[asy]draw((0,0)--(0,6)--(8,6)--(8,3)--(4,3)--(4,0)--cycle); label("6",(0,3),W); label("8",(4,6),N);[/asy] Given that all angles shown are marked, the perimeter of the polygon shown is \[ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ \text{cannot be determined from the information given} \qquad \]

An altitude $h$ of a triangle is increased by a length $m$. How much must be taken from the corresponding base $b$ so that the area of the new triangle is one-half that of the original triangle? ${{ \textbf{(A)}\ \frac{bm}{h+m}\qquad\textbf{(B)}\ \frac{bh}{2h+2m}\qquad\textbf{(C)}\ \frac{b(2m+h)}{m+h}\qquad\textbf{(D)}\ \frac{b(m+h)}{2m+h} }\qquad\textbf{(E)}\ \frac{b(2m+h)}{2(h+m)} } $

Let $L$ be the common point of the symmedians of triangle $ABC$, and $BH$ be its altitude. It is known that $\angle ALH = 180^o -2\angle A$. Prove that $\angle CLH = 180^o - 2\angle C$.

In $ \triangle ABC$, we have $ AC \equal{} BC \equal{} 7$ and $ AB \equal{} 2$. Suppose that $ D$ is a point on line $ AB$ such that $ B$ lies between $ A$ and $ D$ and $ CD \equal{} 8$. What is $ BD$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 2 \sqrt {3}\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 4 \sqrt {2}$

Through a lens that inverts the image we look at the rearview mirror of our car. If it reflects the license plate of the car that follows us, $CS-3965-EN$, draw the image we receive. Also draw the one obtained by permuting previous transformations, that is, reflecting in the mirror the image that the license plate gives the lens. Is the product of both transformations , reflection in the mirror and refraction through the lens, commutative?

Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same? (A Shapovalov)

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$. (a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.

Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$

In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$. Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively. (a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$. (b) Prove the converse of (a).

Using only the ruler, divide the side of a square table into $n$ equal parts. All lines drawn must lie on the surface of the table.

After swimming around the ocean with some snorkling gear, Joshua walks back to the beach where Alexis works on a mural in the sand beside where they drew out symbol lists. Joshua walks directly over the mural without paying any attention. "You're a square, Josh." "No, $\textit{you're}$ a square," retorts Joshua. "In fact, you're a $\textit{cube}$, which is $50\%$ freakier than a square by dimension. And before you tell me I'm a hypercube, I'll remind you that mom and dad confirmed that they could not have given birth to a four dimension being." "Okay, you're a cubist caricature of male immaturity," asserts Alexis. Knowing nothing about cubism, Joshua decides to ignore Alexis and walk to where he stashed his belongings by a beach umbrella. He starts thinking about cubes and computes some sums of cubes, and some cubes of sums: \begin{align*}1^3+1^3+1^3&=3,\\1^3+1^3+2^3&=10,\\1^3+2^3+2^3&=17,\\2^3+2^3+2^3&=24,\\1^3+1^3+3^3&=29,\\1^3+2^3+3^3&=36,\\(1+1+1)^3&=27,\\(1+1+2)^3&=64,\\(1+2+2)^3&=125,\\(2+2+2)^3&=216,\\(1+1+3)^3&=125,\\(1+2+3)^3&=216.\end{align*} Josh recognizes that the cubes of the sums are always larger than the sum of cubes of positive integers. For instance, \begin{align*}(1+2+4)^3&=1^3+2^3+4^3+3(1^2\cdot 2+1^2\cdot 4+2^2\cdot 1+2^2\cdot 4+4^2\cdot 1+4^2\cdot 2)+6(1\cdot 2\cdot 4)\\&>1^3+2^3+4^3.\end{align*} Josh begins to wonder if there is a smallest value of $n$ such that \[(a+b+c)^3\leq n(a^3+b^3+c^3)\] for all natural numbers $a$, $b$, and $c$. Joshua thinks he has an answer, but doesn't know how to prove it, so he takes it to Michael who confirms Joshua's answer with a proof. What is the correct value of $n$ that Joshua found?

Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$. [list=a] [*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$. [*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.

The circumcentre of a triangle $ABC$ is called $O$. The points $A',B'$ and $C'$ are the reflections of $O$ in $BC, CA$, and $AB$, respectively. Show that the three lines $AA' , BB'$, and $CC'$ meet in a common point.

Let there be an acute triangle $ABC$, such that $\angle ABC < \angle ACB$. Let the perpendicular from $A$ to $BC$ hit the circumcircle of $ABC$ at $D$, and let $M$ be the midpoint of $AD$. The tangent to the circumcircle of $ABC$ at $A$ hits the perpendicular bisector of $AD$ at $E$, and the circumcircle of $MDE$ hits the circumcircle of $ABC$ at $F$. Let $G$ be the foot of the perpendicular from $A$ to $BD$, and $N$ be the midpoint of $AG$. Prove that $B, N, F$ are collinear.

p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$. p2. If $a, b$, and $c$ satisfy the system of equations $$ \frac{ab}{a+b}=\frac12$$ $$\frac{bc}{b+c}=\frac13 $$ $$ \frac{ac}{a+c}=\frac17 $$ Determine the value of $(a- c)^b$. p3. Given triangle $ABC$. If point $M$ is located at the midpoint of $AC$, point $N$ is located at the midpoint of $BC$, and the point $P$ is any point on $AB$. Determine the area of ​​the quadrilateral $PMCN$. [img]https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png[/img] p4. Given the rule of motion of a particle on a flat plane $xy$ as following: $N: (m, n)\to (m + 1, n + 1)$ $T: (m, n)\to (m + 1, n - 1)$, where $m$ and $n$ are integers. How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ? p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

[b]p1.[/b] Consider a $4 \times 4$ lattice on the coordinate plane. At $(0,0)$ is Mori’s house, and at $(4,4)$ is Mori’s workplace. Every morning, Mori goes to work by choosing a path going up and right along the roads on the lattice. Recently, the intersection at $(2, 2)$ was closed. How many ways are there now for Mori to go to work? [b]p2.[/b] Given two integers, define an operation $*$ such that if a and b are integers, then a $*$ b is an integer. The operation $*$ has the following properties: 1. $a * a$ = 0 for all integers $a$. 2. $(ka + b) * a = b * a$ for integers $a, b, k$. 3. $0 \le b * a < a$. 4. If $0 \le b < a$, then $b * a = b$. Find $2017 * 16$. [b]p3.[/b] Let $ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, $CA = 15$. Let $A'$, $B'$, $C'$, be the midpoints of $BC$, $CA$, and $AB$, respectively. What is the ratio of the area of triangle $ABC$ to the area of triangle $A'B'C'$? [b]p4.[/b] In a strange world, each orange has a label, a number from $0$ to $10$ inclusive, and there are an infinite number of oranges of each label. Oranges with the same label are considered indistinguishable. Sally has 3 boxes, and randomly puts oranges in her boxes such that (a) If she puts an orange labelled a in a box (where a is any number from 0 to 10), she cannot put any other oranges labelled a in that box. (b) If any two boxes contain an orange that have the same labelling, the third box must also contain an orange with that labelling. (c) The three boxes collectively contain all types of oranges (oranges of any label). The number of possible ways Sally can put oranges in her $3$ boxes is $N$, which can be written as the product of primes: $$p_1^{e_1} p_2^{e_2}... p_k^{e_k}$$ where $p_1 \ne p_2 \ne p_3 ... \ne p_k$ and $p_i$ are all primes and $e_i$ are all positive integers. What is the sum $e_1 + e_2 + e_3 +...+ e_k$? [b]p5.[/b] Suppose I want to stack $2017$ identical boxes. After placing the first box, every subsequent box must either be placed on top of another one or begin a new stack to the right of the rightmost pile. How many different ways can I stack the boxes, if the order I stack them doesn’t matter? Express your answer as $$p_1^{e_1} p_2^{e_2}... p_n^{e_n}$$ where $p_1, p_2, p_3, ... , p_n$ are distinct primes and $e_i$ are all positive integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

Let $C$ be a cube. Let $P$, $Q$, and $R$ be random vertices of $C$, chosen uniformly and independently from the set of vertices of $C$. (Note that $P$, $Q$, and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$, $Q$, and $R$.

Find the locus of the projections of a given point on all planes containing another point $B$.

Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. What is the degree measure of $\angle BAC$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60 $

Given is a rectangular plane coordinate system. (a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates. (b) In the plane the vertices $A, B$ and $C$ lie with integer coordinates in such a way that $AB = AC$. Prove that $\frac{d(A,BC)}{BC}$ is rational.

Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle. Find the necessary and sufficient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.

Let $\vartriangle ABC$ be equilateral. Two points $D$ and $E$ are on side $BC$ (with order $B, D, E, C$), and satisfy $\angle DAE = 30^o$ . If $BD = 2$ and $CE = 3$, what is $BC$? [img]https://cdn.artofproblemsolving.com/attachments/c/8/27b756f84e086fe31b5ea695f51fb6c78b63d0.png[/img]

Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$