In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.

For every point on the plane, one of $ n$ colors are colored to it such that: $ (1)$ Every color is used infinitely many times. $ (2)$ There exists one line such that all points on this lines are colored exactly by one of two colors. Find the least value of $ n$ such that there exist four concyclic points with pairwise distinct colors.

Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying \[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\] Prove that $P(1983) = F_{1983} - 1.$

There are $k$ piles and there are $2019$ stones totally. In every move we split a pile into two or remove one pile. Using finite moves we can reach conclusion that there are $k$ piles left and all of them contain different number of stonws. Find the maximum of $k$.

King Arthur is placing $a + b + c$ knights around a table. $a$ knights are dressed in red, $b$ knights are dressed in brown, and $c$ knights are dressed in orange. Arthur wishes to arrange the knights so that no knight is seated next to someone dressed in the same colour as himself. Show that this is possible if, and only if, there exists a triangle whose sides have lengths $a +\frac12, b +\frac12$, and $c +\frac12$

$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$. $P_{n+4}$ is the midpoint of $P_nP_{n+1}$. $Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$. $A_n$ is the interior of $Q_n$. Find $\cap_{n \geq 0}A_n$.

Let $\mathbb{K}$ be two-dimensional integer lattice. Is there a bijection $f:\mathbb{N} \rightarrow \mathbb{K}$, such that for every distinct $a,b,c \in \mathbb{N}$ we have: \[\gcd(a,b,c)>1 \Rightarrow f(a),f(b),f(c) \mbox{ are not colinear? }\]

Let $F_1, F_2, ...$ be Borel-measurable sets on the plane whose union is the whole plane. Prove that there is a natural number n and circle S for which the set $S \cap F_n$ is dense in S. Also show that the statement is not necessarily true if we omit the condition for the measurability of sets $F_j$.

Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether $$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$

A natural number $n$ is called breakable if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$. Find all breakable numbers.

Philip has $3$ triangles and $6$ pentagons. Let $S$ be the total number of sides of the shapes he has. Let $N$ be the number of shapes he has. What is $S+N$?

[b] Newton's Law of Gravity from Kepler's Laws?[/b] [list=1] [*] Planets in the solar system move in elliptic orbits with the sun at one of the foci. [/*] [*] The line joining the sun and the planet sweeps out equal areas in equal times. [/*] [*] The period of revolution ($T$) and the length of the semi-major axis $(a$) of the ellipse sit in the relation $T^2/a^3 = constant$. [/*] [/list] Now answer the following questions: [list] [*] Starting from Newton's Law of Gravitation and Kepler's first law, derive the second and third law. It is possible to derive the first law but that is beyond the scope of this exam. [/*] [*] For convenience work in the complex (Argand) plane and take the sun to be at the origin $(z=0)$. Show that points on the ellipse may be represented by, \[ z(\theta) = \frac{a(1-\epsilon^2)}{1+\epsilon\cos\theta}\exp(i\theta) = r(\theta) e^{i\theta}\] where $a$ is the length of the semi-major axis, $\epsilon$ is the eccentricity of the ellipse and $\theta$ is called the \emph{true anomaly} in celestial mechanics. [/*] [*] Show that Kepler's second law leads to, \[ \frac{1}{2}r^2 \dot{\theta} = constant\] where $r$ and $\theta$ are defined as in part (b) and a dot $(.)$ over a variable denotes its time derivative. What is this constant in terms of the other variables of the problem? [/*] [*] Using the results of parts (b) and (c) along with Kepler's third law obtain Newton's Law of Gravitation. [/*] [*] Can the above exercise truly be called a "derivation" of Newton's Law of Gravitation? State your reasons. [/*] [/list]

For any given non-negative integer $m$, let $f(m)$ be the number of $1$'s in the base $2$ representation of $m$. Let $n$ be a positive integer. Prove that the integer $$\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)$$ contains at least $n!$ positive divisors.

Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2 \\ x^4 + y^4 + x^2y^2 = 8 \end{cases}$ The value of $P = x^8 + y^8 + x^{2014}y^{2014}$ is: (A): $46$, (B): $48$, (C): $50$, (D): $52$, (E) None of the above.

Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? $\textbf{(A) }8\qquad\textbf{(B) }11\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }18$

Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$. Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.

Let $ABC$ be a triangle and $E$ and $F$ lie on $AC$ and $AB$ such that $AE=AF$. $EF$ intersects $BC$ at $D$ and $(BDF)$ intersects $(CDE)$ at $X$. Let $O_1$ be the center of $(BDF)$ and $O_2$ be the center of $(CDE)$. Let $O$ be the center of $ABC$. Suppose that $XD$ intersects $(XO_1O_2)$ at $Z$. Show that $OZ\parallel BC$. [i](Proposed by Tan Rui Xuen and Yeoh Yi Shuen)[/i]

[b]p1.[/b] Aidan walks into a skyscraper’s first floor lobby and takes the elevator up $50$ floors. After exiting the elevator, he takes the stairs up another $10$ floors, then takes the elevator down $30$ floors. Find the floor number Aidan is currently on. [b]p2.[/b] Jeff flips a fair coin twice and Kaylee rolls a standard $6$-sided die. The probability that Jeff flips $2$ heads and Kaylee rolls a $4$ is $P$. Find $\frac{1}{P}$ . [b]p3.[/b] Given that $a\odot b = a + \frac{a}{b}$ , find $(4\odot 2)\odot 3$. [b]p4.[/b] The following star is created by gluing together twelve equilateral triangles each of side length $3$. Find the outer perimeter of the star. [img]https://cdn.artofproblemsolving.com/attachments/e/6/ad63edbf93c5b7d4c7e5d68da2b4632099d180.png[/img] [b]p5.[/b] In Lexington High School’sMath Team, there are $40$ students: $20$ of whom do science bowl and $22$ of whom who do LexMACS. What is the least possible number of students who do both science bowl and LexMACS? [b]p6.[/b] What is the least positive integer multiple of $3$ whose digits consist of only $0$s and $1$s? The number does not need to have both digits. [b]p7.[/b] Two fair $6$-sided dice are rolled. The probability that the product of the numbers rolled is at least $30$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] At the LHSMath Team Store, $5$ hoodies and $1$ jacket cost $\$13$, and $5$ jackets and $1$ hoodie cost $\$17$. Find how much $15$ jackets and $15$ hoodies cost, in dollars. [b]p9.[/b] Eric wants to eat ice cream. He can choose between $3$ options of spherical ice cream scoops. The first option consists of $4$ scoops each with a radius of $3$ inches, which costs a total of $\$3$. The second option consists of a scoop with radius $4$ inches, which costs a total of $\$2$. The third option consists of $5$ scoops each with diameter $2$ inches, which costs a total of $\$1$. The greatest possible ratio of volume to cost of ice cream Eric can buy is nπ cubic inches per dollar. Find $3n$. [b]p10.[/b] Jen claims that she has lived during at least part of each of five decades. What is the least possible age that Jen could be? (Assume that age is always rounded down to the nearest integer.) [b]p11.[/b] A positive integer $n$ is called Maisylike if and only if $n$ has fewer factors than $n -1$. Find the sum of the values of $n$ that are Maisylike, between $2$ and $10$, inclusive. [b]p12.[/b] When Ginny goes to the nearby boba shop, there is a $30\%$ chance that the employee gets her drink order wrong. If the drink she receives is not the one she ordered, there is a $60\%$ chance that she will drink it anyways. Given that Ginny drank a milk tea today, the probability she ordered it can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the value of $a +b$. [b]p13.[/b] Alex selects an integer $m$ between $1$ and $100$, inclusive. He notices there are the same number of multiples of $5$ as multiples of $7$ betweenm and $m+9$, inclusive. Find how many numbers Alex could have picked. [b]p14.[/b] In LMTown there are only rainy and sunny days. If it rains one day there’s a $30\%$ chance that it will rain the next day. If it’s sunny one day there’s a $90\%$ chance it will be sunny the next day. Over n days, as n approaches infinity, the percentage of rainy days approaches $k\%$. Find $10k$. [b]p15.[/b] A bag of letters contains $3$ L’s, $3$ M’s, and $3$ T’s. Aidan picks three letters at random from the bag with replacement, and Andrew picks three letters at random fromthe bag without replacement. Given that the probability that both Aidan and Andrew pick one each of L, M, and T can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p16.[/b] Circle $\omega$ is inscribed in a square with side length $2$. In each corner tangent to $2$ of the square’s sides and externally tangent to $\omega$ is another circle. The radius of each of the smaller $4$ circles can be written as $(a -\sqrt{b})$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/c76a780ac857f745067a8d6c4433efdace2dbb.png[/img] [b]p17.[/b] In the nonexistent land of Lexingtopia, there are $10$ days in the year, and the Lexingtopian Math Society has $5$ members. The probability that no two of the LexingtopianMath Society’s members share the same birthday can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Let $D(n)$ be the number of diagonals in a regular $n$-gon. Find $$\sum^{26}_{n=3} D(n).$$ [b]p19.[/b] Given a square $A_0B_0C_0D_0$ as shown below with side length $1$, for all nonnegative integers $n$, construct points $A_{n+1}$, $B_{n+1}$, $C_{n+1}$, and $D_{n+1}$ on $A_nB_n$, $B_nC_n$, $C_nD_n$, and $D_nA_n$, respectively, such that $$\frac{A_n A_{n+1}}{A_{n+1}B_n}=\frac{B_nB_{n+1}}{B_{n+1}C_n} =\frac{C_nC_{n+1}}{C_{n+1}D_n}=\frac{D_nD_{n+1}}{D_{n+1}A_n} =\frac34.$$ [img]https://cdn.artofproblemsolving.com/attachments/6/a/56a435787db0efba7ab38e8401cf7b06cd059a.png[/img] The sum of the series $$\sum^{\infty}_{i=0} [A_iB_iC_iD_i ] = [A_0B_0C_0D_0]+[A_1B_1C_1D_1]+[A_2B_2C_2D_2]...$$ where $[P]$ denotes the area of polygon $P$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] Let $m$ and $n$ be two real numbers such that $$\frac{2}{n}+m = 9$$ $$\frac{2}{m}+n = 1$$ Find the sum of all possible values of $m$ plus the sumof all possible values of $n$. [b]p21.[/b] Let $\sigma (x)$ denote the sum of the positive divisors of $x$. Find the smallest prime $p$ such that $$\sigma (p!) \ge 20 \cdot \sigma ([p -1]!).$$ [b]p22.[/b] Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of side $\overline{AB}$. Suppose there exists a point X on the circle passing through points $A$, $M$, and $C$ such that $BMCX$ is a parallelogram and $M$ and $X$ are on opposite sides of line $BC$. Let segments $\overline{AX}$ and $\overline{BC}$ intersect at a point $Y$ . Given that $BY = 8$, find $AY^2$. [b]p23.[/b] Kevin chooses $2$ integers between $1$ and $100$, inclusive. Every minute, Corey can choose a set of numbers and Kevin will tell him how many of the $2$ chosen integers are in the set. How many minutes does Corey need until he is certain of Kevin’s $2$ chosen numbers? [b]p24.[/b] Evaluate $$1^{-1} \cdot 2^{-1} +2^{-1} \cdot 3^{-1} +3^{-1} \cdot 4^{-1} +...+(2015)^{-1} \cdot (2016)^{-1} \,\,\, (mod \,\,\,2017).$$ [b]p25.[/b] In scalene $\vartriangle ABC$, construct point $D$ on the opposite side of $AC$ as $B$ such that $\angle ABD = \angle DBC = \angle BC A$ and $AD =DC$. Let $I$ be the incenter of $\vartriangle ABC$. Given that $BC = 64$ and $AD = 225$, find$ BI$ . [img]https://cdn.artofproblemsolving.com/attachments/b/1/5852dd3eaace79c9da0fd518cfdcd5dc13aecf.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many positive integers $n$ are there such that there are $20$ positive integers that are less than $n$ and relatively prime with $n$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{None}$

If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that $\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$ is also a rational number.

Crisp All, a basketball player, is [i]dropping dimes[/i] and nickels on a number line. Crisp drops a dime on every positive multiple of $10$, and a nickel on every multiple of $5$ that is not a multiple of $10$. Crisp then starts at $0$. Every second, he has a $\frac{2}{3}$ chance of jumping from his current location $x$ to $x+3$, and a $\frac{1}{3}$ chance of jumping from his current location $x$ to $x+7$. When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp [i]stops on a dime[/i]?

We have a positive integer $ n$ such that $ n \neq 3k$. Prove that there exists a positive integer $ m$ such that $ \forall_{k\in N \ k\geq m} \ k$ can be represented as a sum of digits of some multiplication of $ n$.

Let $T_1, T_2, T_3, T_4$ be pairwise distinct collinear points such that $T_2$ lies between $T_1$ and $T_3$, and $T_3$ lies between $T_2$ and $T_4$. Let $\omega_1$ be a circle through $T_1$ and $T_4$; let $\omega_2$ be the circle through $T_2$ and internally tangent to $\omega_1$ at $T_1$; let $\omega_3$ be the circle through $T_3$ and externally tangent to $\omega_2$ at $T_2$; and let $\omega_4$ be the circle through $T_4$ and externally tangent to $\omega_3$ at $T_3$. A line crosses $\omega_1$ at $P$ and $W$, $\omega_2$ at $Q$ and $R$, $\omega_3$ at $S$ and $T$, and $\omega_4$ at $U$ and $V$, the order of these points along the line being $P,Q,R,S,T,U,V,W$. Prove that $PQ + TU = RS + VW$ [i]Geza Kos, Hungary[/i]