A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.

I'd really appreciate help on this. (a) Given a set $X$ of points in the plane, let $f_{X}(n)$ be the largest possible area of a polygon with at most $n$ vertices, all of which are points of $X$. Prove that if $m, n$ are integers with $m \geq n > 2$ then $f_{X}(m) + f_{X}(n) \geq f_{X}(m + 1) + f_{X}(n - 1)$. (b) Let $P_0$ be a $1 \times 2$ rectangle (including its interior) and inductively define the polygon $P_i$ to be the result of folding $P_{i-1}$ over some line that cuts $P_{i-1}$ into two connected parts. The diameter of a polygon $P_i$ is the maximum distance between two points of $P_i$. Determine the smallest possible diameter of $P_{2013}$.

The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.

Consider the graph in $3$-space of \[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\] This graph divides $3$-space into $N$ connected regions. What is $N$?

Let $H$ be the intersection of the altitudes of an acute triangle $ABC$ and $D$ be the midpoint of $[AC]$. Show that $DH$ passes through one of the intersection point of the circumcircle of $ABC$ and the circle with diameter $[BH]$.

Given a regular tetrahedron $ABCD$ with center $O$, find $\sin \angle AOB$.

Suppose $P$ is an interior point of a triangle $ABC$ such that the ratios \[ \frac{d(A,BC)}{d(P,BC)} , \frac{d(B,CA)}{d(P,CA)} , \frac{d(C,AB)}{d(P,AB)} \] are all equal. Find the common value of these ratios. $d(X,YZ)$ represents the perpendicular distance fro $X$ to the line $YZ$.

Let $ABCDEF$ be a convex hexagon with the following properties. (a) $\overline{AC}$ and $\overline{AE}$ trisect $\angle BAF$. (b) $\overline{BE} \parallel \overline{CD}$ and $\overline{CF} \parallel \overline{DE}$. (c) $AB = 2AC = 4AE = 8AF$. Suppose that quadrilaterals $ACDE$ and $ADEF$ have area $2014$ and $1400$, respectively. Find the area of quadrilateral $ABCD$.

Given a triangle $ABC$, $A_1$ divides the length of the path $CAB$ into two equal parts, and define $B_1$ and $C_1$ analogously. Let $l_A$, $l_B$, $l_C$ be the lines passing through $A_1$, $B_1$ and $C_1$ and being parallel to the bisectors of $\angle A$, $\angle B$, and $\angle C$. Show that $l_A$, $l_B$, $l_C$ are concurrent.

Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$ (Nairi Sedrakyan, Ilya Bogdanov)

A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.

[b]p1.[/b] If Bob takes $6$ hours to build $4$ houses, how many hours will he take to build $ 12$ houses? [b]p2.[/b] Compute the value of $\frac12+ \frac16+ \frac{1}{12} + \frac{1}{20}$. [b]p3.[/b] Given a line $2x + 5y = 170$, find the sum of its $x$- and $y$-intercepts. [b]p4.[/b] In some future year, BmMT will be held on Saturday, November $19$th. In that year, what day of the week will April Fool’s Day (April $1$st) be? [b]p5.[/b] We distribute $78$ penguins among $10$ people in such a way that no person has the same number of penguins and each person has at least one penguin. If Mr. Popper (one of the $10$ people) wants to take as many penguins as possible, what is the largest number of penguins that Mr. Popper can take? [b]p6.[/b] A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry? [b]p7. [/b]Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are distinct? [b]p8.[/b] The pages of a book are consecutively numbered from $1$ through $480$. How many times does the digit $8$ appear in this numbering? [b]p9.[/b] A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length $2$, what is the area of the flower? [b]p10.[/b] There are two non-consecutive positive integers $a, b$ such that $a^2 - b^2 = 291$. Find $a$ and $b$. [b]p11.[/b] Let $ABC$ be an equilateral triangle. Let $P, Q, R$ be the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Suppose the area of triangle $PQR$ is $1$. Among the $6$ points $A, B, C, P, Q, R$, how many distinct triangles with area $1$ have vertices from that set of $6$ points? [b]p12.[/b] A positive integer is said to be binary-emulating if its base three representation consists of only $0$s and $1$s. Determine the sum of the first $15$ binary-emulating numbers. [b]p13.[/b] Professor $X$ can choose to assign homework problems from a set of problems labeled $ 1$ to $30$, inclusive. No two problems in his assignment can share a common divisor greater than $ 1$. What is the maximum number of problems that Professor $X$ can assign? [b]p14.[/b] Trapezoid $ABCD$ has legs (non-parallel sides) $BC$ and $DA$ of length $5$ and $6$ respectively, and there exists a point $X$ on $CD$ such that $\angle XBC = \angle XAD = \angle AXB = 90^o$ . Find the area of trapezoid $ABCD$. [b]p15.[/b] Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? (Two games are said to be identical if either player wins/loses rounds in the same order in both games.) [b]p16.[/b] Let $ABC$ be a triangle and M be the midpoint of $BC$. If $AB = AM = 5$ and $BC = 12$, what is the area of triangle $ABC$? [b]p17. [/b] A positive integer $n$ is called good if it can be written as $5x+ 8y = n$ for positive integers $x, y$. Given that $42$, $43$, $44$, $45$ and $46$ are good, what is the largest n that is not good? [b]p18.[/b] Below is a $ 7 \times 7$ square with each of its unit squares labeled $1$ to $49$ in order. We call a square contained in the figure [i]good [/i] if the sum of the numbers inside it is odd. For example, the entire square is [i]good [/i] because it has an odd sum of $1225$. Determine the number of [i]good [/i] squares in the figure. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 [hide][img]https://cdn.artofproblemsolving.com/attachments/9/2/1039c3319ae1eab7102433694acc20fb995ebb.png[/hide] [b]p19.[/b] A circle of integer radius $ r$ has a chord $PQ$ of length $8$. There is a point $X$ on chord $PQ$ such that $\overline{PX} = 2$ and $\overline{XQ} = 6$. Call a chord $AB$ euphonic if it contains $X$ and both $\overline{AX}$ and $\overline{XB}$ are integers. What is the minimal possible integer $ r$ such that there exist $6$ euphonic chords for $X$? [b]p20.[/b] On planet [i]Silly-Math[/i], two individuals may play a game where they write the number $324000$ on a whiteboard and take turns dividing the number by prime powers – numbers of the form $p^k$ for some prime $p$ and positive integer $k$. Divisions are only legal if the resulting number is an integer. The last player to make a move wins. Determine what number the first player should select to divide $324000$ by in order to ensure a win. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In trapezoid $ABCD$ with bases $AD, BC$, $AD = 2BC$ and $M$ is midpoint of $AB$. Prove that line $BD$ passes through the midpoint of segment $CM$.

Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

The incenter of $\triangle A_1A_2A_3$ is $I$, and the center of the $A_i$-excircle is $J_i$ ($i=1,2,3$). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ($A_{i+3}:=A_i$ $\forall i$). Prove that the three lines $B_iJ_i$ are concurrent.

Let $ABC$ be a triangle with $AB \neq AC$ and let $M$ be the middle of $BC$. The bisector of $\angle BAC$ intersects the line $BC$ in $Q$. Let $H$ be the foot of $A$ on $BC$. The perpendicular to $AQ$ passing through $A$ intersects the line $BC$ in $S$. Show that $MH \times QS=AB \times AC$.

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ABCD$ is inscribed quadrilateral. Line, that perpendicular to $BD$ intersects segments $AB$ and $BC$ and rays $DA,DC$ at $P,Q,R,S$ . $PR=QS$. $M$ is midpoint of $PQ$. Prove that $AM=CM$

You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done. Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron are all different for the purpose of the coloring . In this way, two colorings are the same only if the painted edges they are the same.

Consider the graphs of (1): $ y\equal{}x^2\minus{}\frac{1}{2}x\plus{}2$ and (2) $ y\equal{}x^2\plus{}\frac{1}{2}x\plus{}2$ on the same set of axis. These parabolas are exactly the same shape. Then: $ \textbf{(A)}\ \text{the graphs coincide.} \\ \textbf{(B)}\ \text{the graph of (1) is lower than the graph of (2).} \\ \textbf{(C)}\ \text{the graph of (1) is to the left of the graph of (2).} \\ \textbf{(D)}\ \text{the graph of (1) is to the right of the graph of (2).} \\ \textbf{(E)}\ \text{the graph of (1) is higher than the graph of (2).}$

For any pair $(x,y)$ of real numbers, a sequence $(a_{n}(x,y))$ is defined as follows: $$a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0$$ Find the area of the region $\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}$.

In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $ \odot (ABC) $ at $D$(different from point $A$). Prove that $S_{AMDN}=S_{\triangle ABC}.$

Let $ABC$ be a triangle, $H$ the orthocenter and $M$ the midpoint of $AC$. Let $\ell$ be the parallel through $M$ to the bisector of $\angle AHC$. Prove that $\ell$ divides the triangle in two parts of equal perimeters. [i]Pedro Marrone, Panamá[/i]

Let be a triangle $ ABC $ and the points $ E,F,M,N $ positioned in this way: $ E,F $ on the segment $ BC $ (excluding its endpoints), $ M $ on the segment $ AC $ (excluding its endpoints) and $ N $ on the segment $ AC $ (excluding its endpoints). Knowing that $ BAE $ is similar to $ FAC $ and that $ BE=BM,FC=CN,AM=AN, $ show that $ ABC $ is isosceles.

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$