A finite sequence of integers $a_1, a_2, \dots, a_n$ is called [i]regular[/i] if there exists a real number $x$ satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is [i]forced[/i] if the following condition is satisfied: the sequence \[ a_1, a_2, \dots, a_{k-1}, b \] is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.

[b]p1.[/b] What is the value of the sum $2 + 20 + 202 + 2022$? [b]p2.[/b] Find the smallest integer greater than $10000$ that is divisible by $12$. [b]p3.[/b] Valencia chooses a positive integer factor of $6^{10}$ at random. The probability that it is odd can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$. [b]p4.[/b] How many three digit positive integers are multiples of $4$ but not $8$? [b]p5.[/b] At the Jane Street store, Andy accidentally buys $5$ dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining $90$ dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller, $12.5\%$ tip so that he still spends $90$ dollars total. How much percent tip was Andy originally planning on giving? [b]p6.[/b] Let $A,B,C,D$ be four coplanar points satisfying the conditions $AB = 16$, $AC = BC =10$, and $AD = BD = 17$. What is the minimum possible area of quadrilateral $ADBC$? [b]p7.[/b] How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to $30^o$? [b]p8.[/b] Jaeyong rolls five fair $6$-sided die. The probability that the sum of some three rolls is exactly $8$ times the sum of the other two rolls can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p9.[/b] Find the least positive integer n for there exists some positive integer $k > 1$ for which $k$ and $k + 2$ both divide $\underbrace{11...1}_{n\,\,\,1's}$. [b]p10.[/b] For some real constant $k$, line $y = k$ intersects the curve $y = |x^4-1|$ four times: points $A$,$B$,$C$ and $D$, labeled from left to right. If $BC = 2AB = 2CD$, then the value of $k$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Let a be a positive real number and $P(x) = x^2 -8x+a$ and $Q(x) = x^2 -8x+a+1$ be quadratics with real roots such that the positive difference of the roots of $P(x)$ is exactly one more than the positive difference of the roots of $Q(x)$. The value of a can be written as a common fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p12.[/b] Let $ABCD$ be a trapezoid satisfying $AB \parallel CD$, $AB = 3$, $CD = 4$, with area $35$. Given $AC$ and $BD$ intersect at $E$, and $M$, $N$, $P$, $Q$ are the midpoints of segments $AE$,$BE$,$CE$,$DE$, respectively, the area of the intersection of quadrilaterals $ABPQ$ and $CDMN$ can be expressed as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p13.[/b] There are $8$ distinct points $P_1, P_2, ... , P_8$ on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint? [b]p14.[/b] For every positive integer $k$, let $f(k) > 1$ be defined as the smallest positive integer for which $f(k)$ and $f(k)^2$ leave the same remainder when divided by $k$. The minimum possible value of $\frac{1}{x}f(x)$ across all positive integers $x \le 1000$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p15.[/b] In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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).

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$: \begin{align*} &(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\ =&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\ &\qquad \qquad \qquad \qquad \vdots \\ =&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1}) \end{align*} has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.

Determine all sequences of non-negative integers $a_1, \ldots, a_{2016}$ all less than or equal to $2016$ satisfying $i+j\mid ia_i+ja_j$ for all $i, j\in \{ 1,2,\ldots, 2016\}$.

Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coefficients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.

The function $f(x)$, defined on the entire real line, is known but that for any $a > 1 $ the function $f(x)+f(ax)$ is continuous on the entire line. Prove that $f(x)$ is also continuous along the entire line.

Let \( P(x), Q(x) \) be non-constant real polynomials, such that for all positive integer \( m \), there exists a positive integer \( n \) satisfy \( P(m) = Q(n) \). Prove that (1) If \(\deg Q \mid \deg P\), then there exists real polynomial \( h(x) \) \( x \), satisfy \( P(x) = Q(h(x)) \) holds for all real number $x.$ (2) \(\deg Q \mid \deg P\).

Let $A$ be the set of natural numbers $n$ such that the distance of the real number $n\sqrt{2022} - \frac13$ from the nearest integer is at most $\frac1{2022}$. Show that the equation $$20x + 21y = 22z$$ has no solutions over the set $A$.

Real numbers $a,b,c,d$ satisfy the equality $$\frac{1-ab}{a+b}=\frac{bc-1}{b+c}=\frac{1-cd}{c+d}=\sqrt{3}$$ Find all possible values of $ad$.

Let $a_{1},a_{2},\ldots$ be an infinite arithmetic progression. It's known that there exist positive integers $p,q,t$ such that $a_{p}+tp=a_{q}+tq$. If $a_{t}=t$ and the sum of the first $t$ numbers in the sequence is $18$, determine $a_{2008}$.

Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ : $$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$

Prove that $$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$ for any natural number $ n. $ [i]Mihai Piticari[/i]

How many triples $(a,b,c)$ with $a,b,c \in \mathbb{R}$ satisfy the following system? $$\begin{cases} a^4-b^4=c \\ b^4-c^4=a \\ c^4-a^4=b \end{cases}$$.

It is given the equation $x^4-2ax^3+a(a+1)x^2-2ax+a^2=0$. a) Find the greatest value of $a$, such that this equation has at least one real root. b) Find all the values of $a$, such that the equation has at least one real root.

You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

Prove that there is no function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(f(n))=n+1.$ Here $\mathbb{N}$ is the positive integers $\{1,2,3,\dots\}.$

Given a monic quadratic polynomial $Q(x)$, define \[ Q_n (x) = \underbrace{Q(Q(\cdots(Q(x))\cdots))}_{\text{compose $n$ times}} \] for every natural number $n$. Let $a_n$ be the minimum value of the polynomial $Q_n(x)$ for every natural number $n$. It is known that $a_n > 0$ for every natural number $n$ and there exists some natural number $k$ such that $a_k \neq a_{k+1}$. (a) Prove that $a_n < a_{n+1}$ for every natural number $n$. (b) Is it possible to satisfy $a_n < 2021$ for every natural number $n$? [i]Proposed by Fajar Yuliawan[/i]

Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.

Prove that if the equations $x^3+mx-n = 0$ $nx^3-2m^2x^2 -5mnx-2m^3-n^2 = 0$ have one root in common ($n \ne 0$), then the first equation has two equal roots, and find the roots of the equations in terms of $n$.

Let $x, y$, and $z$ be three not necessarily real numbers that satisfy the following system of equations: $x^3 -4 = (2y +1)^2$ $y^3 -4 = (2z +1)^2$ $z^3 -4 = (2x +1)^2$. Find the greatest possible real value of $(x -1)(y -1)(z -1)$.

Find the number of triples of nonnegative integers $(x,y,z)$ such that \[x^2+2xy+y^2-z^2=9.\]

Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that $Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) $ for some complex number $w$ with $||w||=\frac{\sqrt3}{2}$ , $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.

Let be an infinite sequence $ \left( c_n \right)_{n\ge 1} $ of positive real numbers, with $ c_1=1, $ and satisfying $$ c_{n+1}-\frac{1}{c_{n+1}} =c_n+\frac{1}{c_n} , $$ for all natural numbers $ n. $ Prove that: [b]a)[/b] there exists a natural number $ k $ such that the sequence $ \left( c_n^k+\frac{1}{c_n^k} \right)_{n\ge 1} $ is an arithmetic one. [b]b)[/b] there exist two sequences $ \left( u_n \right)_{n\ge 1} ,\left( v_n \right)_{n\ge 1} $ of nonegative integers such that $ c_n=\sqrt{u_n} +\sqrt{v_n} , $ for any natural number $ n. $