Found problems: 3585
Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $AD$ be the bisector of angle $A$ with $D$ on $BC$ . Let the circumcircle of triangle $ACD$ intersect $AB$ again at $E$; and let the circumcircle of triangle $ABD$ intersect $AC$ again at $F$ . Let $K$ be the reflection of $E$ in the line $BC$ . Prove that $FK = BC$.
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
Let $ABC$ be a triangle with $AB=5$, $BC=4$, and $CA=3$. On each side of $ABC$, externally erect a semicircle whose diameter is the corresponding side. Let $X$ be on the semicircular arc erected on side $\overline{BC}$ such that $\angle CBX$ has measure $15^\circ$. Let $Y$ be on the semicircular arc erected on side $\overline{CA}$ such that $\angle ACY$ has measure $15^\circ$. Similarly, let $Z$ be on the semicircular arc erected on side $\overline{AB}$ such that $\angle BAZ$ has measure $15^\circ$. What is the area of triangle $XYZ$?
p1. Let $AB$ be the diameter of the circle and $P$ is a point outside the circle. The lines $PQ$ and $PR$ are tangent to the circles at points $Q$ and $R$. The lines $PH$ is perpendicular on line $AB$ at $H$ . Line $PH$ intersects $AR$ at $S$. If $\angle QPH =40^o$ and $\angle QSA =30^o$, find $\angle RPS$.
p2. There is a meeting consisting of $40$ seats attended by $16$ invited guests. If each invited guest must be limited to at least $ 1$ chair, then determine the number of arrangements.
p3. In the crossword puzzle, in the following crossword puzzle, each box can only be filled with numbers from $ 1$ to $9$.
[img]https://cdn.artofproblemsolving.com/attachments/2/e/224b79c25305e8ed9a8a4da51059f961b9fbf8.png[/img]
Across:
1. Composite factor of $1001$
3. Non-polyndromic numbers
5. $p\times q^3$, with $p\ne q$ and $p,q$ primes
Down:
1. $a-1$ and $b+1$ , $a\ne b$ and $p,q$ primes
2. multiple of $9$
4. $p^3 \times q$, with $p\ne q$ and $p,q$ primes
p4. Given a function $f:R \to R$ and a function $g:R \to R$, so that it fulfills the following figure:
[img]https://cdn.artofproblemsolving.com/attachments/b/9/fb8e4e08861a3572412ae958828dce1c1e137a.png[/img]
Find the number of values of $x$, such that $(f(x))^2-2g(x)-x \in\{-10,-9,-8,…,9,10\}$.
p5. In a garden that is rectangular in shape, there is a watchtower in each corner and in the garden there is a monitoring tower. Small areas will be made in the shape of a triangle so that the corner points are towers (free of monitoring and/or supervisory towers). Let $k(m,n)$ be the number of small areas created if there are $m$ control towers and $n$ monitoring towers.
a. Find the values of $k(4,1)$, $k(4,2)$, $k(4,3)$, and $k(4,4)$
b. Find the general formula $k(m,n)$ with $m$ and $n$ natural numbers .
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle.
Given is the convex quadrilateral $ ABCD$. Assume that there exists a point $ P$ inside the quadrilateral for which the triangles $ ABP$ and $ CDP$ are both isosceles right triangles with the right angle at the common vertex $ P$. Prove that there exists a point $ Q$ for which the triangles $ BCQ$ and $ ADQ$ are also isosceles right triangles with the right angle at the common vertex $ Q$.
Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece
A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.
Let $ABC$ be an acute triangle with orthocenter $H$, and let $P$ be a point on the nine-point circle of $ABC$. Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$, respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$, respectively. Show that as $P$ varies along the nine-point circle of $ABC$, the line $QR$ passes through a fixed point.
[i]Proposed by Brandon Wang[/i]
Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$
Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.
Let $ABC$ be a triangle. The incircle of $ABC$ touches $BC,AC,AB$ at $D,E,F$, respectively. Each pair of the incircles of triangles $AEF, BDF,CED$ has two pair of common external tangents, one of them being one of the sides of $ABC$. Show that the other three tangents divide triangle $DEF$ into three triangles and three parallelograms.
Prove that for all natural $n\ge 6 000$ any convex $1994$-gon can be cut into $n$ such quadrilaterals thata circle can be circumscribed around each of them
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.
[i]Proposed by Max Jiang[/i]
In triangle $ABC$, $\angle BAC= 60^o$. Angle bisector of $\angle ABC$ meets side $AC$ at $X$ and angle bisector of $\angle BCA$ meets side $AB$ at $Y$. Prove that if $I$ is the incenter of triangle $ABC$, then $IX=IY$.
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
Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that \[ AB+AC\geq4DE. \]
Let $ABC$ be a right-angled triangle with $\angle B = 90^\circ$ and let $BD$ be the altitude from $B$ on to $AC$. Draw $DE \perp AB$ and $DF \perp BC$. Let $P, Q, R$ and $S$ be respectively the incentres of triangle $DF C, DBF, DEB$ and $DAE$. Suppose $S, R, Q$ are collinear. Prove that $P, Q, R, D$ lie on a circle.
Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.
Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.
Let $ABCD$ be a convex quadrilateral with perimeter $\tfrac{5}{2}$ and $AC=BD=1$. Determine the maximum possible area of $ABCD$.
In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.