geometry problems from 2nd Round of Iranian Mathematical Olympiads

with aops links in the names

1983 Iran MO 2nd Round P4

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ The bisectors of the angles $B$ and $C$ meet each other in $I$ and meet the sides $AC$ and $AB$ in $D$ and $E$, respectively. Prove that $S_{BCDE}=2S_{BIC},$ where $S$ is the area function.

Let $ABC$ be an acute triangle with sides and area equal to $a, b, c$ and $S$ respectively.

1997 Iran MO 2nd Round P2

Let segments $KN,KL$ be tangent to circle $C$ at points $N,L$, respectively. $M$ is a point on the extension of the segment $KN$ and $P$ is the other meet point of the circle $C$ and the circumcircle of $\triangle KLM$. $Q$ is on $ML$ such that $NQ$ is perpendicular to $ML$. Prove that

$ \angle MPQ=2\angle KML.$

1997 Iran MO 2nd Round P5

In triangle $ABC$, angles $B,C$ are acute. Point $D$ is on the side $BC$ such that $AD\perp{BC}$. Let the interior bisectors of $\angle B,\angle C$ meet $AD$ at $E,F$, respectively. If $BE=CF$, prove that $ABC$ is isosceles.

2000 Iran MO 2nd Round P2

The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if

$\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}$

2000 Iran MO 2nd Round P5

In a tetrahedron we know that sum of angles of all vertices is $180^\circ.$ (e.g. for vertex $A$, we have $\angle BAC + \angle CAD + \angle DAB=180^\circ.$) Prove that faces of this tetrahedron are four congruent triangles.

2010 Iran MO 2nd Round P3

Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.

with aops links in the names

1983 - 2018

The point $M$ moves such that the sum of squares of the lengths from $M$ to faces of a cube, is fixed. Find the locus of $M.$

1984 Iran MO 2nd Round P7

Let $B$ and $C$ be two fixed point on the plane $P.$ Find the locus of the points $M$ on the plane $P$ for which $MB^2 + kMC^2 = a^2.$ ($k$ and $a$ are two given numbers and $k>0.$)
1985 Iran MO 2nd Round G2 P1

Inscribe in the triangle $ABC$ a triangle with minimum perimeter.

1985 Iran MO 2nd Round G2 P2

In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$

1987 Iran MO 2nd Round P3

1988 Iran MO 2nd Round P5

In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that

$\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.$

1989 Iran MO 2nd Round P2

A sphere $S$ with center $O$ and radius $R$ is given. Let $P$ be a fixed point on this sphere. Points $A,B,C$ move on the sphere $S$ such that we have $\angle APB = \angle BPC = \angle CPA = 90^\circ.$ Prove that the plane of triangle $ABC$ passes through a fixed point.

Let $ABCD$ be a parallelogram. The line $\Delta$ meets the lines $AB, BC, CD$ and $DA$ at $M, N, P$ and $Q,$ respectively. Let $R$ be the intersection point of the lines $AB,DN$ and let $S$ be intersection point of the lines $AD, BP.$ Prove that $RS \parallel \Delta.$

Inscribe in the triangle $ABC$ a triangle with minimum perimeter.

In the triangle $ABC$ the length of side $AB$, and height $AH$ are known. also we know that $\angle B = 2 \angle C.$ Plot this triangle.

1986 Iran MO 2nd Round G1

$O$ is a point in the plane. Let $O'$ be an arbitrary point on the axis $Ox$ of the plane and let $M$ be an arbitrary point. Rotate $M$, $90^\circ$ clockwise around $O$ to get the point $M'$ and rotate $M$, $90^\circ$ anticlockwise around $O'$ to get the point $M''.$ Prove that the midpoint of the segment $MM''$ is a fixed point.

1986 Iran MO 2nd Round G2$O$ is a point in the plane. Let $O'$ be an arbitrary point on the axis $Ox$ of the plane and let $M$ be an arbitrary point. Rotate $M$, $90^\circ$ clockwise around $O$ to get the point $M'$ and rotate $M$, $90^\circ$ anticlockwise around $O'$ to get the point $M''.$ Prove that the midpoint of the segment $MM''$ is a fixed point.

In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$

In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that $S_{A'B'C'D'} = \frac 15 S_{ABCD}.$

1988 Iran MO 2nd Round P2

In a cyclic quadrilateral $ABCD$, let $I,J$ be the midpoints of diagonals $AC, BD$ respectively and let $O$ be the center of the circle inscribed in $ABCD.$ Prove that $I, J$ and $O$ are collinear.

In a cyclic quadrilateral $ABCD$, let $I,J$ be the midpoints of diagonals $AC, BD$ respectively and let $O$ be the center of the circle inscribed in $ABCD.$ Prove that $I, J$ and $O$ are collinear.

In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that

$\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.$

1989 Iran MO 2nd Round P2

A sphere $S$ with center $O$ and radius $R$ is given. Let $P$ be a fixed point on this sphere. Points $A,B,C$ move on the sphere $S$ such that we have $\angle APB = \angle BPC = \angle CPA = 90^\circ.$ Prove that the plane of triangle $ABC$ passes through a fixed point.

A line $d$ is called

1990 Iran MO 2nd Round P1*faithful*to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one*faithful*line to both of them, or there exist infinitely*faithful*lines to them.Let $ABCD$ be a parallelogram. The line $\Delta$ meets the lines $AB, BC, CD$ and $DA$ at $M, N, P$ and $Q,$ respectively. Let $R$ be the intersection point of the lines $AB,DN$ and let $S$ be intersection point of the lines $AD, BP.$ Prove that $RS \parallel \Delta.$

(a) Consider the set of all triangles $ABC$ which are inscribed in a circle with radius $R.$ When is $AB^2+BC^2+CA^2$ maximum? Find this maximum.

(b) Consider the set of all tetragonals $ABCD$ which are inscribed in a sphere with radius $R.$ When is the sum of squares of the six edges of $ABCD$ maximum? Find this maximum, and in this case prove that all of the edges are equal.

1991 Iran MO 2nd Round P2

Let $ABCD$ be a tetragonal.

(a) If the plane $(P)$ cuts $ABCD,$ find the necessary and sufficient condition such that the area formed from the intersection of the plane $(P)$ and the tetragonal be a parallelogram. Prove that the problem has three solutions in this case.

(b) Consider one of the solutions of (a). Find the situation of the plane $(P)$ for which the parallelogram has maximum area.

(c) Find a plane $(P)$ for which the parallelogram be a lozenge and then find the length side of his lozenge in terms of the length of the edges of $ABCD.$

1992 Iran MO 2nd Round P1(b) Consider the set of all tetragonals $ABCD$ which are inscribed in a sphere with radius $R.$ When is the sum of squares of the six edges of $ABCD$ maximum? Find this maximum, and in this case prove that all of the edges are equal.

1991 Iran MO 2nd Round P2

Let $ABCD$ be a tetragonal.

(a) If the plane $(P)$ cuts $ABCD,$ find the necessary and sufficient condition such that the area formed from the intersection of the plane $(P)$ and the tetragonal be a parallelogram. Prove that the problem has three solutions in this case.

(b) Consider one of the solutions of (a). Find the situation of the plane $(P)$ for which the parallelogram has maximum area.

(c) Find a plane $(P)$ for which the parallelogram be a lozenge and then find the length side of his lozenge in terms of the length of the edges of $ABCD.$

1991 Iran MO 2nd Round P5

Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that

\[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]

Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that

\[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]

$ IA'+IB'+IC' \geq IA+IB+IC$

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ The bisectors of the angles $B$ and $C$ meet each other in $I$ and meet the sides $AC$ and $AB$ in $D$ and $E$, respectively. Prove that $S_{BCDE}=2S_{BIC},$ where $S$ is the area function.

In triangle $ABC,$ we have $\angle A \leq 90^\circ$ and $\angle B = 2 \angle C.$ The interior bisector of the angle $C$ meets the median $AM$ in $D.$ Prove that $\angle MDC \leq 45^\circ.$ When does equality hold?

Let $ABC$ be an acute triangle with sides and area equal to $a, b, c$ and $S$ respectively.

**Prove or disprove**that a necessary and sufficient condition for existence of a point $P$ inside the triangle $ABC$ such that the distance between $P$ and the vertices of $ABC$ be equal to $x, y$ and $z$ respectively is that there be a triangle with sides $a, y, z$ and area $S_1$, a triangle with sides $b, z, x$ and area $S_2$ and a triangle with sides $c, x, y$ and area $S_3$ where $S_1 + S_2 + S_3 = S.$

1994 Iran MO 2nd Round P2

In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$

In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$

1994 Iran MO 2nd Round P5

The incircle of triangle $ABC$ meet the sides $AB, AC$ and $BC$ in $M,N$ and $P$, respectively. Prove that the orthocenter of triangle $MNP,$ the incenter and the circumcenter of triangle $ABC$ are collinear.

Let $ABC$ be an acute triangle and let $\ell$ be a line in the plane of triangle $ABC.$ We've drawn the reflection of the line $\ell$ over the sides $AB, BC$ and $AC$ and they intersect in the points $A', B'$ and $C'.$ Prove that the incenter of the triangle $A'B'C'$ lies on the circumcircle of the triangle $ABC.$

1995 Iran MO 2nd Round P6

1996 Iran MO 2nd Round P3

The incircle of triangle $ABC$ meet the sides $AB, AC$ and $BC$ in $M,N$ and $P$, respectively. Prove that the orthocenter of triangle $MNP,$ the incenter and the circumcenter of triangle $ABC$ are collinear.

In a quadrilateral $ABCD$ let $A', B', C'$ and $D'$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Denote by $S(X, YZ)$ the plane which passes through the point $X$ and is perpendicular to the line $YZ.$ Prove that if $A', B', C'$ and $D'$ don't lie in a plane, then four planes $S(A, C'D'), S(B, A'D'), S(C, A'B')$ and $S(D, B'C')$ pass through a common point.

Let $N$ be the midpoint of side $BC$ of triangle $ABC$. Right isosceles triangles $ABM$ and $ACP$ are constructed outside the triangle, with bases $AB$ and $AC$. Prove that $\triangle MNP$ is also a right isosceles triangle.

Let segments $KN,KL$ be tangent to circle $C$ at points $N,L$, respectively. $M$ is a point on the extension of the segment $KN$ and $P$ is the other meet point of the circle $C$ and the circumcircle of $\triangle KLM$. $Q$ is on $ML$ such that $NQ$ is perpendicular to $ML$. Prove that

$ \angle MPQ=2\angle KML.$

In triangle $ABC$, angles $B,C$ are acute. Point $D$ is on the side $BC$ such that $AD\perp{BC}$. Let the interior bisectors of $\angle B,\angle C$ meet $AD$ at $E,F$, respectively. If $BE=CF$, prove that $ABC$ is isosceles.

1998 Iran MO 2nd Round P2

Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.

Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.

1998 Iran MO 2nd Round P5

Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that: $ AQ+CQ=BP. $

Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that: $ AQ+CQ=BP. $

1999 Iran MO 2nd Round P2

$ABC$ is a triangle with $\angle{B}>45^{\circ}$ , $\angle{C}>45^{\circ}$. We draw the isosceles triangles $CAM,BAN$ on the sides $AC,AB$ and outside the triangle, respectively, such that $\angle{CAM}=\angle{BAN}=90^{\circ}$. And we draw isosceles triangle $BPC$ on the side $BC$ and inside the triangle such that $\angle{BPC}=90^{\circ}$. Prove that $\Delta{MPN}$ is an isosceles triangle, too, and $\angle{MPN}=90^{\circ}$.

$ABC$ is a triangle with $\angle{B}>45^{\circ}$ , $\angle{C}>45^{\circ}$. We draw the isosceles triangles $CAM,BAN$ on the sides $AC,AB$ and outside the triangle, respectively, such that $\angle{CAM}=\angle{BAN}=90^{\circ}$. And we draw isosceles triangle $BPC$ on the side $BC$ and inside the triangle such that $\angle{BPC}=90^{\circ}$. Prove that $\Delta{MPN}$ is an isosceles triangle, too, and $\angle{MPN}=90^{\circ}$.

Let $ABC$ be a triangle and points $P,Q,R$ be on the sides $AB,BC,AC$, respectively. Now, let $A',B',C'$ be on the segments $PR,QP,RQ$ in a way that $AB||A'B'$ , $BC||B'C'$ and $AC||A'C'$. Prove that: $ \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}.$

Where $S_{XYZ}$ is the surface of the triangle $XYZ$.

The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if

$\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}$

In a tetrahedron we know that sum of angles of all vertices is $180^\circ.$ (e.g. for vertex $A$, we have $\angle BAC + \angle CAD + \angle DAB=180^\circ.$) Prove that faces of this tetrahedron are four congruent triangles.

2001 Iran MO 2nd Round P2

Let $ABC$ be an acute triangle. We draw $3$ triangles $B'AC,C'AB,A'BC$ on the sides of $\Delta ABC$ at the out sides such that:

$\angle{B'AC}=\angle{C'BA}=\angle{A'BC}=30^{\circ} $ , $ \angle{B'CA}=\angle{C'AB}=\angle{A'CB}=60^{\circ} $

If $M$ is the midpoint of side $BC$, prove that $B'M$ is perpendicular to $A'C'$.

Let $ABC$ be an acute triangle. We draw $3$ triangles $B'AC,C'AB,A'BC$ on the sides of $\Delta ABC$ at the out sides such that:

$\angle{B'AC}=\angle{C'BA}=\angle{A'BC}=30^{\circ} $ , $ \angle{B'CA}=\angle{C'AB}=\angle{A'CB}=60^{\circ} $

If $M$ is the midpoint of side $BC$, prove that $B'M$ is perpendicular to $A'C'$.

2001 Iran MO 2nd Round P5

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

2002 Iran MO 2nd Round P3

In a convex quadrilateral $ABCD$ with $\angle ABC = \angle ADC = 135^\circ$, points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD = \angle NCB = 90^\circ$. The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$. Prove that $AK \perp KC.$

In a convex quadrilateral $ABCD$ with $\angle ABC = \angle ADC = 135^\circ$, points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD = \angle NCB = 90^\circ$. The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$. Prove that $AK \perp KC.$

2002 Iran MO 2nd Round P4

Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$

Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$

$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that: $BT+CT\leq{2R}.$

2004 Iran MO 2nd Round P1

$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that:

$\frac{AD}{DI_a}\leq\sqrt{2}-1.$

$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that:

$\frac{AD}{DI_a}\leq\sqrt{2}-1.$

2004 Iran MO 2nd Round P5

The interior bisector of $\angle A$ from $\triangle ABC$ intersects the side $BC$ and the circumcircle of $\Delta ABC$ at $D,M$, respectively. Let $\omega$ be a circle with center $M$ and radius $MB$. A line passing through $D$, intersects $\omega$ at $X,Y$. Prove that $AD$ bisects $\angle XAY$.

The interior bisector of $\angle A$ from $\triangle ABC$ intersects the side $BC$ and the circumcircle of $\Delta ABC$ at $D,M$, respectively. Let $\omega$ be a circle with center $M$ and radius $MB$. A line passing through $D$, intersects $\omega$ at $X,Y$. Prove that $AD$ bisects $\angle XAY$.

2005 Iran MO 2nd Round P2

In triangle $ABC$, $\angle A=60^{\circ}$. The point $D$ changes on the segment $BC$. Let $O_1,O_2$ be the circumcenters of the triangles $\Delta ABD,\Delta ACD$, respectively. Let $M$ be the meet point of $BO_1,CO_2$ and let $N$ be the circumcenter of $\Delta DO_1O_2$. Prove that, by changing $D$ on $BC$, the line $MN$ passes through a constant point.

In triangle $ABC$, $\angle A=60^{\circ}$. The point $D$ changes on the segment $BC$. Let $O_1,O_2$ be the circumcenters of the triangles $\Delta ABD,\Delta ACD$, respectively. Let $M$ be the meet point of $BO_1,CO_2$ and let $N$ be the circumcenter of $\Delta DO_1O_2$. Prove that, by changing $D$ on $BC$, the line $MN$ passes through a constant point.

2005 Iran MO 2nd Round P5

$BC$ is a diameter of a circle and the points $X,Y$ are on the circle such that $XY\perp BC$. The points $P,M$ are on $XY,CY$ (or their stretches), respectively, such that $CY||PB$ and $CX||PM$. Let $K$ be the meet point of the lines $XC,BP$. Prove that $PB\perp MK$.

$BC$ is a diameter of a circle and the points $X,Y$ are on the circle such that $XY\perp BC$. The points $P,M$ are on $XY,CY$ (or their stretches), respectively, such that $CY||PB$ and $CX||PM$. Let $K$ be the meet point of the lines $XC,BP$. Prove that $PB\perp MK$.

2006 Iran MO 2nd Round P1

Let $C_1,C_2$ be two circles such that the center of $C_1$ is on the circumference of $C_2$. Let $C_1,C_2$ intersect each other at points $M,N$. Let $A,B$ be two points on the circumference of $C_1$ such that $AB$ is the diameter of it. Let lines $AM,BN$ meet $C_2$ for the second time at $A',B'$, respectively. Prove that $A'B'=r_1$ where $r_1$ is the radius of $C_1$.

Let $C_1,C_2$ be two circles such that the center of $C_1$ is on the circumference of $C_2$. Let $C_1,C_2$ intersect each other at points $M,N$. Let $A,B$ be two points on the circumference of $C_1$ such that $AB$ is the diameter of it. Let lines $AM,BN$ meet $C_2$ for the second time at $A',B'$, respectively. Prove that $A'B'=r_1$ where $r_1$ is the radius of $C_1$.

2006 Iran MO 2nd Round P5

Let $ABCD$ be a convex cyclic quadrilateral. Prove that:

a) the number of points on the circumcircle of $ABCD$, like $M$, such that $\frac{MA}{MB}=\frac{MD}{MC}$ is $4$.

b) The diagonals of the quadrilateral which is made with these points are perpendicular to each other.

Let $ABCD$ be a convex cyclic quadrilateral. Prove that:

a) the number of points on the circumcircle of $ABCD$, like $M$, such that $\frac{MA}{MB}=\frac{MD}{MC}$ is $4$.

b) The diagonals of the quadrilateral which is made with these points are perpendicular to each other.

2007 Iran MO 2nd Round P1

In triangle $ABC$, $\angle A=90^{\circ}$ and $M$ is the midpoint of $BC$. Point $D$ is chosen on segment $AC$ such that $AM=AD$ and $P$ is the second meet point of the circumcircles of triangles $\Delta AMC,\Delta BDC$. Prove that the line $CP$ bisects $\angle ACB$.

In triangle $ABC$, $\angle A=90^{\circ}$ and $M$ is the midpoint of $BC$. Point $D$ is chosen on segment $AC$ such that $AM=AD$ and $P$ is the second meet point of the circumcircles of triangles $\Delta AMC,\Delta BDC$. Prove that the line $CP$ bisects $\angle ACB$.

2007 Iran MO 2nd Round P5

Two circles $C,D$ are exterior tangent to each other at point $P$. Point $A$ is in the circle $C$. We draw $2$ tangents $AM,AN$ from $A$ to the circle $D$ ($M,N$ are the tangency points.). The second meet points of $AM,AN$ with $C$ are $E,F$, respectively. Prove that $\frac{PE}{PF}=\frac{ME}{NF}$.

Two circles $C,D$ are exterior tangent to each other at point $P$. Point $A$ is in the circle $C$. We draw $2$ tangents $AM,AN$ from $A$ to the circle $D$ ($M,N$ are the tangency points.). The second meet points of $AM,AN$ with $C$ are $E,F$, respectively. Prove that $\frac{PE}{PF}=\frac{ME}{NF}$.

2008 Iran MO 2nd Round P2

Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.

Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.

2008 Iran MO 2nd Round P6

In triangle $ABC$, $H$ is the foot of perpendicular from $A$ to $BC$. $O$ is the circumcenter of $\Delta ABC$. $T,T'$ are the feet of perpendiculars from $H$ to $AB,AC$, respectively. We know that $AC=2OT$. Prove that $AB=2OT'$.

In triangle $ABC$, $H$ is the foot of perpendicular from $A$ to $BC$. $O$ is the circumcenter of $\Delta ABC$. $T,T'$ are the feet of perpendiculars from $H$ to $AB,AC$, respectively. We know that $AC=2OT$. Prove that $AB=2OT'$.

Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $). Prove that $ \angle PAQ\geq\angle BAC $.

Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.

2010 Iran MO 2nd Round P5

In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.

In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.

2011 Iran MO 2nd Round P2

In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.

2011 Iran MO 2nd Round P6

The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.

In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.

The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.

2012 Iran MO 2nd Round P1

Consider a circle $C_1$ and a point $O$ on it. Circle $C_2$ with center $O$, intersects $C_1$ in two points $P$ and $Q$. $C_3$ is a circle which is externally tangent to $C_2$ at $R$ and internally tangent to $C_1$ at $S$ and suppose that $RS$ passes through $Q$. Suppose $X$ and $Y$ are second intersection points of $PR$ and $OR$ with $C_1$. Prove that $QX$ is parallel with $SY$.

Consider a circle $C_1$ and a point $O$ on it. Circle $C_2$ with center $O$, intersects $C_1$ in two points $P$ and $Q$. $C_3$ is a circle which is externally tangent to $C_2$ at $R$ and internally tangent to $C_1$ at $S$ and suppose that $RS$ passes through $Q$. Suppose $X$ and $Y$ are second intersection points of $PR$ and $OR$ with $C_1$. Prove that $QX$ is parallel with $SY$.

2012 Iran MO 2nd Round P6

The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$.

The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$.

by Mehdi E'tesami Fard

2013 Iran MO 2nd Round P3

Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$. Suppose that the altitude drawn from $A$ intersects the circle at $N$. Draw two lines through circumcenter $O$ of $ABC$ paralell to $MB$ and $MC$, which intersect $AB$ and $AC$ at $K$ and $L$, respectively. Prove that $NK=NL$.

Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$. Suppose that the altitude drawn from $A$ intersects the circle at $N$. Draw two lines through circumcenter $O$ of $ABC$ paralell to $MB$ and $MC$, which intersect $AB$ and $AC$ at $K$ and $L$, respectively. Prove that $NK=NL$.

2013 Iran MO 2nd Round P4

Let $P$ be a point out of circle $C$. Let $PA$ and $PB$ be the tangents to the circle drawn from $C$. Choose a point $K$ on $AB$ . Suppose that the circumcircle of triangle $PBK$ intersects $C$ again at $T$. Let ${P}'$ be the reflection of $P$ with respect to $A$. Prove that $ \angle PBT = \angle {P}'KA $

Let $P$ be a point out of circle $C$. Let $PA$ and $PB$ be the tangents to the circle drawn from $C$. Choose a point $K$ on $AB$ . Suppose that the circumcircle of triangle $PBK$ intersects $C$ again at $T$. Let ${P}'$ be the reflection of $P$ with respect to $A$. Prove that $ \angle PBT = \angle {P}'KA $

2014 Iran MO 2nd Round P2

Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that $ \angle BCQ = \dfrac{1}{2} \angle AQP .$

Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that $ \angle BCQ = \dfrac{1}{2} \angle AQP .$

2015 Iran MO 2nd Round P3

Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.

Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.

2015 Iran MO 2nd Round P4

In quadrilateral $ABCD$ , $AC$ is bisector of $\hat{A}$ and $\widehat{ADC}=\widehat{ACB}$. $X$ and $Y$ are feet of perpendicular from $A$ to $BC$ and $CD$,respectively.Prove that orthocenter of triangle $AXY$ is on $BD$.

In quadrilateral $ABCD$ , $AC$ is bisector of $\hat{A}$ and $\widehat{ADC}=\widehat{ACB}$. $X$ and $Y$ are feet of perpendicular from $A$ to $BC$ and $CD$,respectively.Prove that orthocenter of triangle $AXY$ is on $BD$.

2016 Iran MO 2nd Round P2

Let $ABC$ be a triangle such that $\angle C=2\angle B$ and $\omega$ be its circumcircle. a tangent from $A$ to $\omega$ intersect $BC$ at $E$. $\Omega$ is a circle passing throw $B$ that is tangent to $AC$ at $C$. Let $\Omega\cap AB=F$. $K$ is a point on $\Omega$ such that $EK$ is tangent to $\Omega$ ($A,K$ aren't in one side of $BC$). Let $M$ be the midpoint of arc $BC$ of $\omega$ (not containing $A$). Prove that $AFMK$ is a cyclic quadrilateral.

Let $ABC$ be a triangle such that $\angle C=2\angle B$ and $\omega$ be its circumcircle. a tangent from $A$ to $\omega$ intersect $BC$ at $E$. $\Omega$ is a circle passing throw $B$ that is tangent to $AC$ at $C$. Let $\Omega\cap AB=F$. $K$ is a point on $\Omega$ such that $EK$ is tangent to $\Omega$ ($A,K$ aren't in one side of $BC$). Let $M$ be the midpoint of arc $BC$ of $\omega$ (not containing $A$). Prove that $AFMK$ is a cyclic quadrilateral.

2016 Iran MO 2nd Round P5

$ABCD$ is a quadrilateral such that $\angle ACB=\angle ACD$. $T$ is inside of $ABCD$ such that $\angle ADC-\angle ATB=\angle BAC$ and $\angle ABC-\angle ATD=\angle CAD$. Prove that $\angle BAT=\angle DAC$.

$ABCD$ is a quadrilateral such that $\angle ACB=\angle ACD$. $T$ is inside of $ABCD$ such that $\angle ADC-\angle ATB=\angle BAC$ and $\angle ABC-\angle ATD=\angle CAD$. Prove that $\angle BAT=\angle DAC$.

2017 Iran MO 2nd Round P2

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$. Prove that $AB+CD>DA+BC.$

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$. Prove that $AB+CD>DA+BC.$

2017 Iran MO 2nd Round P6

Let $ABC$ be a triangle and $X$ be a point on its circumcircle. $Q,P$ lie on a line $BC$ such that $XQ\perp AC , XP\perp AB$. Let $Y$ be the circumcenter of $\triangle XQP$. Prove that $ABC$ is equilateral triangle if and if only $Y$ moves on a circle when $X$ varies on the circumcircle of $ABC$.

Let $ABC$ be a triangle and $X$ be a point on its circumcircle. $Q,P$ lie on a line $BC$ such that $XQ\perp AC , XP\perp AB$. Let $Y$ be the circumcenter of $\triangle XQP$. Prove that $ABC$ is equilateral triangle if and if only $Y$ moves on a circle when $X$ varies on the circumcircle of $ABC$.

Let $P $ be the intersection of $AC $ and $BD $ in isosceles trapezoid $ABCD $ ($AB\parallel CD$ , $BC=AD $) . The circumcircle of triangle $ABP $ inersects $BC $ for the second time at $X $. Point $Y $ lies on $AX $ such that $DY\parallel BC $. Prove that $\angle YDA =2.\angle YCA $

Two circles $\omega_1,\omega_2$ intersect at $P,Q $. An arbitrary line passing through $P $ intersects $\omega_1 , \omega_2$ at $A,B $ respectively. Another line parallel to $AB $ intersects $\omega_1$ at $D,F $ and $\omega_2$ at $E,C $ such that $E,F $ lie between $C,D $.Let $X\equiv AD\cap BE $ and $Y\equiv BC\cap AF $. Let $R $ be the reflection of $P $ about $CD$. Prove that:

a. $R $ lies on $XY $.

b. PR is the bisector of $\hat {XPY}$.

## Δεν υπάρχουν σχόλια:

## Δημοσίευση σχολίου