Determine all real solutions $x, y, z$ of the following system of equations: $\begin{cases} x^3 - 3x = 4 - y \\ 2y^3 - 6y = 6 - z \\ 3z^3 - 9z = 8 - x\end{cases}$

For a natural number $n>1$ , consider the $n-1$ points on the unit circle $e^{\frac{2\pi ik}{n}}\ (k=1,2,...,n-1) $ . Show that the product of the distances of these points from $1$ is $n$.

[b]p1.[/b] Let $A = \{D,U,K,E\}$ and $B = \{M, A, T,H\}$. How many maps are there from $A$ to $B$? [b]p2.[/b] The product of two positive integers $x$ and $y$ is equal to $3$ more than their sum. Find the sum of all possible $x$. [b]p3.[/b] There is a bag with $1$ red ball and $1$ blue ball. Jung takes out a ball at random and replaces it with a red ball. Remy then draws a ball at random. Given that Remy drew a red ball, what is the probability that the ball Jung took was red? [b]p4.[/b] Let $ABCDE$ be a regular pentagon and let $AD$ intersect $BE$ at $P$. Find $\angle APB$. [b]p5.[/b] It is Justin and his $4\times 4\times 4$ cube again! Now he uses many colors to color all unit-cubes in a way such that two cubes on the same row or column must have different colors. What is the minimum number of colors that Justin needs in order to do so? [b]p6.[/b] $f(x)$ is a polynomial of degree $3$ where $f(1) = f(2) = f(3) = 4$ and $f(-1) = 52$. Determine $f(0)$. [b]p7.[/b] Mike and Cassie are partners for the Duke Problem Solving Team and they decide to meet between $1$ pm and $2$ pm. The one who arrives first will wait for the other for $10$ minutes, the lave. Assume they arrive at any time between $1$ pm and $2$ pm with uniform probability. Find the probability they meet. [b]p8.[/b] The remainder of $2x^3 - 6x^2 + 3x + 5$ divided by $(x - 2)^2$ has the form $ax + b$. Find $ab$. [b]p9.[/b] Find $m$ such that the decimal representation of m! ends with exactly $99$ zeros. [b]p10.[/b] Let $1000 \le n = \overline{DUKE} \le 9999$. be a positive integer whose digits $\overline{DUKE}$ satisfy the divisibility condition: $$1111 | \left( \overline{DUKE} + \overline{DU} \times \overline{KE} \right)$$ Determine the smallest possible value of $n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Non-zero real numbers $a, b$ and $c$ are given such that $ab+bc+ac=0$. Prove that numbers $a+b+c$ and $\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}$ are either both positive or both negative. [i]Proposed by Mykhailo Shtandenko[/i]

Let $M_n$ be the set of permutations $\sigma\in S_n$ for which there exists $\tau\in S_n$ such that the numbers \[\sigma (1)+\tau(1),\, \sigma(2)+\tau(2),\ldots,\sigma(n)+\tau(n),\] are consecutive. Show that \((M_n\neq \emptyset\Leftrightarrow n\text{ is odd})\) and in this case for each $\sigma_1,\sigma_2\in M_n$ the following equality holds: \[\sum_{k=1}^n k\sigma_1(k)=\sum_{k=1}^n k\sigma_2(k).\] [i]Dan Schwarz[/i]

Does there exist a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every $n \ge 2$, \[f (f (n - 1)) = f (n + 1) - f (n)?\]

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

Given are the sequences \[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\] of positive real numbers. For each integer $n$ the following inequalities hold: \[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\] \[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\] \[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\] Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.

Let $n\geqslant 2$ be a fixed integer. Consider $n$ real numbers $a_1,a_2,\ldots,a_n$ not all equal and let\[d:=\max_{1\leqslant i<j\leqslant n}|a_i-a_j|;\qquad s=\sum_{1\leqslant i<j\leqslant n}|a_i-a_j|.\]Determine in terms of $n{}$ the smalest and largest values the quotient $s/d$ may achieve. [i]Selected from the Kvant Magazine[/i]

In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$

A can contains a mixture of two liquids A and B in the ratio $7:5$. When $9$ litres of the mixture are drawn and replaced by the same amount of liquid $B$, the ratio of $A$ and $B$ becomes $7:9$. How many litres of liquid A was contained in the can initially? $\textbf{(A)}~18$ $\textbf{(B)}~19$ $\textbf{(C)}~20$ $\textbf{(D)}~\text{None of the above}$

$(x_{n})_{-\infty<n<\infty}$ is a sequence of real numbers which satisfies $x_{n+1}=\frac{x_{n}^2+10}{7}$ for every $n \in \mathbb{Z}$. If there exist a real upperbound for this sequence, find all the values $x_{0}$ can take.

Let $f(n)$ count the number of values $0\le k\le n^2$ such that $43\nmid\binom{n^2}{k}$. Find the least positive value of $n$ such that $$43^{43}\mid f\left(\frac{43^{n}-1}{42}\right)$$ [i]Proposed by Adam Bertelli[/i]

What is the difference between the median and the mean of the following data set: $12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31$?

Let $P(x)$ and $Q(x)$ be (monic) polynomials with real coefficients (the first coefficient being equal to $1$), and $\deg P(x)=\deg Q(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions, then $ P(x+1)=Q(x-1) $ has a real solution.

If $x$ is a positive real number, and $n$ is a positive integer, prove that \[[ nx] > \frac{[ x]}1 + \frac{[ 2x]}2 +\frac{[ 3x]}3 + \cdots + \frac{[ nx]}n,\] where $[t]$ denotes the greatest integer less than or equal to $t$. For example, $[ \pi] = 3$ and $\left[\sqrt2\right] = 1$.

Let $a,b,$ and $c$ be pairwise distinct complex numbers such that $$a^2=b+6, \quad b^2=c+6, \quad \text{and} \quad c^2=a+6.$$ Compute the two possible values of $a+b+c.$

For all positive real numbers $a,b,c$ with $a+b+c=3$, show that $$a^3b+b^3c+c^3a+9\geq 4(ab+bc+ca).$$

Prove the equality $$\prod_{k=1}^{2m}\cos\frac{k\pi}{2m+1}=\frac{(-1)^m}{4m}.$$

The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$. Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.

Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m,n)$ for which $m^2+an+b$ and $n^2+am+b$ are both perfect squares. Prove that $a$ divides $2b$. [i]Holden Mui[/i]

Is there a function $f(x)$ defined for all $x$ and such that for some $a$ and all $x$ holds the equality $$f(x) + f(2x^2 - 1) = 2x + a?$$

Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.

Suppose real numbers $a, b, c, d$ satisfy $a + b + c + d = 17$ and $ab + bc + cd + da = 46$. If the minimum possible value of $a^2 + b^2 + c^2 + d^2$ can be expressed as a rational number $\frac{p}{q}$ in simplest form, find $p + q$.

There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.