A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$.

Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]

Prove that the equation $$\frac{m^2}{a-x} + \frac{n^2}{b-x} = 1,$$ where $ m \ne 0 $, $ n \ne 0 $, $ a \ne b $, has real roots ($ m $, $ n $, $ a $, $ b $ denote real numbers).

The area of the dark gray triangle depicted below is $35$, and a segment is divided into lengths $14$ and $10$ as shown below. What is the area of the light gray triangle? [asy] size(150); filldraw((0,0)--(0,12)--(24,-60/7)--cycle, lightgray); filldraw((14,0)--(14,5)--(0,12)--cycle, gray); draw((0,0)--(24,0)--(0,12)--cycle); draw((0,0)--(24,0)--(24,-60/7)--cycle); draw((0,12)--(24,-60/7)); draw((14,5)--(14,0)); dot((0,0)); dot((0,12)); dot((14,5)); dot((24,0)); dot((14,0)); dot((24,-60/7)); label("$14$", (7,0), S); label("$10$", (19,0), S); draw((0,2/3)--(2/3,2/3)--(2/3,0)); draw((14,2/3)--(14+2/3,2/3)--(14+2/3,0)); draw((24-2/3,0)--(24-2/3,-2/3)--(24,-2/3)); [/asy] $\textbf{(A) }84\qquad\textbf{(B) }120\qquad\textbf{(C) }132\qquad\textbf{(D) }144\qquad\textbf{(E) }168$

Determine the polygons with $n$ sides $(n \ge 4)$, not necessarily convex, which satisfy the property that the reflection of every vertex of polygon with respect to every diagonal of the polygon does not fall outside the polygon. [b]Note:[/b] Each segment joining two non-neighboring vertices of the polygon is a diagonal. The reflection is considered with respect to the support line of the diagonal.

In triangle $ABC,$ $AB = 13,$ $BC = 14,$ and $CA = 15.$ A circle of radius $r$ passes through point $A$ and is tangent to line $BC$ at $C.$ If $r = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $100m + n.$ [i]Proposed by Michael Tang[/i]

Given two conditions: A: Two triangles have the same area and two corresponding edge equal. B: Two triangles are congruent. Then, which one of the followings are true? $(\text{A})$A is sufficient and necessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient but unnecessary condition of B. $(\text{D})$A is insufficient and unnecessary condition of B.

Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.

An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers. [i]Proposed by Kyle Lee[/i]

Let a quadrilateral $ABCD$ have an inscribed circle and let $K, L, M, N$ be the tangency points of the sides $AB, BC, CD$ and $DA$, respectively. Consider the orthocenters of each of the triangles $\vartriangle AKN, \vartriangle BLK, \vartriangle CML$ and $\vartriangle DNM$. Prove that these four points are the vertices of a parallelogram.

On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated. Which city had the greatest percentage increase in population from 1970 to 1980? [asy] size(300); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(5,0), B=(7,0), C=(10,0), D=(13,0), E=(16,0); pair F=(4,3), G=(5,3), H=(7,3), I=(10,3), J=(12,3); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(I); dot(J); draw((0,0)--(18,0)^^(0,3)--(18,3)); draw((0,0)--(0,.5)^^(5,0)--(5,.5)^^(10,0)--(10,.5)^^(15,0)--(15,.5)); draw((0,3)--(0,2.5)^^(5,3)--(5,2.5)^^(10,3)--(10,2.5)^^(15,3)--(15,2.5)); draw((1,0)--(1,.2)^^(2,0)--(2,.2)^^(3,0)--(3,.2)^^(4,0)--(4,.2)^^(6,0)--(6,.2)^^(7,0)--(7,.2)^^(8,0)--(8,.2)^^(9,0)--(9,.2)^^(10,0)--(10,.2)^^(11,0)--(11,.2)^^(12,0)--(12,.2)^^(13,0)--(13,.2)^^(14,0)--(14,.2)^^(16,0)--(16,.2)^^(17,0)--(17,.2)^^(18,0)--(18,.2)); draw((1,3)--(1,2.8)^^(2,3)--(2,2.8)^^(3,3)--(3,2.8)^^(4,3)--(4,2.8)^^(6,3)--(6,2.8)^^(7,3)--(7,2.8)^^(8,3)--(8,2.8)^^(9,3)--(9,2.8)^^(10,3)--(10,2.8)^^(11,3)--(11,2.8)^^(12,3)--(12,2.8)^^(13,3)--(13,2.8)^^(14,3)--(14,2.8)^^(16,3)--(16,2.8)^^(17,3)--(17,2.8)^^(18,3)--(18,2.8)); label("A", A, S); label("B", B, S); label("C", C, S); label("D", D, S); label("E", E, S); label("A", F, N); label("B", G, N); label("C", H, N); label("D", I, N); label("E", J, N); label("1970", (0,3), W); label("1980", (0,0), W); label("0", (0,1.5)); label("50", (5,1.5)); label("100", (10,1.5)); label("150", (15,1.5)); label("Population", (21,2)); label("in thousands", (21.4,1));[/asy] $ \textbf{(A)}\ A\qquad\textbf{(B)}\ B\qquad\textbf{(C)}\ C\qquad\textbf{(D)}\ D\qquad\textbf{(E)}\ E $

Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)

[u]Round 5[/u] [b]p13.[/b] Two closed disks of radius $\sqrt2$ are drawn centered at the points $(1,0)$ and $(-1, 0)$. Let P be the region belonging to both disks. Two congruent non-intersecting open disks of radius $r$ have all of their points in $P$ . Find the maximum possible value of $r$ . [b]p14.[/b] A rectangle has positive integer side lengths. The sum of the numerical values of its perimeter and area is $2017$. Find the perimeter of the rectangle. [b]p15.[/b] Find all ordered triples of real numbers $(a,b,c)$ which satisfy $$a +b +c = 6$$ $$a \cdot (b +c) = 6$$ $$(a +b) \cdot c = 6$$ [u]Round 6[/u] [b]p16.[/b] A four digit positive integer is called confused if it is written using the digits $2$, $0$, $1$, and $7$ in some order, each exactly one. For example, the numbers $7210$ and $2017$ are confused. Find the sum of all confused numbers. [b]p17.[/b] Suppose $\vartriangle ABC$ is a right triangle with a right angle at $A$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle CAD$. Suppose that $AB = 20$ and $AC = 17$. Compute $AD$. [b]p18.[/b] Let $x$ be a real number. Find the minimum possible positive value of $\frac{|x -20|+|x -17|}{x}$. [u]Round 7[/u] [b]p19.[/b] Find the sum of all real numbers $0 < x < 1$ that satisfy $\{2017x\} = \{x\}$. [b]p20.[/b] Let $a_1,a_2, ,,, ,a_{10}$ be real numbers which sum to $20$ and satisfy $\{a_i\} <0.5$ for $1 \le i\le 10$. Find the sum of all possible values of $\sum_{ 1 \le i <j\le 10} \lfloor a_i +a_j \rfloor .$ Here, $\lfloor x \rfloor$ denotes the greatest integer $x_0$ such that $x_0 \le x$ and $\{x\} =x -\lfloor x \rfloor$. [b]p21.[/b] Compute the remainder when $20^{2017}$ is divided by $17$. [u]Round 8[/u] [b]p22.[/b] Let $\vartriangle ABC$ be a triangle with a right angle at $B$. Additionally, letM be the midpoint of $AC$. Suppose the circumcircle of $\vartriangle BCM$ intersects segment $AB$ at a point $P \ne B$. If $CP = 20$ and $BP = 17$, compute $AC$. [b]p23.[/b] Two vertices on a cube are called neighbors if they are distinct endpoints of the same edge. On a cube, how many ways can a nonempty subset $S$ of the vertices be chosen such that for any vertex $v \in S$, at least two of the three neighbors of $v$ are also in $S$? Reflections and rotations are considered distinct. [b]p24.[/b] Let $x$ be a real number such that $x +\sqrt[4]{5-x^4}=2$. Find all possible values of $x\sqrt[4]{5-x^4}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is a point $T$ on a circle with the radius $R$. Points $A{}$ and $B$ are on the tangent to the circle that goes through $T$ such that they are on the same side of $T$ and $TA\cdot TB=4R^2$. The point $S$ is diametrically opposed to $T$. Lines $AS$ and $BS$ intersect the circle again in $P{}$ and $Q{}$. Prove that the lines $PQ$ and $AB{}$ are perpendicular.

[b]a)[/b] Determine $ x\in\mathbb{N} $ and $ y\in\mathbb{Q} $ such that $ \sqrt{x+\sqrt{x}}=y. $ [b]b)[/b] Show that there are infinitely many pairs $ (x,y)\in\mathbb{Q}^2 $ such that $ \sqrt{x+\sqrt{x}} =y . $

You are given $N$ such that $ n \ge 3$. We call a set of $N$ points on a plane acceptable if their abscissae are unique, and each of the points is coloured either red or blue. Let's say that a polynomial $P(x)$ divides a set of acceptable points either if there are no red dots above the graph of $P(x)$, and below, there are no blue dots, or if there are no blue dots above the graph of $P(x)$ and there are no red dots below. Keep in mind, dots of both colors can be present on the graph of $P(x)$ itself. For what least value of k is an arbitrary set of $N$ points divisible by a polynomial of degree $k$?

Assign independent standard normally distributed random variables to the vertices of an n-dimensional cube. Say one vertex is greater than another if the assigned number is greater. Define a random walk on the vertices according to the following rules: a) the starting point is chosen from all the vertices with equal probability, b) during our journey, if we reach a vertex such that there are adjacent vertices which have higher values, we choose the next vertex with equal probability, c) if there is none, we stop. Prove that $\forall\varepsilon>0 \,\exists K\, \forall n>1$ $$P(\lambda> K \log n) <\varepsilon$$ where $\lambda$ is the number of steps of the random walk.

$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$

We roll a regular 6-sided dice $n$ times. What is the probabilty that the total number of eyes rolled is a multiple of 5?

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Denote by $P, Q, R, S$ the orthogonal projections of $O$ onto $AB$ , $BC$ ,$CD$ , $DA$, respectively. Prove that $$PA \cdot AB + RC \cdot CD =\frac12 (AD^2 + BC^2)$$ if and only if $$QB \cdot BC + SD \cdot DA = \frac12(AB ^2 + CD^2)$$

The [i]spikiness[/i] of a sequence $a_1, a_2, \ldots, a_n$ of at least two real numbers is the sum $\textstyle\sum_{i=1}^{n-1} |a_{i+1}-a_i|.$ Suppose $x_1, x_2, \ldots, x_9$ are chosen uniformly at random from the set $[0, 1].$ Let $M$ be the largest possible value of the spikiness of a permutation of $x_1, x_2, \ldots, x_9.$ Compute the expected value of $M.$

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

Let $S$ be the set of all positive integers less than $10,000$ whose last four digits in base $2$ are the same as its last four digits in base $5$. What remainder do we get if we divide the sum of all elements of $S$ by $10000$?