Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$

a) Prove that one cannot assign to each vertex of a cube $ 8$ distinct numbers from the set $\{0, 1, 2, 3, . . . , 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is even. b) Prove that one can assign to each vertex of a cube $8$ distinct numbers from the set $\{0, 1, 2, 3, . . . , 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is divisible by $3$.

Let $n$ ($n \ge 1$) be an integer. Consider the equation $2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$, where $x$ is the unknown real variable. (a) Solve the equation for $n = 8$. (b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions. (For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$.)

The integers greater than one are arranged in five columns as follows: \[ \begin{tabular}{c c c c c} \ & 2 & 3 & 4 & 5 \\ 9 & 8 & 7 & 6 & \ \\ \ & 10 & 11 & 12 & 13 \\ 17 & 16 & 15 & 14 & \ \\ \ & . & . & . & . \\ \end{tabular} \] (Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.) In which column will the number $1,000$ fall? $ \textbf{(A)}\ \text{first} \qquad\textbf{(B)}\ \text{second} \qquad\textbf{(C)}\ \text{third} \qquad\textbf{(D)}\ \text{fourth} \qquad\textbf{(E)}\ \text{fifth} $

Let $a$ and $b$ be positive integers with the property that $\frac{a}{b} > \sqrt2$. Prove that $$\frac{a}{b} - \frac{1}{2ab} > \sqrt2$$

Let $F$ be an intersection point of altitude $CD$ and internal angle bisector $AE$ of right angled triangle $ABC$, $\angle ACB = 90^{\circ}$. Let $G$ be an intersection point of lines $ED$ and $BF$. Prove that area of quadrilateral $CEFG$ is equal to area of triangle $BDG$

Prove that the number of triples $(A,B,C)$ where $A, B, C$ are subsets of $\{ 1,2, \ldots , n \}$ such that $A \cap B \cap C = \phi$ , $A \cap B \not= \phi$, $C \cap B \not= \phi$ is $7^n - 2 . 6^n + 5^n$.

The faces of a cube contain the number 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, we multiply the three numbers on the faces incident to that corner, and write down its value. (In the diagram, the value of the indicated corner is 1 x 2 x 3 = 6.) What is the sum of the eight values assigned to the cube’s corners?

[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].

On the extension of the angle bisector $AL$ of the triangle $ABC$, a point $P$ is placed such that $P L = AL$. Prove that the perimeter of triangle $PBC$ does not exceed the perimeter of triangle $ABC$.

Let $ a,\ b$ be real constants. Find the minimum value of the definite integral: $ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$

Let $n\geqslant 3$ be an integer and $x_1>x_2>\cdots>x_n$ be real numbers. Suppose that $x_k>0\geqslant x_{k+1}$ for an index $k{}$. Prove that \[\sum_{i=1}^k\left(x_i^{n-2}\prod_{j\neq i}\frac{1}{x_i-x_j}\right)\geqslant 0.\]

The mean of a set of $k$ consecutive integers is $k$. If the smallest integer is $23$, what is the largest integer?

For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1, s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \neq s_2$, and $s_1 + s_2 = n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$ for all $n$?

Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Fix integers $m \geq 3$ and $n \geq 3$. Each cell of an array with $m$ rows and $n$ columns is coloured one of two colours such that: [b](1)[/b] Both colours occur on every column; and [b](2)[/b] On every two rows the cells on the same column share colour on exactly $k$ columns. Show that, if $m$ is odd, then \[\frac{n(m-1)}{2m}\leq k\leq \frac{n(m-2)}{m}\] [i]The Problem Selection Committee[/i]

Is it possible to place $1995$ different natural numbers along a circle so that for any two of these numbers, the ratio of the greatest to the least is a prime? I feel that my solution's wording and notation is awkward (and perhaps unnecessarily complicated), so please feel free to critique it: [hide] Suppose that we do have such a configuration $a_{1},a_{2},...a_{1995}$. WLOG, $a_{2}=p_{1}a_{1}$. Then \[\frac{a_{2}}{a_{3}}= p_{2}, \frac{1}{p_{2}}\] \[\frac{a_{3}}{a_{4}}= p_{3}, \frac{1}{p_{3}}\] \[... \] \[\frac{a_{1995}}{a_{1}}= p_{1995}, \frac{1}{p_{1995}}\] Multiplying these all together, \[\frac{a_{2}}{a_{1}}= \frac{\prod p_{k}}{\prod p_{j}}= p_{1}\] Where $\prod p_{k}$ is some product of the elements in a subset of $\{ p_{2},p_{3}, ...p_{1995}\}$. We clear denominators to get \[p_{1}\prod p_{j}= \prod p_{k}\] Now, by unique prime factorization, the set $\{ p_{j}\}\cup \{ p_{1}\}$ is equal to the set $\{ p_{k}\}$. However, since there are a total of $1995$ primes, this is impossible. We conclude that no such configuration exists. [/hide]

Find the locus of centers of regular triangles such that three given points $A, B, C$ lie respectively on three lines containing sides of the triangle.

Suppose $A_1,A_2,A_3,B_1,B_2,B_3$ are points in the plane such that for each $i,j\in\{1,2,3\}$ it holds that $A_iB_j=i+j$. What can be said about these six points?

For any positive integer $n$ let $n! = 1\times 2\times 3\times ... \times n$. Do there exist infinitely many triples $(p, q, r)$, of positive integers with $p > q > r > 1$ such that the product $p! \cdot q! \cdot r!$$ is a perfect square?

We have a $9$ by $9$ chessboard with $9$ kings (which can move to any of $8$ adjacent squares) in the bottom row. What is the minimum number of moves, if two pieces cannot occupy the same square at the same time, to move all the kings into an $X$ shape (a $5\times5$ region where there are $5$ kings along each diagonal of the $X$, as shown below)? \begin{tabular}{ c c c c c } O & & & & O \\ & O & & O & \\ & & O & & \\ & O & & O & \\ O & & & & O \\ \end{tabular} [i]Proposed by David Tang[/i]

Let $ABCD$ be a given convex quadrilateral in a plane. Prove that there exist a line with four different points $P,Q,R,S$ on it and a square $A’B’C’D’$ such that $P$ lies on both line $AB$ and $A’B’,$ $Q$ lies on both line $BC$ and $B’C’,$ $R$ lies on both line $CD$ and $C’D’,$ $S$ lies on both line $DA$ and $D’A’.$

Let $ ABC $ be a right triangle in $ A, $ and let be a point $ D $ on $ BC. $ The bisectors of $ \angle ADB $ and $ \angle ADC $ intersect $ AB $ and $ AC $ (respectively) in $ M $ and $ N $ (respectively). Show that the small angle between $ BC $ and $ MN $ is equal to $ \frac{1}{2}\cdot\left| \angle ABC -\angle BCA \right| $ if and only if $ D $ is the feet of the perpendicular from $ A. $ [i]Bogdan Enescu[/i]