Let \[ T \equal{} \frac1{3\minus{}\sqrt8} \minus{} \frac1{\sqrt8 \minus{} \sqrt7} \plus{} \frac1{\sqrt7\minus{}\sqrt6} \minus{} \frac1{\sqrt6\minus{}\sqrt5} \plus{} \frac1{\sqrt5\minus{}2}\] then $ \textbf{(A)}\ T<1 \qquad \textbf{(B)}\ T\equal{}1 \qquad \textbf{(C)}\ 1<T<2 \qquad \textbf{(D)}\ T>2 \qquad$ $ \textbf{(E)}\ T \equal{} \frac1{(3\minus{}\sqrt8)(\sqrt8\minus{}\sqrt7)(\sqrt7\minus{}\sqrt6)(\sqrt6\minus{}\sqrt5)(\sqrt5\minus{}2)}$

Of all parallelograms of a given area find the one with the shortest possible longer diagonal.

Let $m$, $n$ be positive integers. On an $m\times{n}$ checkerboard, divided into $1\times1$ squares, we consider all paths that go from upper right vertex to the lower left vertex, travelling exclusively on the grid lines by going down or to the left. We define the area of a path as the number of squares on the checkerboard that are below this path. Let $p$ be a prime such that $r_{p}(m)+r_{p}(n)\geq{p}$, where $r_{p}(m)$ denotes the remainder when $m$ is divided by $p$ and $r_{p}(n)$ denotes the remainder when $n$ is divided by $p$. How many paths have an area that is a multiple of $p$?

Let $x,y,z$ be positive real numbers. Prove that \[ \sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z} \] [i](Proposed by Šefket Arslanagić, Bosnia and Herzegovina)[/i]

Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products? [i](C. Wu)[/i]

Let $a$, $b$, $c \in (0,+\infty)$ . Then the following inequality is true:$$\sqrt{(a+b)(b+c)}+\sqrt{(b+c)(c+a)}+\sqrt{(c+a)(a+b)}+a+b+c\leq \left(ab+bc+ca\right)\left(\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\right)$$

Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number. [b]a)[/b] Prove the inequality for $ k\equal{}1$. [b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.

In a $\triangle XYZ$, points $A,B$ lie on segment $ZX, C,D$ lie on segment $XY , E, F$ lie on segment $YZ$. $A, B, C, D$ lie on a circle, and $\frac{AZ \cdot EY \cdot ZB \cdot Y F}{EZ \cdot CY \cdot ZF \cdot Y D}= 1$ . Let $L = ZX \cap DE$, $M = XY \cap AF$, $N = Y Z \cap BC$. Prove that $L,M,N$ are collinear.

Let $ABC$ be a triangle, with $AB<AC$, and let $\Gamma$ be its circumcircle. Let $D$, $E$ and $F$ be the tangency points of the incircle with $BC$, $CA$ and $AB$ respectively. Let $R$ be the point in $EF$ such that $DR$ is an altitude in the triangle $DEF$, and let $S$ be the intersection of the external bisector of $\angle BAC$ with $\Gamma$. Prove that $AR$ and $SD$ intersect on $\Gamma$.

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 $\frac{S}{S'} \ge 3 +2\sqrt2$