The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.

Points $A, B, C, D,$ and $F$ lie on a sphere with radius $\sqrt{10}$ such that lines $AD, BE,$ and $CF$ are concurrent at point $P$ inside the sphere and are pairwise perpendicular. If $PA=\sqrt{6}, PB=\sqrt{10},$ and $PC=\sqrt{15},$ what is the volume of tetrahedron $DEFP$?

In a triangle $ ABC$, take point $ D$ on $ BC$ such that $ DB \equal{} 14, DA \equal{} 13, DC \equal{} 4$, and the circumcircle of $ ADB$ is congruent to the circumcircle of $ ADC$. What is the area of triangle $ ABC$?

The point ${D}$ inside the equilateral triangle ${\triangle ABC}$ satisfies ${\angle ADC = 150^o}$. Prove that a triangle with side lengths ${|AD|, |BD|, |CD|}$ is necessarily a right-angled triangle.

How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right?

On the sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$, we erect (outwardly) the squares $BB_aC_aC$, $CC_bA_bA$, $AA_cB_cB$, respectively. On the sides $B_cB_a$ and $C_aC_b$ of the triangles $BB_cB_a$ and $CC_aC_b$, we erect (outwardly) the squares $B_cB_vB_uB_a$ and $C_aC_uC_vC_b$. Prove that $B_uC_u\parallel BC$. [i]Comment.[/i] This problem originates in the 68th Moscow MO 2005, and a solution was posted in http://www.mathlinks.ro/Forum/viewtopic.php?t=30184 . However ingenious this solution is, there is a different one which shows a bit more: $B_uC_u=4\cdot BC$. Darij

Consider $27$ unit-cubes assembled into one $3 \times 3 \times 3$ cube. Let $A$ and $B$ be two opposite corners of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube to get from $A$ to $B$. [img]https://cdn.artofproblemsolving.com/attachments/0/5/d3aa802eae40cfe717088445daabd5e7194691.png[/img]

An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.

Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals). The perpendicular from $A$ to $BC$ meets $CD$ at $K$. The perpendicular from $A$ to $CD$ meets $BC$ at $L$. The perpendicular from $C$ to $AB$ meets $AD$ at $M$. The perpendicular from $C$ to $AD$ meets $AB$ at $N$. 1. Prove that $KL$ is parallel to $MN$. 2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.

An angle whose degree measure is equal to $108^o$ is given . Describe how with help compass and ruler can divide this angle into three equal parts. (Yukhim Rabinovych)

The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that \[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\] and find the conditions for equality.

On square $ABCD,$ points $E,F,G,$ and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34.$ Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P,$ and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$. [asy] size(200); defaultpen(linewidth(0.8)+fontsize(10.6)); pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("$P$",P,dir(60)); label("$w$", (H+E)/2,fontsize(15)); label("$x$", (E+F)/2,fontsize(15)); label("$y$", (G+F)/2,fontsize(15)); label("$z$", (H+G)/2,fontsize(15)); label("$w:x:y:z=269:275:405:411$",(sqrt(850)/2,-4.5),fontsize(11)); [/asy]

Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$

Four points in space $A,B,C,D$ satisfy that $|AB|=3,|BC|=7,|CD|=11,|DA|=9$, then the number of values of $\overrightarrow{AC}\cdot\overrightarrow{BD}$ is $\text{(A)}$ Only one. $\text{(B)}$ Two. $\text{(C)}$ Three. $\text{(D)}$ Infinitely many.

Suppose that $ ABCD$ is a cyclic quadrilateral. Let $ E \equal{} AC\cap BD$ and $ F \equal{} AB\cap CD$. Denote by $ H_{1}$ and $ H_{2}$ the orthocenters of triangles $ EAD$ and $ EBC$, respectively. Prove that the points $ F$, $ H_{1}$, $ H_{2}$ are collinear. Original formulation: Let $ ABC$ be a triangle. A circle passing through $ B$ and $ C$ intersects the sides $ AB$ and $ AC$ again at $ C'$ and $ B',$ respectively. Prove that $ BB'$, $CC'$ and $ HH'$ are concurrent, where $ H$ and $ H'$ are the orthocentres of triangles $ ABC$ and $ AB'C'$ respectively.

Let $ P$ be any point in the interior of a $ \triangle ABC$.Prove That $ \frac{PA}{a}\plus{}\frac{PB}{b}\plus{}\frac{PC}{c}\ge \sqrt{3}$.

Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$. In the triangle $EBC$, let N be the foot of the altitude relative to $EC$. Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$, and the lines $NF$ and $DC$ intersect at point $T$. Prove that the line $DP$ is perpendicular to the line $BT$.

Prove that there are no integers $x,y$ for that it is $x^3+y^3=4\cdot(x^2y+xy^2+1)$.

Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $S_{AOB}+ S_{COD} \le 2(S_{AOD}+ S_{BOC})$.

Given a quadrilateral \(ABCD\), point \(M\) is the midpoint of the side \(CD\). It turns out that \(\angle BMA = 90^{\circ}\) and \(\angle MAB = \angle CBD\). Prove that \(AC = AB\). [i]Proposed by Anton Trygub[/i]

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$, circumradius $R$ and orthocenter $H$. Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$. Define $B_1$ and $C_1$ similarly. If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that $\triangle ABC$ is equilateral.

For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers. a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty. b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers. c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.

Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that: $1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively. $2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively. Prove that $KL$ and $ST$ intersect on the line $BC$.