$O$ is a point inside triangle $ABC$ such that $OA=OB+OC$. Suppose $B',C'$ be midpoints of arcs $\overarc{AOC}$ and $AOB$. Prove that circumcircles $COC'$ and $BOB'$ are tangent to each other.

ABCD=cyclic quadrilateral,$AC\cap BD=X$ AA'$\perp $BD,A'$\in$BD CC'$\perp $BD,C'$\in$BD BB'$\perp $AC,B'$\in$AC DD'$\perp $AC,D'$\in$AC Prove that: a)Prove that perpendiculars from midpoints of the sides to the opposite sides are concurrent.The point is called Mathot Point b)A',B',C',D' are concyclic c)If O'=circumcenter of (A'B'C') prove that O'=midpoint of the line that connects the orthocente of triangle XAB and XCD d)O' is the Mathot Point

The incircle of $ \triangle A_{0}B_{0}C_{0}$, meets legs $B_{0}C_{0}$, $C_{0}A_{0}$, $A_{0}B_{0}$, respectively on points $A$, $B$, $C$, and the incircle of $ \triangle ABC$, with center $I$, meets legs $BC$, $CA$, $AB$, on points $A_{1}$, $B_{1}$, $C_{1}$, respectively. We write with $ \sigma (ABC)$, and $ \sigma (A_{1}B_{1}C_{1})$ the areas of $ \triangle ABC$, and $ \triangle A_{1}B_{1}C_{1}$ respectively. Prove that if $ \sigma (ABC)=2 \sigma (A_{1}B_{1}C_{1})$, then lines $AA_{0}$, $BB_{0}$, $CC_{0}$ are concurrent.

In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that: $$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$ For what kind of triangles does the equality hold?

$ABCD$ is inscribed in unit circle $\Gamma$. Let $\Omega_1$, $\Omega_2$ be the circumcircles of $\vartriangle ABD$, $\vartriangle CBD$ respectively. Circles $\Omega_1$, $\Omega_2$ are tangent to segment $BD$ at $M$,$N$ respectively. Line A$M$ intersects $\Gamma$, $\Omega_1$ again at points $X_1$,$X_2$ respectively (different from $A$, $M$). Let $\omega_1$ be the circle passing through $X_1$, $X_2$ and tangent to $\Omega_1$. Line $CN$ intersects $\Gamma$, $\Omega_2$ again at points $Y_1$, $Y_2$ respectively (different from $C$, $N$). Let $\omega_2$ be the circle passing through $Y_1$, $Y_2$ and tangent to $\Omega_2$. Circles $\Omega_1$,$\Omega_2$, $\omega_1$, $\omega_2$ have radii $R_1$, $R_2$, $r_1$, $r_2$ respectively. Prove that $$r_1+r_2-R_1-R_2=1.$$ [img]https://cdn.artofproblemsolving.com/attachments/1/5/70471f2419fadc4b2183f5fe74f0c7a2e69ed4.png[/img] [url=https://www.geogebra.org/m/vxx8ghww]geogebra file[/url]

Construct a triangle $ABC$ given its side $a = BC$, its circumradius $R \ (2R \geq a)$, and the difference $\frac{1}{k} = \frac{1}{c}-\frac{1}{b}$, where $c = AB$ and $ b = AC.$

Given a ball $K$. Find the locus of the vertices $A$ of all parallelograms $ABCD$ such that $ AC \leq BD$, and the diagonal $BD$ lies completely inside the ball $K$.

In an acute-angled triangle $ABC$, $AB>AC$. $M,N$ are distinct points on side $BC$ such that $\angle BAM=\angle CAN$. Let $O_1,O_2$ be the circumcentres of $\triangle ABC, \triangle AMN$, respectively. Prove that $O_1,O_2,A$ are collinear.

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

Let $ABC$ be an equilateral triangle with centre $O$. A line through $C$ meets the circumcircle of triangle $AOB$ at points $D$ and $E$. Prove that points $A, O$ and the midpoints of segments $BD, BE$ are concyclic.

Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered $70\%$ of the arithmetic, $40\%$ of the algebra, and $60\%$ of the geometry problems correctly, she did not pass the test because she got less than $60\%$ of the problems right. How many more problems would she have needed to answer correctly to earn a $60\%$ passing grade? $ \text{(A)}\ 1\qquad\text{(B)}\ 5\qquad\text{(C)}\ 7\qquad\text{(D)}\ 9\qquad\text{(E)}\ 11 $

One bisector of a given triangle is parallel to one sideline of its Nagel triangle. Prove that one of two remaining bisectors is parallel to another sideline of the Nagel triangle. Proposed by:L.Emelyanov

Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$ Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$. a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$. b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.

In a convex pentagon, let the perpendicular line from a vertex to the opposite side be called an altitude. Prove that if four of the altitudes are concurrent at a point then the fifth altitude also passes through this point.

Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $\angle MO_1K$ . Find the area of quadrangle $MO_1KO_2$ .

In the quadrilateral $ABCD $, $AB = BC $, the point $K $ is the midpoint of the side $CD $, the rays $BK $ and $AD $ intersect at the point $M $ , the circumscribed circle $ \Delta ABM $ intersects the line $AC $ for the second time at the point $P $. Prove that $\angle BKP = 90 {} ^ \circ $. (Anton Trygub)

Let $S$ be area of the given parallelogram $ABCD$ and the points $E,F$ belong to $BC$ and $AD$, respectively, such that $BC = 3BE, 3AD = 4AF$. Let $O$ be the intersection of $AE$ and $BF$. Each straightline of $AE$ and $BF$ meets that of $CD$ at points $M$ and $N$, respectively. Determine area of triangle $MON$.

Let $ABC$ be an equilateral triangle of side 1. Draw three circles $O_a$, $O_b$, $O_c$ with diameters $BC$, $CA$, and $AB$, respectively. Let $S_a$ denote the area of the region inside $O_a$ and outside of $O_b$ and $O_c$. Define $S_b$ and $S_c$ similarly, and let $S$ be the area of intersection between the three circles. Find $S_a+S_b+S_c-S$.

Prove that if in the tetrahedron $ ABCD $ we have $ AB = CD $, $ AC = BD $, $ AD = BC $, then all faces of the tetrahedron are acute-angled triangles.

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 trapezoid $ABCD$ has sides $AB$ and $CD$ that are parallel $\hat{DAB} = 6^{\circ}$ and $\hat{ABC} = 42^{\circ}$. Point $X$ lies on the side $AB$ , such that $\hat{AXD} = 78^{\circ}$ and $\hat{CXB} = 66^{\circ}$. The distance between $AB$ and $CD$ is $1$ unit . Prove that $AD + DX - (BC + CX) = 8$ units. (Heir of Ramanujan)

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

Let $ABCD$ be a cyclic quadrilateral and $\omega_1, \omega_2$ the incircles of triangles $ABC$ and $BCD$. Show that the common external tangent line of $\omega_1$ and $\omega_2$, the other one than $BC$, is parallel with $AD$

A convex pentagon $ABCDE$ has rational sides and equal angles. Show that it is regular.

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.