The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.

Luis' friends decided to play a prank on him in his geometry homework. They erased most of a triangle and, instead, drew an equivalent triangle with the sum of its three side lengths. Help Luis complete his homework by reconstructing the original triangle using only a straightedge and compass. Since Luis' method involves measurements, prove that his method results in a triangle longer than the sum of its three sides.

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]R4.16 / P1.4[/b] Adam and Becky are building a house. Becky works twice as fast as Adam does, and they both work at constant speeds for the same amount of time each day. They plan to finish building in $6$ days. However, after $2$ days, their friend Charlie also helps with building the house. Because of this, they finish building in just $5$ days. What fraction of the house did Adam build? [b]R4.17[/b] A bag with $10$ items contains both pencils and pens. Kanye randomly chooses two items from the bag, with replacement. Suppose the probability that he chooses $1$ pen and $1$ pencil is $\frac{21}{50}$ . What are all possible values for the number of pens in the bag? [b]R4.18 / P2.8[/b] In cyclic quadrilateral $ABCD$, $\angle ABD = 40^o$, and $\angle DAC = 40^o$. Compute the measure of $\angle ADC$ in degrees. (In cyclic quadrilaterals, opposite angles sum up to $180^o$.) [b]R4.19 / P2.6[/b] There is a strange random number generator which always returns a positive integer between $1$ and $7500$, inclusive. Half of the time, it returns a uniformly random positive integer multiple of $25$, and the other half of the time, it returns a uniformly random positive integer that isn’t a multiple of $25$. What is the probability that a number returned from the generator is a multiple of $30$? [b]R4.20 / P2.7[/b] Julia is shopping for clothes. She finds $T$ different tops and $S$ different skirts that she likes, where $T \ge S > 0$. Julia can either get one top and one skirt, just one top, or just one skirt. If there are $50$ ways in which she can make her choice, what is $T - S$? [u]Set 5[/u] [b]R5.21[/b] A $5 \times 5 \times 5$ cube’s surface is completely painted blue. The cube is then completely split into $ 1 \times 1 \times 1$ cubes. What is the average number of blue faces on each $ 1 \times 1 \times 1$ cube? [b]R5.22 / P2.10[/b] Find the number of values of $n$ such that a regular $n$-gon has interior angles with integer degree measures. [b]R5.23[/b] $4$ positive integers form an geometric sequence. The sum of the $4$ numbers is $255$, and the average of the second and the fourth number is $102$. What is the smallest number in the sequence? [b]R5.24[/b] Let $S$ be the set of all positive integers which have three digits when written in base $2016$ and two digits when written in base $2017$. Find the size of $S$. [b]R5.25 / P3.12[/b] In square $ABCD$ with side length $13$, point $E$ lies on segment $CD$. Segment $AE$ divides $ABCD$ into triangle $ADE$ and quadrilateral $ABCE$. If the ratio of the area of $ADE$ to the area of $ABCE$ is $4 : 11$, what is the ratio of the perimeter of $ADE$ to the perimeter of $ABCE$? [u]Set 6[/u] [b]R6.26 / P6.25[/b] Submit a decimal n to the nearest thousandth between $0$ and $200$. Your score will be $\min (12, S)$, where $S$ is the non-negative difference between $n$ and the largest number less than or equal to $n$ chosen by another team (if you choose the smallest number, $S = n$). For example, 1.414 is an acceptable answer, while $\sqrt2$ and $1.4142$ are not. [b]R6.27 / P6.27[/b] Guang is going hard on his YNA project. From $1:00$ AM Saturday to $1:00$ AM Sunday, the probability that he is not finished with his project $x$ hours after $1:00$ AM on Saturday is $\frac{1}{x+1}$ . If Guang does not finish by 1:00 AM on Sunday, he will stop procrastinating and finish the project immediately. Find the expected number of minutes $A$ it will take for him to finish his project. An estimate of $E$ will earn $12 \cdot 2^{-|E-A|/60}$ points. [b]R6.28 / P6.28[/b] All the diagonals of a regular $100$-gon (a regular polygon with $100$ sides) are drawn. Let $A$ be the number of distinct intersection points between all the diagonals. Find $A$. An estimate of $E$ will earn $12 \cdot \left(16 \log_{10}\left(\max \left(\frac{E}{A},\frac{A}{E}\right)\right)+ 1\right)^{-\frac12}$ or $0$ points if this expression is undefined. [b]R6.29 / P6.29 [/b]Find the smallest positive integer $A$ such that the following is true: if every integer $1, 2, ..., A$ is colored either red or blue, then no matter how they are colored, there are always 6 integers among them forming an increasing arithmetic progression that are all colored the same color. An estimate of $E$ will earn $12 min \left(\frac{E}{A},\frac{A}{E}\right)$ points or $0$ points if this expression is undefined. [b]R6.30 / P6.30[/b] For all integers $n \ge 2$, let $f(n)$ denote the smallest prime factor of $n$. Find $A =\sum^{10^6}_{n=2}f(n)$. In other words, take the smallest prime factor of every integer from $2$ to $10^6$ and sum them all up to get $A$. You may find the following values helpful: there are $78498$ primes below $10^6$, $9592$ primes below $10^5$, $1229$ primes below $10^4$, and $168$ primes below $10^3$. An estimate of $E$ will earn $\max \left(0, 12-4 \log_{10}(max \left(\frac{E}{A},\frac{A}{E}\right)\right)$ or $0$ points if this expression is undefined. PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]carpeting[/b] suppose that $S$ is a figure in the plane such that it's border doesn't contain any lattice points. suppose that $x,y$ are two lattice points with the distance $1$ (we call a point lattice point if it's coordinates are integers). suppose that we can cover the plane with copies of $S$ such that $x,y$ always go on lattice points ( you can rotate or reverse copies of $S$). prove that the area of $S$ is equal to lattice points inside it. time allowed for this question was 1 hour.

In the triangle $ABC$, let $AM$ and $CN$ internal bisectors, with $M$ in $BC$ and $N$ in $AB$. Prove that if $$\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}$$ then $ABC$ is isosceles.

$AB=CD,AD \parallel BC$ and $AD>BC$. $\Omega$ is circumcircle of $ABCD$. Point $E$ is on $\Omega$ such that $BE \perp AD$. Prove that $AE+BC>DE$

If possible, construct an equilateral triangle whose three vertices are on three given circles.

Let $ABC$ be a triangle with $b\neq c$. Points $D$ is the midpoint of $BC$ and let $E$ be the foot of angle $A$ bisector. In the exterior of the triangle we construct the similar triangles $AMB$ and $ANC$ . Prove: a) $MN\bot AD \Longleftrightarrow MA \bot AB$ b) $MN\bot AE \Longleftrightarrow M,A,N$ are colinear.

Inside a convex quadrilateral $ABCD$ there is a point $P$ such that the triangles $PAB, PBC, PCD, PDA$ have equal areas. Prove that the area of $ABCD$ is bisected by one of the diagonals.

Let $ ABCDEF$ be a convex hexagon such that $ AD \equal{} BC \plus{} EF$, $ BE \equal{} AF \plus{} CD$, $ CF \equal{} DE \plus{} AB$. Prove that: \[ \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}. \]

Let \(AOB\) be a given angle less than \(180^{\circ}\) and let \(P\) be an interior point of the angular region determined by \(\angle AOB\). Show, with proof, how to construct, using only ruler and compass, a line segment \(CD\) passing through \(P\) such that \(C\) lies on the way \(OA\) and \(D\) lies on the ray \(OB\), and \(CP:PD=1:2\).

Let $AK$ and $BL$ be the altitudes of an acute-angled triangle $ABC$, and let $\omega$ be the excircle of $ABC$ touching side $AB$. The common internal tangents to circles $CKL$ and $\omega$ meet $AB$ at points $P$ and $Q$. Prove that $AP =BQ$. [i]Proposed by I.Frolov[/i]

Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$

Suppose two convex quadrangles are such that the sides of each of them lie on the perpendicular bisectors of the sides of the other one. Determine their angles,

(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that $$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in the circle with center $A_1$ and radius $R$. Prove that there is a vertex $S$ of the pentagon for which $$SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.$$

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$. (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$. (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$

Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$

Let $I$ be the incenter of triangle $ABC$, and let $A'$, $B'$, and $C'$ be the circumcenters of triangles $IBC$, $ICA$, and $IAB$, respectively. Prove that the circumcircles of triangles $ABC$ and $A'B'C'$ are concentric.

Let $ABC$ be a triangle inscribed in circle $(I)$ that is tangent to the sides $BC,CA,AB$ at points $D,E, F$ respectively. Assume that $L$ is the intersection of $BE$ and $CF,G$ is the centroid of triangle $DEF,K$ is the symmetric point of $L$ about $G$. If $DK$ meets $EF$ at $P, Q$ is on $EF$ such that $QF = PE$, prove that $\angle DGE + \angle FGQ = 180^o$. Nguyễn Minh Hà

Let be two quadrilaterals $ ABCD,A'B'C'D' $ with $ AB,BC,CD,AC,BD $ being perpendicular to $ A'B',B'C',C'D',A'C',B'D', $ respectively. Show that $ AD $ is perpendicular to $ A'D'. $

Bisectors of $\angle BAC$, $\angle CAD$ in a rectangle $ABCD$ , intersect the sides $BC$, $CD$ at points $M$ and $N$ resp. Prove that $\frac{(MB)}{(MC)}+\frac{(ND)}{(NC)}>1$

Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$. $\Gamma_1$ is a circle tangent to the lines $\Delta$, $(AB)$ and $(BC)$, and $E$ its touchpoint with the line $(AB)$. Let $\Gamma_2$ be a circle tangent to the lines $\Delta$, $(AC)$ and $(BC)$, and $F$ its touchpoint with the line $(AC)$. We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$, and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$. Show that the lines $(BC)$ and $(EF)$ are parallel.

Construct a square, if you know its center and two points that lie on adjacent sides.