The sequence ($a_n$) is defined by $a_1 = a_2 = 1, a_3 = 199$ and $a_{n+1} =\frac{1989+a_na_{n-1}}{a_{n-2}}$ for all $n \ge 3$. Prove that all terms of the sequence are positive integers

Let $ M $ be a finite set of integers, and let be a function $ \varphi :\mathbb{Z}\longrightarrow\mathbb{Z} $ whose restriction to $ \mathbb{Z}\setminus M $ evaluates to a constant $ c, $ such that $$ 2\le |\varphi (M)|=|M|\neq \frac{1}{c}\cdot \sum_{\iota \in \varphi (M) } \iota . $$ Prove that $ \varphi $ is not a sum between an injective function and a surjective function. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

Given $n^2$ numbers $x_{i,j}$ ($i,j=1,2,...,n$) satisfying the system of $n^3$ equations $$x_{i,j}+x_{j,k}+x_{k,i}=0 \,\,\, (i,j,k = 1,...,n)$$Prove that there exist such numbers $a_1,a_2,...,a_n$, that $x_{i,j}=a_i-a_j$ for all $i,j=1,...n$.

$A=\{1,2,3,4,5,6,7,8\} .$ How many functions $f: A \rightarrow A$ are there such that $f(1)<f(2)<f(3)$ [list=1] [*] ${{8}\choose{3}}$ [*] ${{8}\choose{3}}5^{8}$ [*] ${{8}\choose{3}} 8^{5}$ [*] $\frac{8 !}{3 !} $ [/list]

Let f be a ternary operation on a set of at least four elements for which (1) $f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x$ (2) $f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}$ for pairwise distinct x,y,z. Prove that f is a nontrivial composition of g such that g is not a composition of f. (The n-variable operation g is trivial if $g(x_1, ..., x_n) \equiv x_i$ for some i ($1 \leq i \leq n$) )

Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$.

Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)

Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$

Let $a$ be a positive integer. How many non-negative integer solutions x does the equation $\lfloor \frac{x}{a}\rfloor = \lfloor \frac{x}{a+1}\rfloor$ have? $\lfloor ~ \rfloor$ ---> [url=http://en.wikipedia.org/wiki/Floor_function]Floor Function[/url].

Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge 1$ for all real numbers $x$, where $0 < x < 1$

Starting with four given integers $a_1, b_1, c_1, d_1$ is defined recursively for all positive integers $n$: $$a_{n+1} := |a_n - b_n|, b_{n+1} := |b_n - c_n|, c_{n+1} := |c_n - d_n|, d_{n+1} := |d_n - a_n|.$$ Prove that there is a natural number $k$ such that all terms $a_k, b_k, c_k, d_k$ take the value zero.

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

[b]p1.[/b] Two trains, $A$ and $B$, travel between cities $P$ and $Q$. On one occasion $A$ started from $P$ and $B$ from $Q$ at the same time and when they met $A$ had travelled $120$ miles more than $B$. It took $A$ four $(4)$ hours to complete the trip to $Q$ and B nine $(9)$ hours to reach $P$. Assuming each train travels at a constant speed, what is the distance from $P$ to $Q$? [b]p2.[/b] If $a$ and $b$ are integers, $b$ odd, prove that $x^2 + 2ax + 2b = 0$ has no rational roots. [b]p3.[/b] A diameter segment of a set of points in a plane is a segment joining two points of the set which is at least as long as any other segment joining two points of the set. Prove that any two diameter segments of a set of points in the plane must have a point in common. [b]p4.[/b] Find all positive integers $n$ for which $\frac{n(n^2 + n + 1) (n^2 + 2n + 2)}{2n + 1}$ is an integer. Prove that the set you exhibit is complete. [b]p5.[/b] $A, B, C, D$ are four points on a semicircle with diameter $AB = 1$. If the distances $\overline{AC}$, $\overline{BC}$, $\overline{AD}$, $\overline{BD}$ are all rational numbers, prove that $\overline{CD}$ is also rational. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.

Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Find all functions $f: \mathbb R^+ \to \mathbb R^+$ such that \[ f \left( x+\frac{f(xy)}{x} \right) = f(xy) f \left( y + \frac 1y \right) \] holds for all $x,y\in\mathbb R^+.$

The roots of the equation $ax^2 - 4bx + 4c = 0$ with $ a > 0$ belong to interval $[2, 3]$. Prove that: a) $a \le b \le c < a + b.$ b) $\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} .$

Find maximum of the expression $(a -b^2)(b - a^2)$, where $0 \le a,b \le 1$.

Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties: (1) No number occurs more than once in the sequence. (2) The sum of two different elements of the sequence is never a power of two. (3) For all positive integers $n$, we have $a_n<r \cdot n$.

Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.

Let $k$ be a positive integer not less than 3 and $x$ a real number. Prove that if $\cos (k-1)x$ and $\cos kx$ are rational, then there exists a positive integer $n>k$, such that both $\cos (n-1)x$ and $\cos nx$ are rational.

A function $f$ with domain $A$ and range $B$ is called [i]injective[/i] if every input in $A$ maps to a unique output in $B$ (equivalently, if $x, y \in A$ and $x \neq y$, then $f(x) \neq f(y)$). With $\mathbb{C}$ denoting the set of complex numbers, let $P$ be an injective polynomial with domain and range $\mathbb{C}$. Suppose further that $P(0) = 2021$ and that when $P$ is written in standard form, all coefficients of $P$ are integers. Compute the smallest possible positive integer value of $P(10)/P(1)$.

If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$

For which natural numbers $n$ do there exist $n$ natural numbers $a_i\ (1\le i\le n)$ such that $\sum_{i=1}^n a_i^{-2}=1$?