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

Points $A, B, C, D$ are located in the plane as follows: sections $AB$ and $CD$ are perpendicular to each other and are of equal length, moreover, D is just the trisection point of segment $AB$ closer to $A$. The perpendicular from point $D$ on segment $BC$ intersects it at $E$. Let the trisection point of segment $DE$ closer to $E$ be $H$. Prove that segments $CH$ and the sections $AE$ are perpendicular to each other.

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 $n$.

Two circles centered $O_1,O_2$ intersects at two points $M$ and $N$. $O_1M$ line intersects with $O_1$ centered circle and $O_2$ centered circle at $A_1$ and $A_2$, $O_2M$ line intersects with $O_1$ centered circle and $O_2$ centered circle at $B_1$ and $B_2$ respectively. Let $K$ is intersection point of the $A_1B_1$ and $A_2B_2$. Prove that $N,M,K$ collinear.

Two circles $\omega_1$ and $\omega_2$ meeting at point $A$ and a line $a$ are given. Let $BC$ be an arbitrary chord of $\omega_2$ parallel to $a$, and $E$, $F$ be the second common points of $AB$ and $AC$ respectively with $\omega_1$. Find the locus of common points of lines $BC$ and $EF$.

Let $S=\{(x,y)| x,y\in \mathbb{Q} , 0\le x,y\le 1\}$, where $\mathbb{Q}$ is the set of all rational numbers. Given a set of lines and a set of marked points in $S$, Euclid can do one of two moves: (i) Draw a line connecting two marked points, or (ii) Mark a point in $S$ which lies on at least two drawn lines. At first, the five distinct points $A(0,0), B(1,0), C(1,1), D(0,1)$ and $P\in S$ are marked. Find all such points $P$ such that Euclid can mark any point in $S$ after finitely many moves. [i]Glen Lim[/i]

[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].

Let $ABCD$ be an inscribed quadrilateral. Denote the midpoints of the sides $AB, BC, CD$ and $DA$ through $M, L, N$ and $K$, respectively. It turned out that $\angle BM N = \angle MNC$. Prove that: i) $\angle DKL = \angle CLK$. ii) in the quadrilateral $ABCD$ there is a pair of parallel sides.

If $x,y,z > 0, k>2$ and $a=x+ky+kz, b=kx+y+kz, c=kx+ky+z$, show that $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} \ge \frac{3}{2k+1}$.

Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$

Compute \[\cot\left(\sum_{n=1}^{23}\cot^{-1}\left(1+\sum_{k=1}^n2k\right)\right).\]

assume that X is a set of n number.and $0\leq k\leq n$.the maximum number of permutation which acting on $X$ st every two of them have at least k component in common,is $a_{n,k}$.and the maximum nuber of permutation st every two of them have at most k component in common,is $b_{n,k}$. a)proeve that :$a_{n,k}\cdot b_{n,k-1}\leq n!$ b)assume that p is prime number,determine the exact value of $a_{p,2}$.

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]

Compute the number of positive integers $n$ satisfying the inequalities \[ 2^{n-1} < 5^{n-3} < 3^n. \][i]Proposed by Isabella Grabski[/i]

Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that \[PA^2 + PB^2 \geq 2r^2.\]

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]

Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.

Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.

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

There are $6$ pieces of cheese of different weights. For any two pieces we can identify the heavier piece. Given that it is possible to divide them into two groups of equal weights with three pieces in each. Give the explicit way to find these groups by performing two weightings on a regular balance.

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]

In space, a point $O$ and a finite set of vectors $ \overrightarrow{v_1},\ldots,\overrightarrow{v_n} $ are given . We consider the set of points $ P $ for which the vector $ \overrightarrow{OP} $can be represented as a sum $ a_1 \overrightarrow{v_1} + \ldots + a_n\overrightarrow{v_n} $with coefficients satisfying the inequalities $ 0 \leq a_i \leq 1 $ $( i = 1, 2, \ldots, n $). Decide whether this set can be a tetrahedron.

Points $P$ and $Q$ are taken sides $AB$ and $AC$ of a triangle $ABC$ respectively such that $\hat{APC}=\hat{AQB}=45^{0}$. The line through $P$ perpendicular to $AB$ intersects $BQ$ at $S$, and the line through $Q$ perpendicular to $AC$ intersects $CP$ at $R$. Let $D$ be the foot of the altitude of triangle $ABC$ from $A$. Prove that $SR\parallel BC$ and $PS,AD,QR$ are concurrent.

An evil witch is making a potion to poison the people of PUMAClandia. In order for the potion to work, the number of poison dart frogs cannot exceed $5$, the number of wolves’ teeth must be an even number, and the number of dragon scales has to be a multiple of $6$. She can also put in any number of tiger nails. Given that the stew has exactly $2021$ ingredients, in how many ways can she add ingredients for her potion to work?

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 $a_0 = 26, b_0 = 6, c_0 = 2004$.