Let $ABC$ be a triangle, satisfying $2AC=AB+BC$. If $O$ and $I$ are its circumcenter and incenter, show that $\angle OIB=90^{\circ}$.

Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle? $ \textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad $

Let $I$ be the incenter of triangle $ABC$, and $K$ be the common point of $BC$ with the external bisector of angle $A$. The line $KI$ meets the external bisectors of angles $B$ and $C$ at points $X$ and $Y$ . Prove that $\angle BAX = \angle CAY$

The altitude of the acute triangle $ABC$ drawn from $A$ , intersects the side $BC$ at $A_1$ and the circumscribed circle at $A_2$ (different from $A$). Similarly, we get the points $B_1$, $B_2$, $C_1$, $C_2$. Prove that $$\frac{AA_2}{AA_1}+\frac{BB_2}{BB_1}+\frac{CC_2}{CC_1}= 4.$$

[b]p1.[/b] Find the largest k such that the equation $x^2 - 2x + k = 0$ has at least one real root. [b]p2.[/b] Indiana Jones needs to cross a flimsy rope bridge over a mile long gorge. It is so dark that it is impossible to cross the bridge without a flashlight. Furthermore, the bridge is so weak that it can only support the weight of two people. The party has only one flashlight, which has a weak beam so whenever two people cross, they are constrained to walk together, at the speed of the slower person. Indiana Jones can cross the bridge in $5$ minutes. His girlfriend can cross in $10$ minutes. His father needs $20$ minutes, and his father’s side kick needs $25$ minutes. They need to get everyone across safely in on hour to escape the bay guys. Can they do it? [b]p3.[/b] There are ten big bags with coins. Nine of them contain fare coins weighing $10$ g. each, and one contains counterfeit coins weighing $9$ g. each. By one weighing on a digital scale find the bag with counterfeit coins. [b]p4.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 5x + 5$. [b]p5.[/b] (a) In the $x - y$ plane, analytically determine the length of the path $P \to A \to C \to B \to P$ around the circle $(x - 6)^2 + (y - 8)^2 = 25$ from the point $P(12, 16)$ to itself. [img]https://cdn.artofproblemsolving.com/attachments/f/b/24888b5b478fa6576a54d0424ce3d3c6be2855.png[/img] (b) Determine coordinates of the points $A$ and $B$. [b]p6.[/b] (a) Let $ABCD$ be a convex quadrilateral (it means that diagonals are inside the quadrilateral). Prove that $$Area\,\, (ABCD) \le \frac{|AB| \cdot |AD| + |BC| \cdot |CD|}{2}$$ (b) Let $ABCD$ be an arbitrary quadrilateral (not necessary convex). Prove the same inequality as in part (a). (c) For an arbitrary quadrilateral $ABCD$ prove that $Area\,\, (ABCD) \le \frac{|AB| \cdot |CD| + |BC| \cdot |AD|}{2}$ PS. You should use hide for answers.

The insphere of a tetrahedron $ABCD$ touches the faces $ABC$, $BCD$, $CDA$, $DAB$ at $D^{\prime}$, $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ respectively. Denote by $S_{AB}$ the area of the triangle $AC^{\prime}B^{\prime}$. Define similarly $S_{AC}$, $S_{BC},$ $S_{AD}$, $S_{BD}$, $S_{CD}$. Prove that there exists a triangle with sidelengths $\sqrt{S_{AB}S_{CD}}$, $\sqrt{S_{AC}S_{BD}}$ , $\sqrt{S_{AD}S_{BC}}$. Proposed by: S.Arutyunyan

We have an equilateral triangle with circumradius $1$. We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$, is maximum. (The total distance of the point P from the sides of an equilateral triangle is fixed ) [i]Proposed by Erfan Salavati[/i]

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.

Three spheres $s_1$, $s_2$, $s_3$ intersect along one circle $\omega$. Let $A $be an arbitrary point lying on the circle $\omega$. Ray $AB$ intersects spheres $s_1$, $s_2$, $s_3$ at points $B_1$, $B_2$, $B_3$, respectively, ray $AC$ intersects spheres $s_1$, $s_2$, $s_3$ at points $C_1$, $C_2$, $C_3$, respectively ($B_i \ne A_i$, $C_i \ne A_i$, $i=1,2,3$). It is known that $B_2$ is the midpoint of the segment $B_1B_3$. Prove that $C_2$ is the midpoint of the segment $C_1C_3$.

In triangle $ABC$, $90^{o}> \angle A> \angle B> \angle C$. Let the circumcenter and orthocenter of the triangle be $O$ and $H$. $OH$ intersects $BC$ at $T$ and the circumcenter of $(AHO)$ is $X$. Prove that the reflection of $H$ over $XT$ lies on the circumcircle of triangle $ABC$.

A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis. Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.

Let $P$ be a point inside the acute triangle $ABC$ with $m(\widehat{PAC})=m(\widehat{PCB})$. $D$ is the midpoint of the segment $PC$. $AP$ and $BC$ intersect at $E$, and $BP$ and $DE$ intersect at $Q$. Prove that $\sin\widehat{BCQ}=\sin\widehat{BAP}$.

Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.

Let $\Gamma_1$ and $\Gamma_2$ be two circles intersecting at $P$ and $Q$. The common tangent, closer to $P$, of $\Gamma_1$ and $\Gamma_2$ touches $\Gamma_1$ at $A$ and $\Gamma_2$ at $B$. The tangent of $\Gamma_1$ at $P$ meets $\Gamma_2$ at $C$, which is different from $P$, and the extension of $AP$ meets $BC$ at $R$. Prove that the circumcircle of triangle $PQR$ is tangent to $BP$ and $BR$.

Circles with centres $O_1$ and $O_2$ intersect in two points, let one of which be $A$. The common tangent of these circles touches them respectively in points $P$ and $Q$. It is known that points $O_1, A$ and $Q$ are on a common straight line and points $O_2, A$ and $P$ are on a common straight line. Prove that the radii of the circles are equal.

Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$. Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$. Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.

Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.

In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.

In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.

From point $M$ in triangle $ABC$ perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the foot of the perpendicular drawn to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.

In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.

[u]Round 5[/u] [b]5.1.[/b] A triangle has lengths such that one side is $12$ less than the sum of the other two sides, the semi-perimeter of the triangle is $21$, and the largest and smallest sides have a difference of $2$. Find the area of this triangle. [b]5.2.[/b] A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus? [b]5.3.[/b] A drink from YAKSHAY’S SHAKE SHOP is served in a container that consists of a cup, shaped like an upside-down truncated cone, and a semi-spherical lid. The ratio of the radius of the bottom of the cup to the radius of the lid is $\frac23$ , the volume of the combined cup and lid is $296\pi$, and the height of the cup is half of the height of the entire drink container. What is the volume of the liquid in the cup if it is filled up to half of the height of the entire drink container? [u]Round 6[/u] [i]Each answer in the next set of three problems is required to solve a different problem within the same set. There is one correct solution to all three problems; however, you will receive points for any correct answer regardless whether other answers are correct.[/i] [b]6.1.[/b] Let the answer to problem $2$ be $b$. There are b people in a room, each of which is either a truth-teller or a liar. Person $1$ claims “Person $2$ is a liar,” Person $2$ claims “Person $3$ is a liar,” and so on until Person $b$ claims “Person $1$ is a liar.” How many people are truth-tellers? [b]6.2.[/b] Let the answer to problem $3$ be $c$. What is twice the area of a triangle with coordinates $(0, 0)$, $(c, 3)$ and $(7, c)$ ? [b]6.3.[/b] Let the answer to problem $ 1$ be $a$. Compute the smaller zero to the polynomial $x^2 - ax + 189$ which has $2$ integer roots. [u]Round 7[/u] [b]7.1. [/b]Sir Isaac Neeton is sitting under a kiwi tree when a kiwi falls on his head. He then discovers Neeton’s First Law of Kiwi Motion, which states: [i]Every minute, either $\left\lfloor \frac{1000}{d} \right\rfloor$ or $\left\lceil \frac{1000}{d} \right\rceil$ kiwis fall on Neeton’s head, where d is Neeton’s distance from the tree in centimeters.[/i] Over the next minute, $n$ kiwis fall on Neeton’s head. Let $S$ be the set of all possible values of Neeton’s distance from the tree. Let m and M be numbers such that $m < x < M$ for all elements $x$ in $S$. If the least possible value of $M - m$ is $\frac{2000}{16899}$ centimeters, what is the value of $n$? Note that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the least integer greater than or equal to $x$. [b]7.2.[/b] Nithin is playing chess. If one queen is randomly placed on an $ 8 \times 8$ chessboard, what is the expected number of squares that will be attacked including the square that the queen is placed on? (A square is under attack if the queen can legally move there in one move, and a queen can legally move any number of squares diagonally, horizontally or vertically.) [b]7.3.[/b] Nithin is writing binary strings, where each character is either a $0$ or a $1$. How many binary strings of length $12$ can he write down such that $0000$ and $1111$ do not appear? [u]Round 8[/u] [b]8.[/b] What is the period of the fraction $1/2018$? (The period of a fraction is the length of the repeated portion of its decimal representation.) Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2765571p24215461]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given $n \ge 4$ points in the plane such that any four of them are the vertices of a convex quadrilateral, prove that these points are the vertices of a convex polygon.

Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.

In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfies $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.