Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$. a) Prove that: For all $m_1,m_2\in \mathbb{Z^+}$, we have:$$k([m_1,m_2])=[k(m_1),k(m_2)].$$(Here $[a,b]$ is the least common multiple of $a,b.$) b) Determine $k(2),k(4),k(5),k(10).$

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

Let $a,b,c>0$ satisfy for all integers $n$, we have $$\lfloor an\rfloor+\lfloor bn\rfloor=\lfloor cn\rfloor$$Prove that at least one of $a,b,c$ is an integer.

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

Determine the (real) domain of a function $$y=\sqrt{1-\frac{x}{4}|x|+\sqrt{1-\frac{x}{2}|x|\,}\,}-\sqrt{1-\frac{x}{4}|x|-\sqrt{1-\frac{x}{2}|x|\,}\,}$$ and draw its graph.

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

Let $P, Q,R$ be three polynomials with real coefficients such that \[P(Q(x)) + P(R(x))=\text{constant}\] for all $x$. Prove that $P(x)=\text{constant}$ or $Q(x)+R(x)=\text{constant}$ for all $x$.

Determine all nonzero integers $a$ for which there exists two functions $f,g:\mathbb Q\to\mathbb Q$ such that \[f(x+g(y))=g(x)+f(y)+ay\text{ for all } x,y\in\mathbb Q.\] Also, determine all pairs of functions with this property. [i]Vasile Pop[/i]

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $ Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $

[u]Round 1[/u] [b]p1.[/b] How many ways are there to arrange the letters in the word $NEVERLAND$ such that the $2$ $N$’s are adjacent and the two $E$’s are adjacent? Assume that letters that appear the same are not distinct. [b]p2.[/b] In rectangle $ABCD$, $E$ and $F$ are on $AB$ and $CD$, respectively such that $DE = EF = FB$ and $\angle CDE = 45^o$. Find $AB + AD$ given that $AB$ and $AD$ are relatively prime positive integers. [b]p3.[/b] Maisy Airlines sees $n$ takeoffs per day. Find the minimum value of $n$ such that theremust exist two planes that take off within aminute of each other. [u]Round 2[/u] [b]p4.[/b] Nick is mixing two solutions. He has $100$ mL of a solution that is $30\%$ $X$ and $400$ mL of a solution that is $10\%$ $X$. If he combines the two, what percent $X$ is the final solution? [b]p5.[/b] Find the number of ordered pairs $(a,b)$, where $a$ and $b$ are positive integers, such that $$\frac{1}{a}+\frac{2}{b}=\frac{1}{12}.$$ [b]p6.[/b] $25$ balls are arranged in a $5$ by $5$ square. Four of the balls are randomly removed from the square. Given that the probability that the square can be rotated $180^o$ and still maintain the same configuration can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime, find $m+n$. [u]Round 3[/u] [b]p7.[/b] Maisy the ant is on corner $A$ of a $13\times 13\times 13$ box. She needs to get to the opposite corner called $B$. Maisy can only walk along the surface of the cube and takes the path that covers the least distance. Let $C$ and $D$ be the possible points where she turns on her path. Find $AC^2 + AD^2 +BC^2 +BD^2 - AB^2 -CD^2$. [b]p8.[/b] Maisyton has recently built $5$ intersections. Some intersections will get a park and some of those that get a park will also get a chess school. Find how many different ways this can happen. [b]p9.[/b] Let $f (x) = 2x -1$. Find the value of $x$ that minimizes $| f ( f ( f ( f ( f (x)))))-2020|$. [u]Round 4[/u] [b]p10.[/b] Triangle $ABC$ is isosceles, with $AB = BC > AC$. Let the angle bisector of $\angle A$ intersect side $\overline{BC}$ at point $D$, and let the altitude from $A$ intersect side $\overline{BC}$ at point $E$. If $\angle A = \angle C= x^o$, then the measure of $\angle DAE$ can be expressed as $(ax -b)^o$, for some constants $a$ and $b$. Find $ab$. [b]p11[/b]. Maisy randomly chooses $4$ integers $w$, $x$, $y$, and $z$, where $w, x, y, z \in \{1,2,3, ... ,2019,2020\}$. Given that the probability that $w^2 + x^2 + y^2 + z^2$ is not divisible by $4$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p12.[/b] Evaluate $$-\log_4 \left(\log_2 \left(\sqrt{\sqrt{\sqrt{...\sqrt{16}}}} \right)\right),$$ where there are $100$ square root signs. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

[b]p1. [/b](a) Show that for every positive integer $m > 1$, there are positive integers $x$ and $y$ such that $x^2 - y^2 = m^3$. (b) Find all pairs of positive integers $(x, y)$ such that $x^6 = y^2 + 127$. [b]p2.[/b] (a) Let $P(x)$ be a polynomial with integer coefficients. Suppose that $P(0)$ is an odd integer and that $P(1)$ is also an odd integer. Show that if $c$ is an integer then $P(c)$ is not equal to $0$. (b) Let P(x) be a polynomial with integer coefficients. Suppose that $P(1,000) = 1,000$ and $P(2,000) = 2,000.$ Explain why $P(3,000)$ cannot be equal to $1,000$. [b]p3.[/b] Triangle $\vartriangle ABC$ is created from points $A(0, 0)$, $B(1, 0)$ and $C(1/2, 2)$. Let $q, r$, and $s$ be numbers such that $0 < q < 1/2 < s < 1$, and $q < r < s$. Let D be the point on $AC$ which has $x$-coordinate $q$, $E$ be the point on AB which has $x$-coordinate $r$, and $F$ be the point on $BC$ that has $x$-coordinate $s$. (a) Find the area of triangle $\vartriangle DEF$ in terms of $q, r$, and $s$. (b) If $r = 1/2$, prove that at least one of the triangles $\vartriangle ADE$, $\vartriangle CDF$, or $\vartriangle BEF$ has an area of at least $1/4$. [b]p4.[/b] In the Gregorian calendar: (i) years not divisible by $4$ are common years, (ii) years divisible by $4$ but not by $100$ are leap years, (iii) years divisible by $100$ but not by $400$ are common years, (iv) years divisible by $400$ are leap years, (v) a leap year contains $366$ days, a common year $365$ days. From the information above: (a) Find the number of common years and leap years in $400$ consecutive Gregorian years. Show that $400$ consecutive Gregorian years consists of an integral number of weeks. (b) Prove that the probability that Christmas falls on a Wednesday is not equal to $1/7$. [b]p5.[/b] Each of the first $13$ letters of the alphabet is written on the back of a card and the $13$ cards are placed in a row in the order $$A,B,C,D,E, F, G,H, I, J,K, L,M$$ The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order $$B,M, A,H, G,C, F,E,D, L, I,K, J$$ What was the order of the letters after the cards were rearranged the first time? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

[u]Part 1[/u] You might think this round is broken after solving some of these problems, but everything is intentional. [b]1.1.[/b] The number $n$ can be represented uniquely as the sum of $6$ distinct positive integers. Find $n$. [b]1.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = 5$ and $AC^2 = 1188$. Find $AB$. [b]1.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a \$45 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 12$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $15 + 5k$ cents. The amount it made in sales for the previous month was $\$40.50$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]1.4.[/b] Let $A$ be the number of dierent ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 1$, $1 \times 3$, and $1 \times 6$. Let B be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 2$ and $1 \times 5$, where there are 2 different colors available for the $1 \times 2$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]1.5.[/b] An integer $n \ge 0$ is such that $n$ when represented in base $2$ is written the same way as $2n$ is in base $5$. Find $n$. [b]1.6.[/b] Let $x$ be a positive integer such that $3$, $ \log_6(12x)$, $\log_6(18x)$ form an arithmetic progression in some order. Find $x$. [u]Part 2[/u] Oops, it looks like there were some [i]intentional [/i] printing errors and some of the numbers from these problems got removed. Any $\blacksquare$ that you see was originally some positive integer, but now its value is no longer readable. Still, if things behave like they did for Part 1, maybe you can piece the answers together. [b]2.1.[/b] The number $n$ can be represented uniquely as the sum of $\blacksquare$ distinct positive integers. Find $n$. [b]2.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = \blacksquare$ and $AC^2 = 1536$. Find $AB$. [b]2.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a $\$50$ dollar budget to divide between production and advertising. If it spent k dollars on production, it could make $2k - 2$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $25 + 5k$ cents. The amount it made in sales for the previous month was $\$\blacksquare$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]2.4.[/b] Let $A$ be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times \blacksquare$, $1 \times \blacksquare$, and $1 \times \blacksquare$. Let $B$ be the number of different ways to tile a $1\times n$ rectangle with tiles of size $1 \times \blacksquare$ and $1 \times \blacksquare$, where there are $\blacksquare$ different colors available for the $1 \times \blacksquare$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]2.5.[/b] An integer $n \ge \blacksquare$ is such that $n$ when represented in base $9$ is written the same way as $2n$ is in base $\blacksquare$. Find $n$. [b]2.6.[/b] Let $x$ be a positive integer such that $1$, $\log_{96}(6x)$, $\log_{96}(\blacksquare x)$ form an arithmetic progression in some order. Find $x$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

Solve in real numbers the system $$\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}$$

For a given real number $a$ and a positive integer $n$, prove that: i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that \[\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}\] ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$. [i]Liang Yengde[/i]

Prove that if the numbers $ p $, $ q $, $ r $ satisfy the equality $$ p+q + r=1$$ $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0$$ then for any numbers $ a $, $ b $, $ c $ equality holds $$a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.$$

Let $x$, $y$, and $z$ be positive real numbers. Prove that $\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$. When does equality hold?

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]