$ chance Brian will miss the target, but otherwise the chances of hitting each target are proportional to the area of the region. The chance that after three shots, Brian will have scored $15$ points can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p21.[/b] What is the largest possible integer value of $n$ such that $\frac{2n^3+n^2+7n-15}{2n+1}$ is an integer? [b]p22.[/b] Let $f(x, y) = x^3 + x^2y + xy^2 + y^3$. Compute $f(0, 2) + f(1, 3) +... f(9, 11).$ [b]p23.[/b] Let $\vartriangle ABC$ be a triangle. Let $AM$ be a median from $A$. Let the perpendicular bisector of segment $\overline{AM}$ meet $AB$ and $AC$ at $D$, $E$ respectively. Given that $AE = 7$, $ME = MC$, and $BDEC$ is cyclic, then compute $AM^2$. [b]p24.[/b] Compute the number of ordered triples of positive integers $(a, b, c)$ such that $a \le 10$, $b \le 11$, $c \le 12$ and $a > b - 1$ and $b > c - 1$. [b]p25.[/b] For a positive integer $n$, denote by $\sigma (n)$ the the sum of the positive integer divisors of $n$. Given that $n + \sigma (n)$ is odd, how many possible values of $n$ are there from $1$ to $2022$, inclusive? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b, c$ be nonzero integers, with $1$ as their only positive common divisor, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 0$. Find the number of such triples $(a, b, c)$ with $50 \ge |a| \ge |b| \ge |c| 1$.

Alice has an integer $N > 1$ on the blackboard. Each minute, she deletes the current number $x$ on the blackboard and writes $2x+1$ if $x$ is not the cube of an integer, or the cube root of $x$ otherwise. Prove that at some point of time, she writes a number larger than $10^{100}$. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

For which integers $n \ge 2$ can we arrange numbers $1,2, \ldots, n$ in a row, such that for all integers $1 \le k \le n$ the sum of the first $k$ numbers in the row is divisible by $k$?

Find all prime numbers p such that $2^p+p^2 $ is also a prime number.

Find all quadruples \((a,b,c,d)\) of positive integers such that \(\displaystyle \frac{ac+bd}{a+c}\) and \(\displaystyle \frac{bc-ad}{b-d}\) are equal to the prime number \(90121\).

The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$

A positive integer \(n\) is called perfect if the sum of its positive divisors \(\sigma(n)\) is twice \(n\), that is, \(\sigma(n) = 2n\). For example, \(6\) is a perfect number since the sum of its positive divisors is \(1 + 2 + 3 + 6 = 12\), which is twice \(6\). Prove that if \(n\) is a positive perfect integer, then: \[ \sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1} \] where the sums are taken over all prime divisors \(p\) of \(n\).

Find all ordered 6-tuples satisfy following system of modular equation: $ab+a'b' \equiv 1 $(mod 15) $bc+b'c' \equiv 1 $(mod 15) $ca+c'a' \equiv 1 $(mod 15) Given that $a,b,c,a',b',c' \epsilon (0;1;2;...;14)$

A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence? $ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $

Can a set of six numbers $\left\{a, b,c, \frac{a^2}{b} , \frac{b^2}{c} , \frac{c^2}{a} \right\}$ , where $a, b, c$ positive numbers, turn out to be exactly exactly three different numbers?

Find $$\sum^{100}_{i=1}i \gcd(i ,100).$$

Call a positive integer number $n$ [i]poor[/i] if equation \[x_1x_2 \dots x_{101}=(n-x_1)(n-x_2)\dots (n-x_{101}) \] has no solutions in positive integers $1<x_i<n$. Does there exist poor number, which has more than $100 \ 000$ distinct prime divisors?

We work in the decimal system and the following operations are allowed to be done with a positive integer: a) append $4$ at the end of the number. b) append $0$ at the end of the number. c) divide the number by $2$ if it's even. Show that starting with $4$, we can reach every positive integer by a finite number of these operations

Simplify the fraction $\frac{123456788...87654321}{1234567899...987654321}$’ if the digit $8$ in the numerator occurs $2000$ times, and the digit $9$ in the denominator $1999$ occurs times (as a result you need to get an irreducible fraction).

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

A sequence of integers $1 = x_1 < x_2 < x_3 <...$ satisfies $x_{n+1} \le 2n$ for all $n$. Show that every positive integer $k$ can be written as $x_j -x_i$ for some $i, j$.

Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$

Find all positive integers $n$ such that $\sum_{k=1}^{n}\frac{n}{k!}$ is an integer.