Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Let $\phi (n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Evaluate $$\sum \limits _{n=1} ^{64} (-1)^{n} \left \lfloor \frac{64}{n} \right \rfloor \phi (n).$$

Compute the smallest positive integer $k$ such that $49$ divides $\tbinom{2k}{k}.$

Let $a_1,a_2,a_3,\ldots$ be the sequence of real numbers defined by $a_1=1$ and $$a_m=\frac1{a_1^2+a_2^2+\ldots+a_{m-1}^2}\qquad\text{for }m\ge2.$$ Determine whether there exists a positive integer $N$ such that $$a_1+a_2+\ldots+a_N>2017^{2017}.$$

The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\le b\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively. Let $s=AA'+BB'+CC'$. Then, for all positions of point $P$, $s$ is less than: $\textbf{(A) }2a+b\qquad\textbf{(B) }2a+c\qquad\textbf{(C) }2b+c\qquad\textbf{(D) }a+2b\qquad \textbf{(E) }$ $a+b+c$ [asy] import math; defaultpen(fontsize(11pt)); pair A = (0,0), B = (1,3), C = (5,0), P = (1.5,1); pair X = extension(B,C,A,P), Y = extension(A,C,B,P), Z = extension(A,B,C,P); draw(A--B--C--cycle); draw(A--X); draw(B--Y); draw(C--Z); dot(P); dot(A); dot(B); dot(C); label("$A$",A,dir(210)); label("$B$",B,dir(90)); label("$C$",C,dir(-30)); label("$A'$",X,dir(-100)); label("$B'$",Y,dir(65)); label("$C'$",Z,dir(20)); label("$P$",P,dir(70)); label("$a$",X,dir(80)); label("$b$",Y,dir(-90)); label("$c$",Z,dir(110)); //Credit to bobthesmartypants for the diagram [/asy]

Let $ABC$ be an acute triangle with orthocenter $H$. Segments $AH$ and $CH$ intersect segments $BC$ and $AB$ in points $A_1$ and $C_1$ respectively. The segments $BH$ and $A_1C_1$ meet at point $D$. Let $P$ be the midpoint of the segment $BH$. Let $D'$ be the reflection of the point $D$ in $AC$. Prove that quadrilateral $APCD'$ is cyclic. [i]Proposed by Matko Ljulj.[/i]

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 $n \ge 2$, \[f (f (n - 1)) = f (n + 1) - f (n)?\]