Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Let $ABCD$ be an isosceles trapezoid with bases $AB = 92$ and $CD = 19$. Suppose $AD = BC = x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2 = $ $ \textbf{(A)}\ 1369\qquad\textbf{(B)}\ 1679\qquad\textbf{(C)}\ 1748\qquad\textbf{(D)}\ 2109\qquad\textbf{(E)}\ 8825 $

As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\] where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$? [asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("$A$",(-3,-1),S); dot((-2,0)); label("$E$",(-2,0),NW); dot((-1,-1)); label("$B$",(-1,-1),S); dot((0,0)); label("$F$",(0,0),N); dot((1,-1)); label("$C$",(1,-1), S); dot((2,0)); label("$G$", (2,0),NE); dot((3,-1)); label("$D$", (3,-1), S); [/asy] $\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$

[asy] import cse5; size(180); real a=4, b=3; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0)))); //Credit to chezbgone2 for the diagram[/asy] In $\triangle ABC$, $AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The length of segment $QR$ is $\textbf{(A) }4.75\qquad\textbf{(B) }4.8\qquad\textbf{(C) }5\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}$

Let $G$ be a connected simple graph. When we add an edge to $G$ (between two unconnected vertices), then using at most $17$ edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.

Let $n>1$ be a given integer. Let $G$ be a finite simple graph with the property that each of its edges is contained in at most $n$ cycles. Prove that the chromatic number of the graph is at most $n+1$.

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

[b]p1. [/b] At Duke, $1/2$ of the students like lacrosse, $3/4$ like football, and $7/8$ like basketball. Let $p$ be the proportion of students who like at least all three of these sports and let $q$ be the difference between the maximum and minimum possible values of $p$. If $q$ is written as $m/n$ in lowest terms, find the value of $m + n$. [b]p2.[/b] A [i]dukie [/i]word is a $10$-letter word, each letter is one of the four $D, U, K, E$ such that there are four consecutive letters in that word forming the letter $DUKE$ in this order. For example, $DUDKDUKEEK$ is a dukie word, but $DUEDKUKEDE$ is not. How many different dukie words can we construct in total? [b]p3.[/b] Rectangle $ABCD$ has sides $AB = 8$, $BC = 6$. $\vartriangle AEC$ is an isosceles right triangle with hypotenuse $AC$ and $E$ above $AC$. $\vartriangle BFD$ is an isosceles right triangle with hypotenuse $BD$ and $F$ below $BD$. Find the area of $BCFE$. [b]p4.[/b] Chris is playing with $6$ pumpkins. He decides to cut each pumpkin in half horizontally into a top half and a bottom half. He then pairs each top-half pumpkin with a bottom-half pumpkin, so that he ends up having six “recombinant pumpkins”. In how many ways can he pair them so that only one of the six top-half pumpkins is paired with its original bottom-half pumpkin? [b]p5.[/b] Matt comes to a pumpkin farm to pick $3$ pumpkins. He picks the pumpkins randomly from a total of $30$ pumpkins. Every pumpkin weighs an integer value between $7$ to $16$ (including $7$ and $16$) pounds, and there’re $3$ pumpkins for each integer weight between $7$ to $16$. Matt hopes the weight of the $3$ pumpkins he picks to form the length of the sides of a triangle. Let $m/n$ be the probability, in lowest terms, that Matt will get what he hopes for. Find the value of $m + n$ [b]p6.[/b] Let $a, b, c, d$ be distinct complex numbers such that $|a| = |b| = |c| = |d| = 3$ and $|a + b + c + d| = 8$. Find $|abc + abd + acd + bcd|$. [b]p7.[/b] A board contains the integers $1, 2, ..., 10$. Anna repeatedly erases two numbers $a$ and $b$ and replaces it with $a + b$, gaining $ab(a + b)$ lollipops in the process. She stops when there is only one number left in the board. Assuming Anna uses the best strategy to get the maximum number of lollipops, how many lollipops will she have? [b]p8.[/b] Ajay and Joey are playing a card game. Ajay has cards labelled $2, 4, 6, 8$, and $10$, and Joey has cards labelled $1, 3, 5, 7, 9$. Each of them takes a hand of $4$ random cards and picks one to play. If one of the cards is at least twice as big as the other, whoever played the smaller card wins. Otherwise, the larger card wins. Ajay and Joey have big brains, so they play perfectly. If $m/n$ is the probability, in lowest terms, that Joey wins, find $m + n$. [b]p9.[/b] Let $ABCDEFGHI$ be a regular nonagon with circumcircle $\omega$ and center $O$. Let $M$ be the midpoint of the shorter arc $AB$ of $\omega$, $P$ be the midpoint of $MO$, and $N$ be the midpoint of $BC$. Let lines $OC$ and $PN$ intersect at $Q$. Find the measure of $\angle NQC$ in degrees. [b]p10.[/b] In a $30 \times 30$ square table, every square contains either a kit-kat or an oreo. Let $T$ be the number of triples ($s_1, s_2, s_3$) of squares such that $s_1$ and $s_2$ are in the same row, and $s_2$ and $s_3$ are in the same column, with $s_1$ and $s_3$ containing kit-kats and $s_2$ containing an oreo. Find the maximum value of $T$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The girl continues the sequence of letters $ASSARA... $, adding one of the three letters $A$, $R$ or $S$. When adding the next letter, the girl makes sure that no two written sevens of consecutive letters coincide. At some point it turned out that it was impossible to add a new letter according to these rules. What letter could be written last?

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

In the plane, $n$ lines are drawn in general position (that is, there are neither two of them parallel nor three of them passing through the same point). Prove that it is possible to put a positive integer in each region (finite or infinite) determined by these lines so that for each line the sum of the numbers in the regions of a sdemiplane is equal to the sum of the numbers in the regions of the other semiplane. Note: A region is a set of points such that the straight line connecting any two of them it does not intersect any of the lines. For example, a line divides the plane into $2$ infinite regions and three lines into general position divide the plane into $7$ regions, some finite(s) and others infinite.

Is there a convex pentagon in which each diagonal is equal to a side?

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

The sequence of natural numbers is based on the following rule: each term, starting with the second, is obtained from the previous addition works of all its various simple divisors (for example, after the number $12$ should be the number $18$, and after the number $125$ , the number $130$). Prove that any two sequences constructed in this way have a common member.

Let $a_1 , a_2 ,... , a_n$ be an arrangement of the integers $1,2,..., n$. Let $$S=\frac{a_1}{1}+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{1}.$$ Find a natural number $n$ such that among the values of $S$ for all arrangements $a_1 , a_2 ,... , a_n$ , all the integers from $n$ to $n + 100$ appear .

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

[asy] size(2.5inch); pair A = (-2,0), B = 2dir(150), D = (2,0), C; draw(A..(0,2)..D--cycle); C = intersectionpoint(A..(0,2)..D,Circle(B,arclength(A--B))); draw(A--B--C--D--cycle); label("$A$",A,W); label("$B$",B,NW); label("$C$",C,N); label("$D$",D,E); label("$4$",A--D,S); label("$1$",A--B,E); label("$1$",B--C,SE); //Credit to chezbgone2 for the diagram[/asy] Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length $\textbf{(A) }\frac{7}{2}\qquad\textbf{(B) }\frac{5\sqrt{2}}{2}\qquad\textbf{(C) }\sqrt{11}\qquad\textbf{(D) }\sqrt{13}\qquad \textbf{(E) }2\sqrt{3}$

Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$. Proposed by Jeremy King, UK

Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$ $$f(x + y -xy) = f(x) + f(y) - f(xy).$$

A sign at the fish market says, '50\% off, today only: half-pound packages for just \$3 per package.' What is the regular price for a full pound of fish, in dollars? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$

In Miskolc there are two tram lines: line $1$ runs between Tiszai railway station and UpperMajláth, while line $2$ runs between Tiszai railway station and the Ironworks. The timetable for trams leaving Tiszai railway station is as follows: tram $ 1$ leaves at every minute ending in a $0$ or $6$, and tram $2$ leaves at every minute ending in a $3$. There are three types of passengers waiting for the trams: those who will take tram $ 1$ only, those who will take tram $2$ only and those who will take any tram. Every minute there is a constant number of passengers of each type arriving at the station. (This number is not necessarily the same for the different types.) Also, every tram departs with an equal number of passengers from Tiszai railway station. How many passengers are there on a departing tram, if we know that every minute there are $3$ passengers arriving at the station who will take tram $2$ only?

For how many integer value of $ m$ does the lines $ 13x\plus{}11y \equal{} 700$ and $ y \equal{} mx\minus{}1$ intersect in a point with integer valued coordinats? A. None B. 1 C. 2 D. 3 E. Infinitely many

Triangle $ABC$ with incircle $(I)$ touches the sides $AB, BC, AC$ at $F, D, E$, res. $I_b, I_c$ are $B$- and $C-$ excenters of $ABC$. $P, Q$ are midpoints of $I_bE, I_cF$. $(PAC)\cap AB=\{ A, R\}$, $(QAB)\cap AC=\{ A,S\}$. a. Prove that $PR, QS, AI$ are concurrent. b. $DE, DF$ cut $I_bI_c$ at $K, J$, res. $EJ\cap FK=\{ M\}$. $PE, QF$ cut $(PAC), (QAB)$ at $X, Y$ res. Prove that $BY, CX, AM$ are concurrent.

$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.

Let $a_1, a_2,...,a_{2n}$ be positive real numbers such that $a_i + a_{n+i} = 1$, for all $i = 1,...,n$. Prove that there exist two different integers $1 \le j, k \le 2n$ for which $$\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}$$