Found problems: 410
Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$.
[i]Liubomir Chiriac, Moldova[/i]
Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$.
[i]Liubomir Chiriac, Moldova[/i]
et $ABCDEF$ be a convex hexagon which has an inscribed circle and a circumcribed. Denote by $\omega_{A}, \omega_{B},\omega_{C},\omega_{D},\omega_{E}$ and $\omega_{F}$ the inscribed circles of the triangles $FAB, ABC, BCD, CDE, DEF$ and $EFA$, respecitively. Let $l_{AB}$, be the external of $\omega_{A}$ and $\omega_{B}$; lines $l_{BC}$, $l_{CD}$, $l_{DE}$, $l_{EF}$, $l_{FA}$ are analoguosly defined. Let $A_1$ be the intersection point of the lines $l_{FA}$ and $l_{AB}$, $B_1, C_1, D_1, E_1, F_1$ are analogously defined.
Prove that $A_1D_1, B_1E_1, C_1F_1$ are concurrent.
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$.
[i]Proposed by Zuming Feng[/i]
Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.
(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
In the right triangle $ABC$ ( $\angle A = 90^o$), $D$ is the intersection of the bisector of the angle $A$ with the side $(BC)$, and $P$ and $Q$ are the projections of the point $D$ on the sides $(AB),(AC)$ respectively . If $BQ \cap DP=\{M\}$, $CP \cap DQ=\{N\}$, $BQ\cap CP=\{H\}$, show that:
a) $PM = DN$
b) $MN \parallel BC$
c) $AH \perp BC$.
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.
In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.
On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$, respectively. Let $F$ ($F \neq C$) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$, and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$. If $DG=EH$, prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$.
[i]Proposed by Milos Milosavljevic[/i]
Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$.
[i]Proposed by Ankan Bhattacharya[/i]
In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$.
[img]https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png[/img]
Let $ABC$ be an acute and scalene triangle. Points $D$ and $E$ are in the interior of the triangle so that $<$ $DAB$ $=$ $<$ $DCB$, $<$ $DAC$ $=$ $<$ $DBC$, $<$ $EAB$ $=$ $<$ $EBC$ and $<$ $EAC$ $=$ $<$ $ECB$. Prove that the triangle $ADE$ is a right triangle.
Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent.
[i]Proposed by Alireza Dadgarnia[/i]
Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.
The circle $ \omega $ inscribed in triangle $ABC$ touches its sides $AB, BC, CA$ at points $K, L, M$ respectively. In the arc $KL$ of the circle $ \omega $ that does not contain the point $M$, we select point $S$. Denote by $P, Q, R, T$ the intersection points of straight $AS$ and $KM, ML$ and $SC, LP$ and $KQ, AQ$ and $PC$ respectively. It turned out that the points $R, S$ and $M$ are collinear. Prove that the point $T$ also lies on the line $SM$.
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all positive real $x, y$ holds
$$xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)$$.
[i]Fedir Yudin[/i]
Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$. Prove $BN=LC$.
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$
Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$.
[i]Laurențiu Panaitopol[/i]
[b]Q8.[/b] Given a triangle $ABC$ and $2$ point $K \in AB, \; N \in BC$ such that $BK=2AK, \; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$. Compute $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$