A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.

Let $\alpha,\beta,\gamma$ be the angles of a triangle opposite to its sides with lengths $a,b,c$ respectively. Prove the inequality \[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]

Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius $1$ is regular (i.e., has equal sides).

On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $BC$. The point $D$ is the reflection of $A$ over $BC$, and the point $E$ is the intersection of $\Gamma$ and the ray $MD$. Let $S$ be the circumcentre of the triangle $ADE$. Prove that the points $A$, $E$, $M$, $O$, and $S$ lie on the same circle.

Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$

In the plane we have a line $r$ and two points $A$ and $B$ outside the line and in the same half plane. Determine a point $M$ on the line such that the angle of $r$ with $AM$ is double that of $r$ with $BM$. (Consider the smaller angle of two lines of the angles they form).

We consider the inscriptible pentagon $ABCDE$ in which $AB=BC=CD$ and the centroid of the pentagon coincides with the circumcenter. Prove that the pentagon $ABCDE$ is regular. [i]The centroid of a pentagon is the point in the plane of the pentagon whose position vector is equal to the average of the position vectors of the vertices.[/i]

In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$. [i]Proposed by Lewis Chen[/i]

In right triangle $ABC$, $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$. $M$ is the midpoint of $AB$, $P$ is the foot of the altitude from $C$, and $N$ is the midpoint of $CP$. Line $AB$ meets the circumcircle of $CNB$ again at $Q$. $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$ lies on ray $CA$ past $A$ such that $BR = RS$, and $V$ lies on segment $SP$ such that $AV = VP$. Line $SP$ meets the circumcircle of $CPB$ again at $T$. $W$ lies on ray $VA$ past $A$ such that $2AW = ST$, and $O$ is the circumcenter of $SPM$. Prove that lines $OM$ and $BW$ are perpendicular.

The circle $\omega$ lies inside the circle $\Omega$ and touches it internally at $T.$ Let $XY{}$ be a variable chord of the circle $\Omega$ touching $\omega.$ Denote by $X'$ and $Y'$ the midpoints of the arcs $TY{}$ and $TX{}$ which do not contain $X{}$ and $Y{}$ respectively. Prove that all possible lines $X'Y'$ pass through a fixed point. [i]Proposed by F. Petrov[/i]

Circles $k_1$ and $k_2$ intersect at $P$ and $Q$, and $A$ and $B$ are the tangency points of their common tangent that is closer to $P$ (where $A$ is on $k_1$ and $B$ on $k_2$). The tangent to $k_1$ at $P$ intersects $k_2$ again at $C$. The lines $AP$ and $BC$ meet at $R$. Show that the lines $BP$ and $BC$ are tangent to the circumcircle of triangle $PQR$.

$ABCD$ is a cyclic quadrilateral inscribed in circle $O$. Let $O_1$ be the $A$-excenter of $ABC$ and $O_2$ the $A$-excenter of $ABD$. Show that $A, B, O_1, O_2$ is concyclic, and $O$ passes through the center of $(ABO_1O_2)$. Recall that for concyclic $X, Y, Z, W$, the notation $(XYZW)$ denotes the circumcircle of $XYZW$.

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

The area of a square is $144$. An equilateral triangle has the same perimeter as the square. The area of a regular hexagon is $6$ times the area of the equilateral triangle. What is the perimeter of the hexagon?

Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers: $a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.

Let $\Omega_1$ and $\Omega_2$ be two circles intersecting at distinct points $P$ and $Q$. The line tangent to $\Omega_1$ at $P$ passes through $\Omega_2$ at a second point $A$, and the line tangent to $\Omega_2$ at $P$ passes through $\Omega_1$ at a second point $B$. Ray $AQ$ intersects $\Omega_1$ at a second point $C$, and ray $BQ$ intersects $\Omega_2$ at a second point $D$. Suppose that $\angle CPD > \angle APB$ (measuring both angles as the non-reflex angle) and that \[ \frac{\text{Area}(CPD)}{PA \cdot PB} = \frac{1}{4}. \] Find the sum of all possible measures of $\angle APB$ in degrees.

The polygon ABCDEFG (shown on the right) is a regular octagon. Prove that the area of the rectangle $ADEH$ is one half the area of the whole polygon $ABCDEFGH$. [asy] draw((0,1.414)--(1.414,0)--(3.414,0)--(4.828,1.414)--(4.828,3.414)--(3.414,4.828)--(1.414,4.828)--(0,3.414)--(0,1.414)); fill((0,1.414)--(0,3.414)--(4.828,3.414)--(4.828,1.414)--cycle, mediumgrey); label("$B$",(1.414,0),SW); label("$C$",(3.414,0),SE); label("$D$",(4.828,1.414),SE); label("$E$",(4.828,3.414),NE); label("$F$",(3.414,4.828),NE); label("$G$",(1.414,4.828),NW); label("$H$",(0,3.414),NW); label("$A$",(0,1.414),SW); [/asy]

[u]Round 1[/u] [b]p1.[/b] Suppose that gold satisfies the relation $p = v + v^2$, where $p$ is the price and $v$ is the volume. How many pieces of gold with volume $1$ can be bought for the price of a piece with volume $2$? [b]p2.[/b] Find the smallest prime number with each digit greater or equal to $8$. [b]p3.[/b] What fraction of regular hexagon $ZUMING$ is covered by both quadrilateral $ZUMI$ and quadrilateral$ MING$? [u]Round 2[/u] [b]p4.[/b] The two smallest positive integers expressible as the sum of two (not necessarily positive) perfect cubes are $1 = 1^3 +0^3$ and $2 = 1^3 +1^3$. Find the next smallest positive integer expressible in this form. [b]p5.[/b] In how many ways can the numbers $1, 2, 3,$ and $4$ be written in a row such that no two adjacent numbers differ by exactly $1$? [b]p6.[/b] A real number is placed in each cell of a grid with $3$ rows and $4$ columns. The average of the numbers in each column is $2016$, and the average of the numbers in each row is a constant $x$. Compute $x$. [u]Round 3[/u] [b]p7.[/b] Fardin is walking from his home to his oce at a speed of $1$ meter per second, expecting to arrive exactly on time. When he is halfway there, he realizes that he forgot to bring his pocketwatch, so he runs back to his house at $2$ meters per second. If he now decides to travel from his home to his office at $x$ meters per second, find the minimum $x$ that will allow him to be on time. [b]p8.[/b] In triangle $ABC$, the angle bisector of $\angle B$ intersects the perpendicular bisector of $AB$ at point $P$ on segment $AC$. Given that $\angle C = 60^o$, determine the measure of $\angle CPB$ in degrees. [b]p9.[/b] Katie colors each of the cells of a $6\times 6$ grid either black or white. From top to bottom, the number of black squares in each row are $1$, $2$, $3$, $4$, $5$, and $6$, respectively. From left to right, the number of black squares in each column are $6$, $5$, $4$, $3$, $2$, and $1$, respectively. In how many ways could Katie have colored the grid? [u]Round 4[/u] [b]p10.[/b] Lily stands at the origin of a number line. Each second, she either moves $2$ units to the right or $1$ unit to the left. At how many different places could she be after $2016$ seconds? [b]p11.[/b] There are $125$ politicians standing in a row. Each either always tells the truth or always lies. Furthermore, each politician (except the leftmost politician) claims that at least half of the people to his left always lie. Find the number of politicians that always lie. [b]p12.[/b] Two concentric circles with radii $2$ and $5$ are drawn on the plane. What is the side length of the largest square whose area is contained entirely by the region between the two circles? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2934055p26256296]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$. Determine the length of $PQ$.

$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? [img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]