A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$. Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.

Let $n \in \mathbb{N}$ and $S_{n}$ the set of all permutations of $\{1,2,3,...,n\}$. For every permutation $\sigma \in S_{n}$ denote $I(\sigma) := \{ i: \sigma (i) \le i \}$. Compute the sum $\sum_ {\sigma \in S_{n}} \frac{1}{|I(\sigma )|} \sum_ {i \in I(\sigma)} (i+ \sigma(i))$.

Let $k, n$ be positive integers, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ all be $k$-th roots of unity, satisfying: \[ \alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k). \] Prove that among $\alpha_1, \alpha_2, \ldots, \alpha_n$, each $k$-th root of unity appears the same number of times.

Let $f : (0,1) \to R$ be a function such that $f(1/n) = (-1)^n$ for all n ∈ N. Prove that there are no increasing functions $g,h : (0,1) \to R$ such that $f = g - h$.

The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

Let $\{ a_n\}$ be a series consisting of positive integers such that $n^2 \mid \sum_{i=1}^{n}{a_i}$ and $a_n\leq (n+2016)^2$ for all $n\geq 2016$. Define $b_n=a_{n+1}-a_n$. Prove that the series $\{ b_n\}$ is eventually constant.

Let $p$ be a polynomial with integer coefficients and let $a_1<a_2<\cdots <a_k$ be integers. Given that $p(a_i)\ne 0\forall\; i=1,2,\cdots, k$. [list] (a) Prove $\exists\; a\in \mathbb{Z}$ such that \[ p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k \] (b) Does there exist $a\in \mathbb{Z}$ such that \[ \prod_{i=1}^{k}p(a_i)\mid p(a) \][/list]

Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]

For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

Let $z\ne0$ be a complex number such that $z^8=\overline z$. What are the possible values of $z^{2001}$?

Prove that for every non-negative integer $ n, $ there exists a non-negative integer $ m $ such that $$ \left( 1+\sqrt{2} \right)^n=\sqrt m +\sqrt{m+1} . $$

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.

Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$ Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.

Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.

Let \( n \geq 2 \) be a positive integer. Find all \( n \)-tuples \( (a_1, \ldots, a_n) \) of complex numbers such that the numbers \( a_1 - 2a_2 \), \( a_2 - 2a_3 \), $\ldots$ , \( a_{n-1} - 2a_n \), \( a_n - 2a_1 \) form a permutation of the numbers \( a_1, \ldots, a_n \).

Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.

Find all functions $f : R \to R$ such that for all $x, y \in R$: $$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$

Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$ over the integers for every $i$.

The diagonals of a convex quadrilateral $ABCD$ intersect in $E$. Let $S_1, S_2, S_3$ and $S_4$ the areas of the triangles $AEB, BEC, CED, DEA$ respectively. Prove that, if exists real numbers $w, x, y$ and $z$ such that $$S_1=x+y+xy, S_2=y+z+yz, S_3=w+z+wz, S_4=w+x+wx,$$ then $E$ is the midpoint of $AC$ or $E$ is the midpoint of $BD$.

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

[u]Round 1[/u] [b]p1.[/b] Every angle of a regular polygon has degree measure $179.99$ degrees. How many sides does it have? [b]p2.[/b] What is $\frac{1}{20} + \frac{1}{1}+ \frac{1}{5}$ ? [b]p3.[/b] If the area bounded by the lines $y = 0$, $x = 0$, and $x = 3$ and the curve $y = f(x)$ is $10$ units, what is the area bounded by $y = 0$, $x = 0$, $x = 6$, and $y = f(x/2)$? [u]Round 2[/u] [b]p4.[/b] How many ways can $42$ be expressed as the sum of $2$ or more consecutive positive integers? [b]p5.[/b] How many integers less than or equal to $2015$ can be expressed as the sum of $2$ (not necessarily distinct) powers of two? [b]p6.[/b] $p,q$, and $q^2 - p^2$ are all prime. What is $pq$? [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 + ax + a$ be a polynomial with integer roots, where $a$ is an integer. What are all the possible values of $a$? [b]p8.[/b] In a given right triangle, the perimeter is $30$ and the sum of the squares of the sides is $338$. Find the lengths of the three sides. [b]p9.[/b] Each of the $6$ main diagonals of a regular hexagon is drawn, resulting in $6$ triangles. Each of those triangles is then split into $4$ equilateral triangles by connecting the midpoints of the $3$ sides. How many triangles are in the resulting figure? [u]Round 4[/u] [b]p10.[/b] Let $f = 5x+3y$, where $x$ and $y$ are positive real numbers such that $xy$ is $100$. Find the minimum possible value of $f$. [b]p11.[/b] An integer is called "Awesome" if its base $8$ expression contains the digit string $17$ at any point (i.e. if it ever has a $1$ followed immediately by a $7$). How many integers from $1$ to $500$ (base $10$) inclusive are Awesome? [b]p12.[/b] A certain pool table is a rectangle measuring $15 \times 24$ feet, with $4$ holes, one at each vertex. When playing pool, Joe decides that a ball has to hit at least $2$ sides before getting into a hole or else the shot does not count. What is the minimum distance a ball can travel after being hit on this table if it was hit at a vertex (assume it only stops after going into a hole) such that the shot counts? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].