Found problems: 433
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 $AD$ be a median of $ABC$ and $S$ its midpoint. Let $E$ be a intersection point of $AB$ and $CS$. Prove that $BE=2AE$
Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD \equal{} \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$
Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$. Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$, and the line $MN$ intersects the line $CD$ at the point $O$. Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$. Prove that $CD=PQ$. [i](Plamen Koshlukov)[/i]
Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the intersection point of the lines $AB$ and $CD$ and $N$ is the intersection point of the lines $BC$ and $AD$. It is known that $BM = DN$. Prove that $CM = CN$.
(F Nazarov)
Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.
Let $ ABC$ be a triangle. Circle $ \omega$ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of $ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.
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]
In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
In a triangle $ABC$, points $D$ and $E$ are on segments $BC$ and $AC$ such that $BD=3DC$ and $AE=4EC$. Point $P$ is on line $ED$ such that $D$ is the midpoint of segment $EP$. Lines $AP$ and $BC$ intersect at point $S$. Find the ratio $BS/SD$.
Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$. A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$. Let $X$ be an arbitrary point of line $BC$. Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$
In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.
There is a convex quadrilateral $ ABCD $ which satisfies $ \angle A=\angle D $.
Let the midpoints of $ AB, AD, CD $ be $ L,M,N $.
Let's say the intersection point of $ AC, BD $ be $ E $ .
Let's say point $ F $ which lies on $ \overrightarrow{ME} $ satisfies $ \overline{ME}\times \overline{MF}=\overline{MA}^{2} $.
Prove that $ \angle LFM=\angle MFN $. :)
Let $E$ be the midpoint of side $[AB]$ of square $ABCD$. Let the circle through $B$ with center $A$ and segment $[EC]$ meet at $F$. What is $|EF|/|FC|$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ \dfrac{3}{2}
\qquad\textbf{(C)}\ \sqrt{5}-1
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \sqrt{3}
$
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $N$ denote the second point of intersection of line $AI$ and $\omega$. The line through $I$ perpendicular to $AI$ intersects line $BC$, segment $[AB]$, and segment $[AC]$ at the points $D$, $E$, and $F$, respectively. The circumcircle of triangle $AEF$ meets $\omega$ again at $P$, and lines $PN$ and $BC$ intersect at $Q$. Prove that lines $IQ$ and $DN$ intersect on $\omega$.
[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 $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$
Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that \[\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.\]
In an inscribed quadrilateral $ABCD$, we have $BC=CD$ but $AB\neq AD$. Points $I$ and $J$ are the incenters of triangles $ABC$ and $ACD$ respectively. Point $K$ was chosen on segment $AC$ so that $IK=JK$. Points $M$ and $N$ are the incenters of triangles $AIK$ and $AJK$. Prove that the perpendicular to $CD$ at $D$ and the perpendicular to $KI$ at $I$ intersect on the circumcircle of $MAN$.
In triangle $ABC$, with $AB<AC$, $I$ is the incenter, $E$ is the intersection of $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ=90$. Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$. if $M$ is the Midpoint od the arc $BC$, which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$.
Proposed by Amirmahdi Mohseni
Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
Let $\triangle ABC$ be a scalene triangle, and let $D$ and $E$ be points on sides $AB$ and $AC$ respectively such that the circumcircles of triangles $\triangle ACD$ and $\triangle ABE$ are tangent to $BC$. Let $F$ be the intersection point of $BC$ and $DE$. Prove that $AF$ is perpendicular to the Euler line of $\triangle ABC$.
Let $ABC$ be an equilateral triangle and $D$ a point on side $AC$. Let $E$ be a point on $BC$ such that $DE \perp BC$, $F$ on $AB$ such that $EF \perp AB$, and $G$ on $AC$ such that $FG \perp AC$. Lines $FG$ and $DE$ intersect in $P$. If $M$ is the midpoint of $BC$, show that $BP$ bisects $AM$.
Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.