Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| \equal{} |PB|$.

Right triangle $\vartriangle ABC$ with $\angle A = 30^o$ and $\angle B = 90^o$ is inscribed in a circle $\omega_1$ with radius $4$. Circle $\omega_2$ is drawn to be the largest circle outside of $\vartriangle ABC$ that is tangent to both $\overline{BC}$ and $\omega_1$, and circles $\omega_3$ and $\omega_4$ are drawn this same way for sides $\overline{AC}$ and $\overline{AB}$, respectively. Suppose that the intersection points of these smaller circles with the bigger circle are noted as points $D$, $E$, and $F$. Compute the area of triangle $\vartriangle DEF$.

Let $ABC$ be a triangle with incenter $I$. Points $E$ and $F$ are on segments $AC$ and $BC$ respectively such that, $AE=AI$ and $BF=BI$. If $EF$ is the perpendicular bisector of $CI$, then $\angle{ACB}$ in degrees can be written as $\frac{m}{n}$ where $m$ and $n$ are co-prime positive integers. Find the value of $m+n$.

Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $. a) Prove that the lines $ EF $ and $ AB $ are perpendicular. b) Determine the locus of the point $ E $. c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.

If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ \sqrt{50}\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ \sqrt{200}\qquad \textbf{(E)}\ \text{none of these}$

In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}, \overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, $PF = 2$. [i]Proposed by Eugene Chen[/i]

The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$. a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$. Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $ P$. b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I $ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D16.[/b] The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes? [b]D17.[/b] Each digit in a $3$ digit integer is either $1, 2$, or $4$ with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit? [b]D18 / Z11.[/b] How many two digit numbers are there such that the product of their digits is prime? [b]D19 / Z9.[/b] In the coordinate plane, a point is selected in the rectangle defined by $-6 \le x \le 4$ and $-2 \le y \le 8$. What is the largest possible distance between the point and the origin, $(0, 0)$? [b]D20 / Z10.[/b] The sum of two numbers is $6$ and the sum of their squares is $32$. Find the product of the two numbers. [u]Set 5[/u] [b]D21 / Z12.[/b] Triangle $ABC$ has area $4$ and $\overline{AB} = 4$. What is the maximum possible value of $\angle ACB$? [b]D22 / Z13.[/b] Let $ABCD$ be an iscoceles trapezoid with $AB = CD$ and M be the midpoint of $AD$. If $\vartriangle ABM$ and $\vartriangle MCD$ are equilateral, and $BC = 4$, find the area of trapezoid $ABCD$. [b]D23 / Z14.[/b] Let $x$ and $y$ be positive real numbers that satisfy $(x^2 + y^2)^2 = y^2$. Find the maximum possible value of $x$. [b]D24 / Z17.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]D25.[/b] The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$? [u]Set 6[/u] [b]D26 / Z26.[/b] For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get $n$. Estimate the greatest integer $a$ such that $2^a$ evenly divides $n$. [b]D27 / Z27.[/b] Circles of radius $5$ are centered at each corner of a square with side length $6$. If a random point $P$ is chosen randomly inside the square, what is the probability that $P$ lies within all four circles? [b]D28 / Z28.[/b] Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s $4$th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class? [b]D29 / Z29. [/b]Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are $10$ meters west from a roast turkey. Beard, can turn exactly $0.7^o$ and Bored can turn exactly $0.2^o$ degrees. Driving at a consistent $2$ meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey. Suppose Beard gets to the Turkey in about $818.5$ seconds. Estimate the amount of time it will take Bored. [b]D30 / Z30.[/b] Let a be the probability that $4$ randomly chosen positive integers have no common divisor except for $1$. Estimate $300a$. Note that the integers $1, 2, 3, 4$ have no common divisor except for $1$. Remark. This problem is asking you to find $300 \lim_{n\to \infty} a_n$, if $a_n$ is defined to be the probability that $4$ randomly chosen integers from $\{1, 2, ..., n\}$ have greatest common divisor $1$. PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be the set of the points $(x_1, x_2, . . . , x_{2012})$ in $2012$-dimensional space such that $|x_1|+|x_2|+...+|x_{2012}| \le 1$. Let $T$ be the set of points in $2012$-dimensional space such that $\max^{2012}_{i=1}|x_i| = 2$. Let $p$ be a randomly chosen point on $T$. What is the probability that the closest point in $S$ to $p$ is a vertex of $S$?

Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that \[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]

[u]Round 1[/u] [b]p1.[/b] What is $2 + 22 + 1 + 3 - 31 - 3$? [b]p2.[/b] Let $ABCD$ be a rhombus. Given $AB = 5$, $AC = 8$, and $BD = 6$, what is the perimeter of the rhombus? [b]p3.[/b] There are $2$ hats on a table. The first hat has $3$ red marbles and 1 blue marble. The second hat has $2$ red marbles and $4$ blue marbles. Jordan picks one of the hats randomly, and then randomly chooses a marble from that hat. What is the probability that she chooses a blue marble? [u]Round 2[/u] [b]p4.[/b] There are twelve students seated around a circular table. Each of them has a slip of paper that they may choose to pass to either their clockwise or counterclockwise neighbor. After each person has transferred their slip of paper once, the teacher observes that no two students exchanged papers. In how many ways could the students have transferred their slips of paper? [b]p5.[/b] Chad wants to test David's mathematical ability by having him perform a series of arithmetic operations at lightning-speed. He starts with the number of cubic centimeters of silicon in his 3D printer, which is $109$. He has David perform all of the following operations in series each second: $\bullet$ Double the number $\bullet$ Subtract $4$ from the number $\bullet$ Divide the number by $4$ $\bullet$ Subtract $5$ from the number $\bullet$ Double the number $\bullet$ Subtract $4$ from the number Chad instructs David to shout out after three seconds the result of three rounds of calculations. However, David computes too slowly and fails to give an answer in three seconds. What number should David have said to Chad? [b]p6.[/b] Points $D, E$, and $F$ lie on sides $BC$, $CA$, and $AB$ of triangle $ABC$, respectively, such that the following length conditions are true: $CD = AE = BF = 2$ and $BD = CE = AF = 4$. What is the area of triangle $ABC$? [u]Round 3[/u] [b]p7.[/b] In the $2, 3, 5, 7$ game, players count the positive integers, starting with $1$ and increasing, which do not contain the digits $2, 3, 5$, and $7$, and also are not divisible by the numbers $2, 3, 5$, and $7$. What is the fifth number counted? [b]p8.[/b] If A is a real number for which $19 \cdot A = \frac{2014!}{1! \cdot 2! \cdot 2013!}$ , what is $A$? Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ...\cdot 2 \cdot 1$. [b]p9.[/b] What is the smallest number that can be written as both $x^3 + y^2$ and $z^3 + w^2$ for positive integers $x, y, z,$ and $w$ with $x \ne z$? [u]Round 4[/u] [i]Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. In addition, it is given that the answer to each of the following problems is a positive integer less than or equal to the problem number. [/i] [b]p10.[/b] Let $B$ be the answer to problem $11$ and let $C$ be the answer to problem $12$. What is the sum of a side length of a square with perimeter $B$ and a side length of a square with area $C$? [b]p11.[/b] Let $A$ be the answer to problem $10$ and let $C$ be the answer to problem $12$. What is $(C - 1)(A + 1) - (C + 1)(A - 1)$? [b]p12.[/b] Let $A$ be the answer to problem $10$ and let $B$ be the answer to problem $11$. Let $x$ denote the positive difference between $A$ and $B$. What is the sum of the digits of the positive integer $9x$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2915810p26040675]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

Let $AX$ and $BZ$ be altitudes of the triangle $ABC$. Let $AY$ and $BT$ be its angle bisectors. It is given that angles $XAY$ and $ZBT$ are equal. Does this necessarily imply that $ABC$ is isosceles? [i]The Jury[/i]

In a convex quadrilateral $ABCD$, point $M$ is the midpoint of the diagonal $AC$. Prove that if $\angle BAD=\angle BMC=\angle CMD$, then a circle can be inscribed in quadrilateral $ABCD$.

[hide=Note][i]The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.[/i][/hide] For each integer $n\ge2$, consider a regular polygon with $2n$ sides, all of length $1$. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length $1$. Determine the limit inferior and the limit superior of the sequence defined by $$\frac1{n^2}\log_2C(n).$$

Show that in a triangle the altitude of the longest side is at most as long as it the the sum of the lengths of the perpendicular segments drawn from any point on the longest side on the other two sides.

$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.

Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.

A square in the [i]xy[/i]-plane has area [i]A[/i], and three of its vertices have [i]x[/i]-coordinates $2,0,$ and $18$ in some order. Find the sum of all possible values of [i]A[/i].

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

Points $X$ and $Y$ lie on side $\overline{AB}$ of $\vartriangle ABC$ such that $AX = 20$, $AY = 28$, and $AB = 42$. Suppose $XC = 26$ and $Y C = 30$. Find $AC + BC$.

A point $P$ is chosen inside a convex quadrilateral $ABCD$. Could it happen that$$PA = AB, \quad PB = BC, \quad PC = CD \quad \text{and} \quad PD = DA?$$

Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.

(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?