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] $\textbf{(A) }84\qquad\textbf{(B) }120\qquad\textbf{(C) }132\qquad\textbf{(D) }144\qquad\textbf{(E) }168$

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.

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.

[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.

A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: i) No line passes through any point of the configuration. ii) No region contains points of both colors. Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines. Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]

Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? $\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$ [asy] draw((0,0)--(0,8)--(6,8)--(6,0)--cycle); draw((0,8)--(5,9)--(5,8)); draw((3,-1.5)--(3,10.3),dashed); draw((0,5.5)..(-.75,4.75)..(0,4)); draw((0,4)--(1.5,4),EndArrow); [/asy]

The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

Let $ABCD$ be a rectangle and let $M, N$ and $P, Q$ be the points of intersections of some line $e$ with the sides $AB, CD$ and $AD, BC$, respectively (or their extensions). Given the points $M, N, P, Q$ and the length $p$ of side $AB$, construct the rectangle. Under what conditions can this problem be solved, and how many solutions does it have?

Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.

Let $ABCDEF$ be a regular hexagon with side length $1,$ and let $G$ be the midpoint of side $\overline{CD},$ and define $H$ to be the unique point on side $\overline{DE}$ such that $AGHF$ is a trapezoid. Find the length of the altitude dropped from point $H$ to $\overline{AG}.$

In all triangles $ABC$ does it hold that: $$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.

Let $ABCD$ be a quadrilateral $\angle B = \angle D$ and $AD = CD$. The incircle of triangle $ABC$ touches the sides $BC$ and $AB$ at points $E$ and $F$ respectively. Prove that the midpoints of segments $AC, BD, AE,$ and $CF$ are concyclic.

Let $ABC$ be an acute-angled triangle and $D$ a point on the altitude through $C$. Let $E$, $F$, $G$ and $H$ be the midpoints of the segments $AD$, $BD$, $BC$ and $AC$. Show that $E$, $F$, $G$, and $H$ form a rectangle. (G. Anegg, Innsbruck)

The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.

Let $ABC$ be an acute triangle. Let $M_A, M_B$ and $M_C$ be the midpoints of sides $BC,CA$, respectively $AB$. Let $M'_A , M'_B$ and $M'_C$ be the the midpoints of the arcs $BC, CA$ and $AB$ respectively of the circumscriberd circle of triangle $ABC$. Let $P_A$ be the intersection of the straight line $M_BM_C$ and the perpendicular to $M'_BM'_C$ through $A$. Define $P_B$ and $P_C$ similarly. Show that the straight line $M_AP_A, M_BP_B$ and $M_CP_C$ intersect at one point.

In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.

The incircle of $ \triangle ABC$ touches $ BC$, $ AC$, and $ AB$ at $ A_1$, $ B_1$, and $ C_1$, respectively. The line $ AA_1$ intersects the incircle at $ Q$, again. $ A_1C_1$ and $ A_1B_1$ intersect the line, passing through $ A$ and parallel to $ BC$, at $ P$ and $ R$, respectively. If $ \angle PQC_1 \equal{} 45^\circ$ and $ \angle RQB_1 \equal{} 65^\circ$, then $ \angle PQR$ will be ? $\textbf{(A)}\ 110^\circ \qquad\textbf{(B)}\ 115^\circ \qquad\textbf{(C)}\ 120^\circ \qquad\textbf{(D)}\ 125^\circ \qquad\textbf{(E)}\ 130^\circ$

In space there are given four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of the segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint.

Let $ABCD$ be a rectangle, $AB = a, BC = b$. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being $d$) that are at an the angle $\phi, 0 \leq \phi \leq 90^{\circ},$ with respect to $AB$. Let $L$ be the sum of the lengths of all the segments intersecting the rectangle. Find: [i](a)[/i] how $L $ varies, [i](b)[/i] a necessary and sufficient condition for $L$ to be a constant, and [i](c)[/i] the value of this constant.

$ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then $AO + AQ + AR$ equals [asy] size(200); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair O=origin, A=2*dir(90), B=2*dir(18), C=2*dir(306), D=2*dir(234), E=2*dir(162), P=(C+D)/2, Q=C+3.10*dir(C--B), R=D+3.10*dir(D--E), S=C+4.0*dir(C--B), T=D+4.0*dir(D--E); draw(A--B--C--D--E--A^^E--T^^B--S^^R--A--Q^^A--P^^rightanglemark(A,Q,S,7)^^rightanglemark(A,R,T,7)); dot(O); label("$O$",O,dir(B)); label("$1$",(O+P)/2,W); label("$A$",A,dir(A)); label("$B$",B,dir(B)); label("$C$",C,dir(C)); label("$D$",D,dir(D)); label("$E$",E,dir(E)); label("$P$",P,dir(P)); label("$Q$",Q,dir(Q-A)); label("$R$",R,dir(R-A)); [/asy] $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 1 + \sqrt{5}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 2 + \sqrt{5}\qquad\textbf{(E)}\ 5 $

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

Let $ABC$ be a triangle with incenter $I$. Two distinct points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that \[ \angle API = \angle AQI = 45^\circ. \] Lines $PQ$ and $BC$ meet at $S$. Let $H$ denote the foot of the altitude from $A$ to $BC$. Prove that $\angle AHI = \angle ISH$. [i]Matsvei Zorka[/i]