The Fibonacci sequence is defined by $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}1} \equal{} F_n \plus{}F_{n\minus{}1}$ for $ n > 1$. Let $ f(x) \equal{} 1985x^2 \plus{} 1956x \plus{} 1960$. Prove that there exist infinitely many natural numbers $ n$ for which $ f(F_n)$ is divisible by $ 1989$. Does there exist $ n$ for which $ f(F_n) \plus{} 2$ is divisible by $ 1989$?

Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

Prove that for all primes $p$ true is equality $$\sum_{k=1}^{p-1}\left\lfloor\frac{k^3}{p}\right\rfloor=\frac{(p-2)(p-1)(p+1)}{4}$$

Determine the increasing geometric progressions, with three integer terms, such that the sum of these terms is $57$

Given a finite set $X$, two positive integers $n,k$, and a map $f:X\to X$. Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$,$i=1,2,3,\ldots$. It is known that for any $x\in X$,$f^{(n)}(x)=x$. Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$. Prove that: (1)$\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$ (2)$\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.

[b]p1.[/b] Prove that for every point inside a regular polygon, the average of the distances to the sides equals the radius of the inscribed circle. The distance to a side means the shortest distance from the point to the line obtained by extending the side. [b]p2.[/b] Suppose we are given positive (not necessarily distinct) integers $a_1, a_2,..., a_{1997}$ . Show that it is possible to choose some numbers from this list such that their sum is a multiple of $1997$. [b]p3.[/b] You have Blue blocks, Green blocks and Red blocks. Blue blocks and green blocks are $2$ inches thick. Red blocks are $1$ inch thick. In how many ways can you stack the blocks into a vertical column that is exactly $12$ inches high? (For example, for height $3$ there are $5$ ways: RRR, RG, GR, RB, BR.) [b]p4.[/b] There are $1997$ nonzero real numbers written on the blackboard. An operation consists of choosing any two of these numbers, $a$ and $b$, erasing them, and writing $a+b/2$ and $b-a/2$ instead of them. Prove that if a sequence of such operations is performed, one can never end up with the initial collection of numbers. [b]p5.[/b] An $m\times n$ checkerboard (m and n are positive integers) is covered by nonoverlapping tiles of sizes $2\times 2$ and $1\times 4$. One $2\times 2$ tile is removed and replaced by a $1\times 4$ tile. Is it possible to rearrange the tiles so that they cover the checkerboard? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a given integer $d$, let us define $S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}$. Suppose that $p, q$ are two elements of $S$ , where $p$ is prime and $p | q$. Prove that $r = q/p$ also belongs to $S$ .

Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \ne 0$. What is the four-digit number $\overline{abcd}$?

Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

For a positive integer $n$, consider the set \[S = \{0, 1, 1 + 2, 1 + 2 + 3, \ldots, 1 + 2 + 3 +\ldots + (n - 1)\}\] Prove that the elements of $S$ are mutually incongruent modulo $n$ if and only if $n$ is a power of $2$.

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non-negative integers. Let $A_k$ (respectively $B_k$) be the set of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of integers such that $0\leq f_i\leq e_i$ for all $i$ and $\sum_{i=1}^k f_i$ is even (respectively odd). Show that $|A_k|-|B_k|=0 \textrm{ or } 1$.

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

What is the minimum distance between $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \in Z$?

Let $N$ be a natural number which can be expressed in the form $a^{2}+b^{2}+c^{2}$, where $a,b,c$ are integers divisible by 3. Prove that $N$ can be expressed in the form $x^{2}+y^{2}+z^{2}$, where $x,y,z$ are integers and any of them are divisible by 3.

Let $p\neq 3$ be a prime number. Show that there is a non-constant arithmetic sequence of positive integers $x_1,x_2,\ldots ,x_p$ such that the product of the terms of the sequence is a cube.

A computer program generates a sequence of numbers with the following rule: the first number is written by Camilo; thereafter, the program calculates the integer division of the last number generated by $18$; thus obtains a quotient and a remainder. The sum of that quotient plus that remainder is the next number generated. For example, if Camilo's number is $5291,$ the computer makes $5291 = 293 \times 18 + 17$, and generates $310 = 293 + 17$. The next number generated will be $21$, since $310 = 17 \times 18 + 4$ and $17 + 4= 21$; etc Whatever Camilo's initial number is, from some point on, the computer always generates the same number. Determine what is that number that will be repeated indefinitely, if Camilo's initial number is equal to $2^{110}.$

Find all nonnegative integers $a, b, c$ such that $$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$

Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form $ x^3\plus{}2y^2$, with $ x, y$ being integers.

For any positive integer $a$, denote $f(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{1,9\}\}|$ and $g(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{3,7\}\}|$. Prove that $f(a)\geq g(a)\forall a\in\mathbb{N}$.

Prove that every prime of the form $4k+1$ is the hypotenuse of a rectangular triangle with integer sides.

Let $\{a_n\}_{n\ge 0}$ be a sequence of rational numbers given by $a_0 = a_1 = a_2 = a_3 = 1$ and for all $n \ge 4$ we have $a_{n-4}a_n = a_{n-3}a_{n-1} + a^2_{n-2}$. Prove that all the terms of the sequence are integers.

Let $N$ be a positive integer. Suppose given any real $x\in (0,1)$ with decimal representation $0.a_1a_2a_3a_4\cdots$, one can color the digits $a_1,a_2,\cdots$ with $N$ colors so that the following hold: 1. each color is used at least once; 2. for any color, if we delete all the digits in $x$ except those of this color, the resulting decimal number is rational. Find the least possible value of $N$. [i]~Sutanay Bhattacharya[/i]

[u]Round 5[/u] [b]p13.[/b] Express the number $3024_8$ in base $2$. [b]p14.[/b] $\vartriangle ABC$ has a perimeter of $10$ and has $AB = 3$ and $\angle C$ has a measure of $60^o$. What is the maximum area of the triangle? [b]p15.[/b] A weighted coin comes up as heads $30\%$ of the time and tails $70\%$ of the time. If I flip the coin $25$ times, howmany tails am I expected to flip? [u]Round 6[/u] [b]p16.[/b] A rectangular box with side lengths $7$, $11$, and $13$ is lined with reflective mirrors, and has edges aligned with the coordinate axes. A laser is shot from a corner of the box in the direction of the line $x = y = z$. Find the distance traveled by the laser before hitting a corner of the box. [b]p17.[/b] The largest solution to $x^2 + \frac{49}{x^2}= 2018$ can be represented in the form $\sqrt{a}+\sqrt{b}$. Compute $a +b$. [b]p18.[/b] What is the expected number of black cards between the two jokers of a $54$ card deck? [u]Round 7[/u] p19. Compute ${6 \choose 0} \cdot 2^0 + {6 \choose 1} \cdot 2^1+ {6 \choose 2} \cdot 2^2+ ...+ {6 \choose 6} \cdot 2^6$. [b]p20.[/b] Define a sequence by $a_1 =5$, $a_{n+1} = a_n + 4 * n -1$ for $n\ge 1$. What is the value of $a_{1000}$? [b]p21.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B = 15^o$ and $\angle C = 30^o$. Let $D$ be the point such that $\vartriangle ADC$ is an isosceles right triangle where $D$ is in the opposite side from $A$ respect to $BC$ and $\angle DAC = 90^o$. Find the $\angle ADB$. [u]Round 8[/u] [b]p22.[/b] Say the answer to problem $24$ is $z$. Compute $gcd (z,7z +24).$ [b]p23.[/b] Say the answer to problem $22$ is $x$. If $x$ is $1$, write down $1$ for this question. Otherwise, compute $$\sum^{\infty}_{k=1} \frac{1}{x^k}$$ [b]p24.[/b] Say the answer to problem $23$ is $y$. Compute $$\left \lfloor \frac{y^2 +1}{y} \right \rfloor$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)$ equal to $12$? $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$ $\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$ $\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$ $\textbf{(J) }9\hspace{14.2em}\textbf{(K) }10\hspace{13.5em}\textbf{(L) }11$ $\textbf{(M) }12\hspace{13.3em}\textbf{(N) }13$