A sequence $a_n$ is defined by $$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.

Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$ [i]Proposed by N. Safaei (Iran)[/i]

Prove that the equation $ z^3\plus{}z^2\plus{}z\minus{}6\equal{}0$ doesn't have roots $ x$ with $ |x|\equal{}2$.

(a) Given $2 + 4 + 6 + \dots + p = 6480$, find $p$. (b) Given $7 + 11 + 15 + \dots + q = 5250$, find $q$. (c) Given $2^2 + 4^2 + 6^2 + \dots + r^2 - 1^2 - 3^2 - 5^2 - \dots - (r-1)^2 = 2485$, compute $r$.

If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).\] The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 $

Prove that if a polynomial with integer coefficients takes a value equal to $1$ in absolute value at three different integer points, then it has no integer zeros.

Let $A$ and $B$ be positive integers. Define the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.

Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$, show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$.

What percentage of twenty thousand is a quarter of a million?

Function $f$ satisfies the equation $f(\cos x) = \cos (17x)$. Prove that it also satisfies the equation $f(\sin x) = \sin (17x)$.

We are given an integer $n \ge 2024$ and a sequence $a_1,a_2,\dots,a_{n^2}$ of real numbers satisfying \[\vert a_k-a_{k-1}\vert \le \frac{1}{k} \quad \text{and} \quad \vert a_1+a_2+\dots+a_k\vert \le 1\] for $k=2,3,\dots,n^2$. Show that $\vert a_{n(n-1)}\vert \le \frac{2}{n}$. [i]Note: Proving $\vert a_{n(n-1)}\vert \le \frac{75}{n}$ will be rewarded partial points.[/i]

Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $. $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.

Let $x_1, x_2, \cdots , x_n$ be positive numbers. Prove that \[\frac{x_1^2}{x_1^2+x_2x_3} + \frac{x_2^2}{x_2^2+x_3x_4} + \cdots +\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1} +\frac{x_n^2}{x_n^2+x_1x_2} \leq n-1\]

Show that the number of 5-tuples ($a$, $b$, $c$, $d$, $e$) such that $abcde = 5(bcde + acde + abde + abce + abcd)$ is odd

In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows: [list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$ - and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list] Prove that in this sequence [b](a) [/b] the number $1$ occurs infinitely many times; [b](b)[/b] each positive integer occurs infinitely many times. (The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )

Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$

Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$. Prove that the inequality $$(a+2)(b+2) \ge cd$$ holds and give four numbers $a$, $b$, $c$ and $d$ such that equality holds. (Walther Janous)

Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

Find the value of $a$ satisfying \begin{align*} a+b&=3\\ b+c&=11\\ c+a&=61 \end{align*}

Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?

[b]p1.[/b] Let $ABCD$ be a square with side length $1$, $\Gamma_1$ be a circle centered at $B$ with radius 1, $\Gamma_2$ be a circle centered at $D$ with radius $1$, $E$ be a point on the segment $AB$ with $|AE| = x$ $(0 < x \le 1)$, and $\Gamma_3$ be a circle centered at $A$ with radius $|AE|$. $\Gamma_3$ intersects $\Gamma_1$ and $\Gamma_2$ inside the square at $G$ and $F$, respectively. Let region $I$ be the region bounded by the segment $GC$ and the minor arc $GC$ of $\Gamma_1$, and region II be the region bounded by the segment $FG$ and the minor arc $FG$ of $\Gamma_3$, as illustrated in the graph below. Let $r(x)$ be the ratio of the area of region I to the area of region II. (i) Find $r(1)$. Justify your answer. (ii) Find an explicit formula of $r(x)$ in terms of $x$ $(0 < x \le 1)$. Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/0/bd2379a1390a578d78dc7e9f4cde756d5f4723.png[/img] [b]p2.[/b] We call a [i]party [/i] any set of people $V$ . If $v_1 \in V$ knows $v_2 \in V$ in a party, we always assume that $v_2$ also knows $v_1$. For a person $v \in V$ in some party, the degree of v, denoted by $deg\,\,(v)$, is the number of people $v$ knows in the party. (i) Suppose that a party has four people with $V = \{v_1, v_2, v_3, v_4\}$, and that $deg\,\,(v_i) = i$ for $i = 1, 2, 3$ show that $deg\,\,(v_4) = 2$. (ii) Suppose that a party is attended by $n = 4k$ ($k \ge 1$) people with $V = \{v_1, v_2, ..., v_{4k}\}$, and that $deg\,\,(v_i) = i$ for $1 \le i \le n - 1$. Show that $deg\,\,(v_n) = \frac{n}{2}$ . [b]p3.[/b] Let $a, b$ be two real number parameters and consider the function $f(x) =\frac{b + \sin x}{a + \cos x}$. (i) Find an example of $(a, b)$ such that $f(x) \ge 2$ for all real numbers $x$. Justify your answer. (ii) If $a > 1$ and the range of the function $f(x)$ (when x varies over the set of all real numbers) is $[-1, 1]$, find the values of $a$ and $b$. Justify your answer. [b]p4.[/b] Let $f$ be the function that assigns to each positive multiple $x$ of $8$ the number of ways in which $x$ can be written as a difference of squares of positive odd integers. (For example, $f(8) = 1$, because $8 = 3^2 -1^2$, and $f(24) = 2$, because $24 = 5^2 - 1^2 = 7^2 - 5^2$.) (a) Determine with proof the value of $f(120)$. (b) Determine with proof the smallest value $x$ for which $f(x) = 8$. (c) Show that the range of this function is the set of all positive integers. [b]p5.[/b] Consider the binomial coefficients $C_{n,r} ={n \choose r}= \frac{n!}{r!(n - r)!}$, for $n \ge 2$. Prove that $C_{n,r}$ are even, for all $1 \le r \le n - 1$, if and only if $n = 2^m$, for some counting number $m$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$