Let $ n$ be a given positive integer. Show that we can choose numbers $ c_k\in\{\minus{}1,1\}$ ($ i\le k\le n$) such that \[ 0\le\sum_{k\equal{}1}^nc_k\cdot k^2\le4.\]

Let $\zeta= e^{2\pi i/99}$ and $\omega e^{2\pi i/101}$. The polynomial $$x^{9999} + a_{9998}x^{9998} + ...+ a_1x + a_0$$ has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + ...+ a_{9998}$.

Find the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} . $$

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form $$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$ ,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

Solve the following system of equations for real $x,y$ and $z$: \begin{eqnarray*} x &=& \sqrt{2y+3}\\ y &=& \sqrt{2z+3}\\ z &=& \sqrt{2x+3}. \end{eqnarray*}

[b]p1.[/b] Find all positive integers $n$ such that $n^3 - 14n^2 + 64n - 93$ is prime. [b]p2.[/b] Let $a, b, c$ be real numbers such that $0 \le a, b, c \le 1$. Find the maximum value of $$\frac{a}{1 + bc}+\frac{b}{1 + ac}+\frac{c}{1 + ab}$$ [b]p3.[/b] Find the maximum value of the function $f(x, y, z) = 4x + 3y + 2z$ on the ellipsoid $16x^2 + 9y^2 + 4z^2 = 1$ [b]p4.[/b] Let $x_1,..., x_n$ be numbers such that $x_1+...+x_n = 2009$. Find the minimum value of $x^2_1+...+x^2_n$ (in term of $n$). [b]p5.[/b] Find the number of odd integers between $1000$ and $9999$ that have at least 3 distinct digits. [b]p6.[/b] Let $A_1,A_2,...,A_{2^n-1}$ be all the possible nonempty subsets of $\{1, 2, 3,..., n\}$. Find the maximum value of $a_1 + a_2 + ... + a_{2^n-1}$ where $a_i \in A_i$ for each $i = 1, 2,..., 2^n - 1$. [b]p7.[/b] Find the rightmost digit when $41^{2009}$ is written in base $7$. [b]p8.[/b] How many integral ordered triples $(x, y, z)$ satisfy the equation $x+y+z = 2009$, where $10 \le x < 31$, $100 < z < 310$ and $y \ge 0$. [b]p9.[/b] Scooby has a fair six-sided die, labeled $1$ to $6$, and Shaggy has a fair twenty-sided die, labeled $1$ to $20$. During each turn, they both roll their own dice at the same time. They keep rolling the die until one of them rolls a 5. Find the probability that Scooby rolls a $5$ before Shaggy does. [b]p10.[/b] Let $N = 1A323492110877$ where $A$ is a digit in the decimal expansion of $N$. Suppose $N$ is divisible by $7$. Find $A$. [b]p11.[/b] Find all solutions $(x, y)$ of the equation $\tan^4(x+y)+\cot^4(x+y) = 1-2x-x^2$, where $-\frac{\pi}{2} \le x; y \le \frac{\pi}{2}$ [b]p12.[/b] Find the remainder when $\sum^{50}_{k=1}k!(k^2 + k - 1)$ is divided by $1008$. [b]p13.[/b] The devil set of a positive integer $n$, denoted $D(n)$, is defined as follows: (1) For every positive integer $n$, $n \in D(n)$. (2) If $n$ is divisible by $m$ and $m < n$, then for every element $a \in D(m)$, $a^3$ must be in $D(n)$. Furthermore, call a set $S$ scary if for any $a, b \in S$, $a < b$ implies that $b$ is divisible by $a$. What is the least positive integer $n$ such that $D(n)$ is scary and has at least $2009$ elements? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that \[ \left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2 \] for any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$. (Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.)

If \[f(x) = (x + 2x^2 +\cdots+ nx^n)^2 = a_2x^2 + a_3x^3 + \cdots+ a_{2n}x^{2n},\] prove that \[a_{n+1} + a_{n+2} + \cdots + a_{2n} =\dbinom{n + 1}{2}\frac{5n^2 + 5n + 2}{12}\]

Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$. 1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers. 2) Is there an $a_0$ for which $a_{2010}$ is an integer?

From a fixed set formed by the first consecutive natural numbers, find the number of subsets having exactly three elements, and these in arithmetic progression.

The function f is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1)=2$ and $f(xy)=f(x)f(y)-f(x+y)+1$ for all $x,y \in \mathbb{Q}$. Determine f (with proof)

Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$. [i]Proposed byMuztaba Syed[/i]

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$. Find the greatest value of $M = abc$

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

$p(x)$ is an irreducible polynomial with integer coefficients, and $q$ is a fixed prime number. Let $a_n$ be a number of solutions of the equation $p(x)\equiv 0\mod q^n$. Prove that we can find $M$ such that $\{a_n\}_{n\ge M}$ is constant.

Find all real values of a for which the equation $ (a \minus{} 3x^2 \plus{} \cos \frac {9\pi x}{2})\sqrt {3 \minus{} ax} \equal{} 0$ has an odd number of solutions in the interval $ [ \minus{} 1,5]$

Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any $n$ $\in$ $\mathbb{N}$. Prove, $a_1=0$

Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.

Let be a $ 3\times 3 $ real matrix $ A. $ Prove the following statements. [b]a)[/b] $ f(A)\neq O_3, $ for any polynomials $ f\in\mathbb{R} [X] $ whose roots are not real. [b]b)[/b] $ \exists n\in\mathbb{N}\quad \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0 $ [i]Laurențiu Panaitopol[/i]

Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$? $ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$

Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there?

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]