How many integers $n$, satisfy $|n|<2020$ and the equation $11^3|n^3+3n^2-107n+1$ (A)$0$ (B)$101$ (C)$367$ (D)$368$

[i]Palindromic partitioning [/i] of the natural number $ A $ is called, when $ A $ is written as the sum of natural the terms $ A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n $ ($ n \geq 1 $), in which $ a_1 = a_n , a_2 = a_ {n-1} $ and in general, $ a_i = a_ {n + 1 - i} $ with $ 1 \leq i \leq n $. For example, $ 16 = 16 $, $ 16 = 2 + 12 + 2 $ and $ 16 = 7 + 1 + 1 + 7 $ are [i]palindromic partitions[/i] of the number $16$. Find the number of all [i]palindromic partitions[/i] of the number $2006$.

Given $f_1(x)=2x-2$ and, for $k\ge2$, defined $f_k(x)=f(f_{k-1}(x))$ to be a real-valued function of $x$. Find the remainder when $f_{2013}(2012)$ is divided by the prime $2011$.

Let \( \mathbb{Z^{+}} \) denote the set of all positive integers. Find all functions \( f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} \) such that for every pair of positive integers \( a \) and \( b \), there exists a positive integer \( c \) satisfying: $$ f(a)f(b) - ab = 2^{c-1} - 1. $$ Proposed by Matin Yousefi

Bob plays a game on the whiteboard. Initially, the numbers $\{1, 2, ...,n\}$ are shown. On each turn, Bob takes two numbers from the board $x$, $y$, erases them both, and writes down $2x + y$ onto the board. In terms of n, what is the maximum possible value that Bob can end up with?

Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

a) Let $f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3)$. Prove that if for some positive integers $a, b$ the number $f(a, b)$ is a cube of an integer then $f(a, b)$ is also a square of an integer. b) Are there infinitely many pairs of positive integers $(a, b)$ for which $f(a, b)$ is a square but not a cube ?

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

Let $k$ be a fixed positive integer. For each integer $1 \leq i \leq 4$, let $x_i$ and $y_i$ be positive integers such that their least common multiple is $k$. Suppose that the four points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$ are the vertices of a non-degenerate rectangle in the Cartesian plane. Prove that $x_1x_2x_3x_4$ is a perfect square. [i]Andrei Chirita[/i]

Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that \[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \] if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$. (Dan Schwarz)

[u]Round 6[/u] [b]p16.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $CA = 5$. There exist two possible points $X$ on $CA$ such that if $Y$ and $Z$ are the feet of the perpendiculars from $X$ to $AB$ and $BC,$ respectively, then the area of triangle $XY Z$ is $1$. If the distance between those two possible points can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $gcd(a, c) = 1$, then find $a +b+ c$. [b]p17.[/b] Let $f(n)$ be the number of orderings of $1,2, ... ,n$ such that each number is as most twice the number preceding it. Find the number of integers $k$ between $1$ and $50$, inclusive, such that $f (k)$ is a perfect square. [b]p18.[/b] Suppose that $f$ is a function on the positive integers such that $f(p) = p$ for any prime p, and that $f (xy) = f(x) + f(y)$ for any positive integers $x$ and $y$. Define $g(n) = \sum_{k|n} f (k)$; that is, $g(n)$ is the sum of all $f(k)$ such that $k$ is a factor of $n$. For example, $g(6) = f(1) + 1(2) + f(3) + f(6)$. Find the sum of all composite $n$ between $50$ and $100$, inclusive, such that $g(n) = n$. [u]Round 7[/u] [b]p19.[/b] AJ is standing in the center of an equilateral triangle with vertices labelled $A$, $B$, and $C$. They begin by moving to one of the vertices and recording its label; afterwards, each minute, they move to a different vertex and record its label. Suppose that they record $21$ labels in total, including the initial one. Find the number of distinct possible ordered triples $(a, b, c)$, where a is the number of $A$'s they recorded, b is the number of $B$'s they recorded, and c is the number of $C$'s they recorded. [b]p20.[/b] Let $S = \sum_{n=1}^{\infty} (1- \{(2 + \sqrt3)^n\})$, where $\{x\} = x - \lfloor x\rfloor$ , the fractional part of $x$. If $S =\frac{\sqrt{a} -b}{c}$ for positive integers $a, b, c$ with $a $ squarefree, find $a + b + c$. [b]p21.[/b] Misaka likes coloring. For each square of a $1\times 8$ grid, she flips a fair coin and colors in the square if it lands on heads. Afterwards, Misaka places as many $1 \times 2$ dominos on the grid as possible such that both parts of each domino lie on uncolored squares and no dominos overlap. Given that the expected number of dominos that she places can be written as $\frac{a}{b}$, for positive integers $a$ and $b$ with $gcd(a, b) = 1$, find $a + b$. PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 4-5 [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that, we find infinite numbers such that, this number writeable $1999k+1$ for $k \in {\mathbb N}$ and all digits are equal.

Richard and Kaarel are taking turns to choose numbers from the set $\{1,\dots,p-1\}$ where $p > 3$ is a prime. Richard is the first one to choose. A number which has been chosen by one of the players cannot be chosen again by either of the players. Every number chosen by Richard is multiplied with the next number chosen by Kaarel. Kaarel wins the game if at any moment after his turn the sum of all of the products calculated so far is divisible by $p$. Richard wins if this does not happen, i.e. the players run out of numbers before any of the sums is divisible by $p$. Can either of the players guarantee their victory regardless of their opponent's moves and if so, which one?

Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.

Find the number of solutions to the equation: $6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $ With {x} we denote the fractional part of the number x.

The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers. [i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

The sum of the reciprocals of three positive integers is equal to $1$. What are all the possible such triples?

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

[b]p1.[/b] Roll two dice. What is the probability that the sum of the rolls is prime? [b]p2. [/b]Compute the sum of the first $20$ squares. [b]p3.[/b] How many integers between $0$ and $999$ are not divisible by $7, 11$, or $13$? [b]p4.[/b] Compute the number of ways to make $50$ cents using only pennies, nickels, dimes, and quarters. [b]p5.[/b] A rectangular prism has side lengths $1, 1$, and $2$. What is the product of the lengths of all of the diagonals? [b]p6.[/b] What is the last digit of $7^{6^{5^{4^{3^{2^1}}}}}$ ? [b]p7.[/b] Given square $ABCD$ with side length $3$, we construct two regular hexagons on sides $AB$ and $CD$ such that the hexagons contain the square. What is the area of the intersection of the two hexagons? [img]https://cdn.artofproblemsolving.com/attachments/f/c/b2b010cdd0a270bc10c6e3bb3f450ba20a03e7.png[/img] [b]p8.[/b] Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of $10$ miles per hour per minute until she reaches the speed limit of $25$ miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of $20$ miles per hour per minute until he reaches the rate of $40$ miles per hour. If the police officer catches up to Brooke after 3 minutes, how fast was Brooke driving initially? [b]p9.[/b] Find the ordered pair of positive integers $(x, y)$ such that $144x - 89y = 1$ and $x$ is minimal. [b]p10.[/b] How many zeroes does the product of the positive factors of $10000$ (including $1$ and $10000$) have? [b]p11.[/b] There is a square configuration of desks. It is known that one can rearrange these desks such that it has $7$ fewer rows but $10$ more columns, with $13$ desks remaining. How many desks are there in the square configuration? [b]p12.[/b] Given that there are $168$ primes with $3$ digits or less, how many numbers between $1$ and $1000$ inclusive have a prime number of factors? [b]p13.[/b] In the diagram below, we can place the integers from $1$ to $19$ exactly once such that the sum of the entries in each row, in any direction and of any size, is the same. This is called the magic sum. It is known that such a configuration exists. Compute the magic sum. [img]https://cdn.artofproblemsolving.com/attachments/3/4/7efaa5ba5ad250e24e5ad7ef03addbf76bcfb4.png[/img] [b]p14.[/b] Let $E$ be a random point inside rectangle $ABCD$ with side lengths $AB = 2$ and $BC = 1$. What is the probability that angles $ABE$ and $CDE$ are both obtuse? [b]p15.[/b] Draw all of the diagonals of a regular $13$-gon. Given that no three diagonals meet at points other than the vertices of the $13$-gon, how many intersection points lie strictly inside the $13$-gon? [b]p16.[/b] A box of pencils costs the same as $11$ erasers and $7$ pencils. A box of erasers costs the same as $6$ erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of pencils and a box of erasers combined? [b]p17.[/b] In the following figure, all angles are right angles and all sides have length $1$. Determine the area of the region in the same plane that is at most a distance of $1/2$ away from the perimeter of the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f53ae3b802618703f04f41546e3990a7d0640e.png[/img] [b]p18.[/b] Given that $468751 = 5^8 + 5^7 + 1$ is a product of two primes, find both of them. [b]p19.[/b] Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color? [b]p20.[/b] Consider a cylinder with height $20$ and radius $14$. Inside the cylinder, we construct two right cones also with height $20$ and radius $14$, such that the two cones share the two bases of the cylinder respectively. What is the volume ratio of the intersection of the two cones and the union of the two cones? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

Define the sequence $\{a_n\}_{n=1}^\infty$ as \[ a_1 = a_2 = 1,\quad a_{n+2} = 14a_{n+1} - a_n \; (n \geq 1) \] Prove that if $p$ is prime and there exists a positive integer $n$ such that $\frac{a_n}p$ is an integer, then $\frac{p-1}{12}$ is also an integer.

[u]Round 1[/u] [b]1.1[/b] If the sum of two non-zero integers is $28$, then find the largest possible ratio of these integers. [b]1.2[/b] If Tom rolls a eight-sided die where the numbers $1$ − $8$ are all on a side, let $\frac{m}{n}$ be the probability that the number is a factor of $16$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]1.3[/b] The average score of $35$ second graders on an IQ test was $180$ while the average score of $70$ adults was $90$. What was the total average IQ score of the adults and kids combined? [u]Round 2[/u] [b]2.1[/b] So far this year, Bob has gotten a $95$ and a 98 in Term $1$ and Term $2$. How many different pairs of Term $3$ and Term $4$ grades can Bob get such that he finishes with an average of $97$ for the whole year? Bob can only get integer grades between $0$ and $100$, inclusive. [b]2.2[/b] If a complement of an angle $M$ is one-third the measure of its supplement, then what would be the measure (in degrees) of the third angle of an isosceles triangle in which two of its angles were equal to the measure of angle $M$? [b]2.3[/b] The distinct symbols $\heartsuit, \diamondsuit, \clubsuit$ and $\spadesuit$ each correlate to one of $+, -, \times , \div$, not necessarily in that given order. Given that $$((((72 \,\, \,\, \diamondsuit \,\, \,\,36) \,\, \,\,\spadesuit \,\, \,\,0 ) \,\, \,\, \diamondsuit \,\, \,\, 32) \,\, \,\, \clubsuit \,\, \,\, 3)\,\, \,\, \heartsuit \,\, \,\, 2 = \,\, \,\, 6,$$ what is the value of $$(((((64 \,\, \,\, \spadesuit \,\, \,\, 8) \heartsuit \,\, \,\, 6) \,\, \,\, \spadesuit \,\, \,\, 5) \,\, \,\, \heartsuit \,\, \,\, 1) \,\, \,\, \clubsuit \,\, \,\, 7) \,\, \,\, \diamondsuit \,\, \,\, 1?$$ [u]Round 3[/u] [b]3.1[/b] How many ways can $5$ bunnies be chosen from $7$ male bunnies and $9$ female bunnies if a majority of female bunnies is required? All bunnies are distinct from each other. [b]3.2[/b] If the product of the LCM and GCD of two positive integers is $2021$, what is the product of the two positive integers? [b]3.3[/b] The month of April in ABMC-land is $50$ days long. In this month, on $44\%$ of the days it rained, and on $28\%$ of the days it was sunny. On half of the days it was sunny, it rained as well. The rest of the days were cloudy. How many days were cloudy in April in ABMC-land? [u]Round 4[/u] [b]4.1[/b] In how many ways can $4$ distinct dice be rolled such that a sum of $10$ is produced? [b]4.2[/b] If $p, q, r$ are positive integers such that $p^3\sqrt{q}r^2 = 50$, find the sum of all possible values of $pqr$. [b]4.3[/b] Given that numbers $a, b, c$ satisfy $a + b + c = 0$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}= 10$, and $ab + bc + ac \ne 0$, compute the value of $\frac{-a^2 - b^2 - a^2}{ab + bc + ac}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2826137p24988781]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].