Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly. Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.

Points $C_1, A_1$ and $B_1$ are marked on the sides $AB, BC$ and $CA$ of a triangle $ABC$ so that the segments $AA_1, BB_1$, and $CC_1$ are concurrent (see the fig.). It is known that the area of the white part of the triangle $ABC$ is equal to the area of its black part. Prove that at least one of the segments $AA_1, BB_1, CC_1$ is a median of the triangle $ABC$. [img]https://1.bp.blogspot.com/-nVVhqdRdf0s/X-WVmt_gyqI/AAAAAAAAM40/943sCRGyCPwT-vqIilTCtXOXHByRLIvPwCLcBGAsYHQ/s0/2014%2Bbelarus%2B11.6.png[/img]

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

An equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that intersects the hexagon at $12$ distinct points, we have $M < a < N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$ .

Let $I$ be the incenter of a right-angled triangle $ABC$, and $M$ be the midpoint of hypothenuse $AB$. The tangent to the circumcircle of $ABC$ at $C$ meets the line passing through $I$ and parallel to $AB$ at point $P$. Let $H$ be the orthocenter of triangle $PAB$. Prove that lines $CH$ and $PM$ meet at the incircle of triangle $ABC$.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$. (a) Prove that $P$, $T$, $S$ are collinear. (b) Prove that $P$, $K$, $L$ are collinear. [i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]

Let $ABC$ be a triangle with $BC = 5$, $CA = 3$, and $AB = 4$. Variable points $P, Q$ are on segments $AB$, $AC$, respectively such that the area of $APQ$ is half of the area of $ABC$. Let $x$ and $y$ be the lengths of perpendiculars drawn from the midpoint of $PQ$ to sides $AB$ and $AC$, respectively. Find the range of values of $2y + 3x$.

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.

Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis. Prove that : \[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ with $AC\ne BC$. The line through the midpoints of the segments $AB$ and $HC$ intersects the bisector of $\angle ACB$ at $D$. Suppose that the line $HD$ contains the circumcenter of $\triangle ABC$. Determine $\angle ACB$.

Consider the set of triangles with side lengths $1 \le x \le y \le z$ such that $x$, $y$, and $z$ are the solutions to the equation $t^3-at^2+bt = 12$ for some real numbers $a$ and $b$. Compute the smallest real number $N$ such that $N > ab$ for any choice of $x$, $y$, and $z$.

Unit square $ABCD$ is drawn on a plane. Point $O$ is drawn outside of $ABCD$ such that lines $AO$ and $BO$ are perpendicular. Square $F ROG$ is drawn with $F$ on $AB$ such that $AF =\frac23$, $R$ is on $\overline{BO}$, and $G$ is on $\overline{AO}$. Extend segment $\overline{OF}$ past $\overline{AB}$ to intersect side $\overline{CD}$ at $E$. Compute $DE$.

[u]Round 1[/u] [b]p1.[/b] A $\$100$ TV has its price increased by $10\%$. The new price is then decreased by $10\%$. What is the current price of the TV? [b]p2.[/b] If $9w + 8x + 7y = 42$ and $w + 2x + 3y = 8$, then what is the value of $100w + 101x + 102y$? [b]p3.[/b] Find the number of positive factors of $37^3 \cdot 41^3$. [u]Round 2[/u] [b]p4.[/b] Three hoses work together to fill up a pool, and each hose expels water at a constant rate. If it takes the first, second, and third hoses 4, 6, and 12 hours, respectively, to fill up the pool alone, then how long will it take to fill up the pool if all three hoses work together? [b]p5.[/b] A semicircle has radius $1$. A smaller semicircle is inscribed in the larger one such that the two bases are parallel and the arc of the smaller is tangent to the base of the larger. An even smaller semicircle is inscribed in the same manner inside the smaller of the two semicircles, and this procedure continues indefinitely. What is the sum of all of the areas of the semicircles? [b]p6.[/b] Given that $P(x)$ is a quadratic polynomial with $P(1) = 0$, $P(2) = 0$, and $P(0) = 2012$, find $P(-1)$. [u]Round 3[/u] [b]p7.[/b] Darwin has a paper circle. He labels one point on the circumference as $A$. He folds $A$ to every point on the circumference on the circle and undoes it. When he folds $A$ to any point $P$, he makes a blue mark on the point where $\overline{AP}$ and the made crease intersect. If the area of Darwin paper circle is 80, then what is the area of the region surrounded by blue? [b]p8.[/b] Α rectangular wheel of dimensions $6$ feet by $8$ feet rolls for $28$ feet without sliding. What is the total distance traveled by any corner on the rectangle during this roll? [b]p9[/b]. How many times in a $24$-hour period do the minute hand and hour hand of a $12$-hour clock form a right angle? [u]Round 4[/u] The answers in this section all depend on each other. Find smallest possible solution set. [b]p10.[/b] Let B be the answer to problem $11$. Right triangle $ACD$ has a right angle at $C$. Squares $ACEF$ and $ADGH$ are drawn such that points $D$ and $E$ do not coincide and points $E$ and $H$ do not coincide. The midpoints of the sides of $ADGH$ are connected to form a smaller square with area $B.$ If the area of $ACEF$ is also $B$, then find the length $CD$ rounded up to the nearest integer. [b]p11.[/b] Let $C$ be the answer to problem $12$. Find the sum of the digits of $C$. [b]p12.[/b] Let $A$ be the answer to problem $10$. Given that $a_0 = 1$, $a_1 = 2$, and that $a_n = 3a_{n-1 }-a_{n-2}$ for $n \ge 2$, find $a_A$. PS. You should use hide for answers.Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle is inscribed in a regular octagon with area $2024$. A second regular octagon is inscribed in the circle, and its area can be expressed as $a + b\sqrt{c}$, where $a, b, c$ are integers and $c$ is square-free. Compute $a + b + c$.

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

Let $M$ be an arbitrary point of a circle circumscribed around an acute-angled triangle $ABC$. Prove that the product of the distances from the point $M$ to the sides $AC$ and $BC$ is equal to the product of the distances from $M$ to the side $AB$ and to the tangent to the circumscribed circle at point $C$.

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 1: 3 \qquad\textbf{(C)}\ 1: \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3}: 2 \qquad\textbf{(E)}\ 2: 3$

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.

Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of Bob's tower, and Bob shines his flashlight at the top of Alice's tower by first reflecting it off the ice. The light from Alice's tower travels $16$ meters to get to Bob's tower, while the light from Bob's tower travels $26$ meters to get to Alice's tower. Assuming that the lights are both shown from exactly the top of their respective towers, what are the possibilities for the height of Alice's tower?

What is the radius of the circle passing through the center of the square $ABCD$ with side length $1$, its corner $A$, and midpoint of its side $[BC]$? $ \textbf{(A)}\ \dfrac {\sqrt 3}4 \qquad\textbf{(B)}\ \dfrac {\sqrt 5}4 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ \sqrt 3 \qquad\textbf{(E)}\ \dfrac {\sqrt {10}}4 $

Let $ABCD$ be a convex quadrilateral such that $\angle A, \angle B, \angle C$ are acute. $AB$ and $CD$ meet at $E$ and $BC,DA$ meet at $F$. Let $K,L,M,N$ be the midpoints of $AB,BC,CD,DA$ repectively. $KM$ meets $BC,DA$ at $X$ and $Y$, and $LN$ meets $AB,CD$ at $Z$ and $W$. Prove that the line passing $E$ and the midpoint of $ZW$ is parallel to the line passing $F$ and the midpoint of $XY$.