A [i]self-avoiding rook walk[/i] on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each begins where the previous move ended and such that no move ever crosses a square that has previously been crossed, i.e., the rook's path is non-self-intersecting. Let $ R(m, n)$ be the number of self-avoiding rook walks on an $ m \times n$ ($ m$ rows, $ n$ columns) chessboard which begin at the lower-left corner and end at the upper-left corner. For example, $ R(m, 1) \equal{} 1$ for all natural numbers $ m$; $ R(2, 2) \equal{} 2$; $ R(3, 2) \equal{} 4$; $ R(3, 3) \equal{} 11$. Find a formula for $ R(3, n)$ for each natural number $ n$.

Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that \[f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.\]

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the question from a) still true if $AB\neq BA$ ?

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Find the value of the sum \[ \left\lfloor 2+\frac{1}{2^{2021}} \right\rfloor+\left\lfloor 2+\frac{1}{2^{2020}} \right\rfloor+\cdots+\left\lfloor 2+\frac{1}{2^1} \right\rfloor+\left\lfloor 2+\frac{1}{2^0} \right\rfloor. \]

Find all quadruples $(a, b, c, d)$ of real numbers such that \[a + bcd = b + cda = c + dab = d + abc.\]

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

For 3 real numbers $a,b,c$ let $s_n=a^{n}+b^{n}+c^{n}$. It is known that $s_1=2$, $s_2=6$ and $s_3=14$. Prove that for all natural numbers $n>1$, we have $|s^2_n-s_{n-1}s_{n+1}|=8$

Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$

Solve the following equation where $x$ is a real number: $\lfloor x^2 \rfloor -10\lfloor x \rfloor + 24 = 0$

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

The discriminant of the equation $ x^2\plus{}2x\sqrt{3}\plus{}3\equal{}0$ is zero. Hence, its roots are: $ \textbf{(A)}\ \text{real and equal} \qquad \textbf{(B)}\ \text{rational and equal} \qquad \textbf{(C)}\ \text{rational and unequal} \\ \textbf{(D)}\ \text{irrational and unequal} \qquad \textbf{(E)}\ \text{imaginary}$

Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros. (i) Prove that $$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$ (ii) Show that, for each $n > 1$, both inequalities can hold as equalities.

Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$? [i]By Omid Hatami[/i]

Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x)=2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?" After some calculations, Jon says, "There is more than one such polynomial." Steve says, "You’re right. Here is the value of $a$." He writes down a positive integer and asks, "Can you tell me the value of $c$?" Jon says, "There are still two possible values of $c$." Find the sum of the two possible values of $c$.

A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$

Let $a,b,c$ real numbers such that $a^n+b^n= c^n$ for three positive integers consecutive of $n$. Prove that $abc= 0$

Prove that there exists a unique polynomial function f with positive integer coefficients such that $f(1) = 6$ and $f(2) = 2016$.

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]

Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$ Find the sum of all the numbers Pedro wrote.

[u]Round 5[/u] [b]p13.[/b] Find the number of six-digit positive integers that satisfy all of the following conditions: (i) Each digit does not exceed $3$. (ii) The number $1$ cannot appear in two consecutive digits. (iii) The number $2$ cannot appear in two consecutive digits. [b]p14.[/b] Find the sum of all distinct prime factors of $103040301$. [b]p15.[/b] Let $ABCA'B'C'$ be a triangular prism with height $3$ where bases $ABC$ and $A'B'C'$ are equilateral triangles with side length $\sqrt6$. Points $P$ and $Q$ lie inside the prism so that $ABCP$ and $A'B'C'Q$ are regular tetrahedra. The volume of the intersection of these two tetrahedra can be expressed in the form $\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers and $m$ is not divisible by the square of any prime. Find $m + n$. [u]Round 6[/u] [b]p16.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a^2_n -a_{n-1}a_{n+1} = a_n -a_{n-1}$ for all positive integers $n$. Given that $a_0 = 1$ and $a_1 = 4$, compute the smallest positive integer $k$ such that $a_k$ is an integer multiple of $220$. [b]p17.[/b] Vincent the Bug is on an infinitely long number line. Every minute, he jumps either $2$ units to the right with probability $\frac23$ or $3$ units to the right with probability $\frac13$ . The probability that Vincent never lands exactly $15$ units from where he started can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$? [b]p18.[/b] Battler and Beatrice are playing the “Octopus Game.” There are $2022$ boxes lined up in a row, and inside one of the boxes is an octopus. Beatrice knows the location of the octopus, but Battler does not. Each turn, Battler guesses one of the boxes, and Beatrice reveals whether or not the octopus is contained in that box at that time. Between turns, the octopus teleports to an adjacent box and secretly communicates to Beatrice where it teleported to. Find the least positive integer $B$ such that Battler has a strategy to guarantee that he chooses the box containing the octopus in at most $B$ guesses. [u]Round 7[/u] [b]p19.[/b] Given that $f(x) = x^2-2$ the number $f(f(f(f(f(f(f(2.5)))))))$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find the greatest positive integer $n$ such that $2^n$ divides $ab+a+b-1$. [b]p20.[/b] In triangle $ABC$, the shortest distance between a point on the $A$-excircle $\omega$ and a point on the $B$-excircle $\Omega$ is $2$. Given that $AB = 5$, the sum of the circumferences of $\omega$ and $\Omega$ can be written in the form $\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Note: The $A$-excircle is defined to be the circle outside triangle $ABC$ that is tangent to the rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and to the side $ BC$. The $B$-excircle is defined similarly for vertex $B$.) [b]p21.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a_0 = 1$, $a_1 = 1$, and there exists two fixed integer constants $x$ and $y$ for which $a_{n+2}$ is the remainder when $xa_{n+1}+ya_n$ is divided by $15$ for all nonnegative integers $n$. Let $t$ be the least positive integer such that $a_t = 1$ and $a_{t+1} = 1$ if such an integer exists, and let $t = 0$ if such an integer does not exist. Find the maximal value of t over all possible ordered pairs $(x, y)$. [u]Round 8[/u] [b]p22.[/b] A mystic square is a $3$ by $3$ grid of distinct positive integers such that the least common multiples of the numbers in each row and column are the same. Let M be the least possible maximal element in a mystic square and let $N$ be the number of mystic squares with $M$ as their maximal element. Find $M + N$. [b]p23.[/b] In triangle $ABC$, $AB = 27$, $BC = 23$, and $CA = 34$. Let $X$ and $Y$ be points on sides $ AB$ and $AC$, respectively, such that $BX = 16$ and $CY = 7$. Given that $O$ is the circumcenter of $BXY$ , the value of $CO^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p24.[/b] Alan rolls ten standard fair six-sided dice, and multiplies together the ten numbers he obtains. Given that the probability that Alan’s result is a perfect square is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, compute $a$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949416p26408251]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$ The sum $|x_1| + |x_2| + |x_3|$ is (A): $4$ (B): $6$ (C): $8$ (D): $14$ (E): None of the above.

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[xf(x+f(y))=(y-x)f(f(x)).\] [i]Proposed by Nikola Velov, Macedonia[/i]

It is known that for irrational numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and for any positive integer $n$ the following is true: $$[n\alpha]+[n\beta]=[n\gamma]+[n\delta]$$ Does this mean that sets $\{\alpha,\beta\}$ and $\{\gamma,\delta\}$ are equal? (As usual $[x]$ means the greatest integer not greater than $x$).