geometry problems from Irish Mathematical Olympiads (IRMO)

with aops links in the names

1988 - 2018

with aops links in the names

1988 - 2018

1988 Irish Paper1 P1

A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.

1988 Irish Paper1 P2

$A, B,C, D$ are the vertices of a square, and $P$ is a point on the arc $CD$ of its circumcircle. Prove that $ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| $

1988 Irish Paper1 P3

$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$. If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$.

1988 Irish Paper2 P1

The triangles $ABG$ and $AEF$ are in the same plane. Between them the following conditions hold:

(a) $E$ is the mid-point of $AB$;

(b) points $A,G$ and $F$ are on the same line;

(c) there is a point $C$ at which $BG$ and $EF$ intersect;

(d) $|CE|=1$ and $|AC|=|AE|=|FG|$.

Show that if $|AG|=x$, then $|AB|=x^3$.

1989 Irish Paper1 P5

Let $ABC$ be a right-angled triangle with right-angle at $A$. Let $X$ be the foot of the perpendicular from $A$ to $BC$, and $Y$ the mid-point of $XC$. Let $AB$ be extended to $D$ so that $|AB|=|BD|$. Prove that $DX$ is perpendicular to $AY$.

A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.

1988 Irish Paper1 P2

$A, B,C, D$ are the vertices of a square, and $P$ is a point on the arc $CD$ of its circumcircle. Prove that $ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| $

$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$. If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$.

1988 Irish Paper2 P1

The triangles $ABG$ and $AEF$ are in the same plane. Between them the following conditions hold:

(a) $E$ is the mid-point of $AB$;

(b) points $A,G$ and $F$ are on the same line;

(c) there is a point $C$ at which $BG$ and $EF$ intersect;

(d) $|CE|=1$ and $|AC|=|AE|=|FG|$.

Show that if $|AG|=x$, then $|AB|=x^3$.

1989 Irish Paper1 P5

Let $ABC$ be a right-angled triangle with right-angle at $A$. Let $X$ be the foot of the perpendicular from $A$ to $BC$, and $Y$ the mid-point of $XC$. Let $AB$ be extended to $D$ so that $|AB|=|BD|$. Prove that $DX$ is perpendicular to $AY$.

1989 Irish Paper2 P1

Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$. Let $k$ be a fixed nonzero real number. For $P$ a point on $L$, let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$. Determine the locus of $Q$ as $P$ varies along the line $L$.

Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$. Let $k$ be a fixed nonzero real number. For $P$ a point on $L$, let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$. Determine the locus of $Q$ as $P$ varies along the line $L$.

1989 Irish Paper2 P3

Suppose $P$ is a point in the interior of a triangle $ABC$, that $x, y, z$ are the distances from $P$ to $A,B, C$, respectively, and that $p,q, r$ are the perpendicular distances from $P$ to the sides $BC,CA,AB$, respectively. Prove that $xyz \geq 8pqr$, with equality implying that the triangle $ABC$ is equilateral.

Suppose $P$ is a point in the interior of a triangle $ABC$, that $x, y, z$ are the distances from $P$ to $A,B, C$, respectively, and that $p,q, r$ are the perpendicular distances from $P$ to the sides $BC,CA,AB$, respectively. Prove that $xyz \geq 8pqr$, with equality implying that the triangle $ABC$ is equilateral.

1990 Irish Paper1 P1

A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$

A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$

1991 Irish Paper1 P1

Three points $X,Y$ and $Z$ are given that are, respectively, the circumcenter of a triangle $ABC$, the mid-point of $BC$, and the foot of the altitude from $B$ on $AC$. Show how to reconstruct the triangle $ABC$.

1992 Irish Paper1 P4

In a triangle $ABC$, the points $A’$, $B’$ and $C’$ on the sides opposite $A$ ,$B$ and $C$, respectively, are such that the lines $AA’$, $BB’$ and $CC’$ are concurrent. Prove that the diameter of the circumscribed circle of the triangle $ABC$ equals the product $|AB’|\cdot |BC’|\cdot |CA’|$ divided by the area of the triangle $A’B’C’$.

1992 Irish Paper2 P4

Three points $X,Y$ and $Z$ are given that are, respectively, the circumcenter of a triangle $ABC$, the mid-point of $BC$, and the foot of the altitude from $B$ on $AC$. Show how to reconstruct the triangle $ABC$.

1992 Irish Paper1 P4

A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.

1993 Irish Paper1 P3

A line $ l$ is tangent to a circle $ S$ at $ A$. For any points $ B,C$ on $ l$ on opposite sides of $ A$, let the other tangents from $ B$ and $ C$ to $ S$ intersect at a point $ P$. If $ B,C$ vary on $ l$ so that the product $ AB \cdot AC$ is constant, find the locus of $ P$.

A line $ l$ is tangent to a circle $ S$ at $ A$. For any points $ B,C$ on $ l$ on opposite sides of $ A$, let the other tangents from $ B$ and $ C$ to $ S$ intersect at a point $ P$. If $ B,C$ vary on $ l$ so that the product $ AB \cdot AC$ is constant, find the locus of $ P$.

1994 Irish Paper1 P2

Let $ A,B,C$ be collinear points on the plane with $ B$ between $ A$ and $ C$. Equilateral triangles $ ABD,BCE,CAF$ are constructed with $ D,E$ on one side of the line $ AC$ and $ F$ on the other side. Prove that the centroids of the triangles are the vertices of an equilateral triangle, and that the centroid of this triangle lies on the line $ AC$.

1995 Irish Paper1 P3

Points $ A,X,D$ lie on a line in this order, point $ B$ is on the plane such that $ \angle ABX>120^{\circ}$, and point $ C$ is on the segment $ BX$. Prove the inequality:

$ 2AD \ge \sqrt{3} (AB+BC+CD)$.

Points $ A,X,D$ lie on a line in this order, point $ B$ is on the plane such that $ \angle ABX>120^{\circ}$, and point $ C$ is on the segment $ BX$. Prove the inequality:

$ 2AD \ge \sqrt{3} (AB+BC+CD)$.

1995 Irish Paper2 P4

Points $ P,Q,R$ are given in the plane. It is known that there is a triangle $ ABC$ such that $ P$ is the midpoint of $ BC$, $ Q$ the point on side $ CA$ with $ \frac{CQ}{QA}=2$, and $ R$ the point on side $ AB$ with $ \frac{AR}{RB}=2$. Determine with proof how the triangle $ ABC$ may be reconstructed from $ P,Q,R$.

1996 Irish Paper1 P4

Let $ F$ be the midpoint of the side $ BC$ of a triangle $ ABC$. Isosceles right-angled triangles $ ABD$ and $ ACE$ are constructed externally on $ AB$ and $ AC$ with the right angles at $ D$ and $ E$. Prove that the triangle $ DEF$ is right-angled and isosceles.

Points $ P,Q,R$ are given in the plane. It is known that there is a triangle $ ABC$ such that $ P$ is the midpoint of $ BC$, $ Q$ the point on side $ CA$ with $ \frac{CQ}{QA}=2$, and $ R$ the point on side $ AB$ with $ \frac{AR}{RB}=2$. Determine with proof how the triangle $ ABC$ may be reconstructed from $ P,Q,R$.

1996 Irish Paper1 P4

Let $ F$ be the midpoint of the side $ BC$ of a triangle $ ABC$. Isosceles right-angled triangles $ ABD$ and $ ACE$ are constructed externally on $ AB$ and $ AC$ with the right angles at $ D$ and $ E$. Prove that the triangle $ DEF$ is right-angled and isosceles.

1996 Irish Paper2 P4

In an acute-angled triangle $ ABC$, $ D,E,F$ are the feet of the altitudes from $ A,B,C$, respectively, and $ P,Q,R$ are the feet of the perpendiculars from $ A,B,C$ onto $ EF,FD,DE$, respectively. Prove that the lines $ AP,BQ,CR$ are concurrent.

1997 Irish Paper1 P2

For a point $ M$ inside an equilateral triangle $ ABC$, let $ D,E,F$ be the feet of the perpendiculars from $ M$ onto $ BC,CA,AB$, respectively. Find the locus of all such points $ M$ for which $ \angle FDE$ is a right angle.

In an acute-angled triangle $ ABC$, $ D,E,F$ are the feet of the altitudes from $ A,B,C$, respectively, and $ P,Q,R$ are the feet of the perpendiculars from $ A,B,C$ onto $ EF,FD,DE$, respectively. Prove that the lines $ AP,BQ,CR$ are concurrent.

1997 Irish Paper1 P2

For a point $ M$ inside an equilateral triangle $ ABC$, let $ D,E,F$ be the feet of the perpendiculars from $ M$ onto $ BC,CA,AB$, respectively. Find the locus of all such points $ M$ for which $ \angle FDE$ is a right angle.

1997 Irish Paper2 P2

A circle $ \Gamma$ is inscribed in a quadrilateral $ ABCD$. If $ \angle A=\angle B=120^{\circ}, \angle D=90^{\circ}$ and $ BC=1$, find, with proof, the length of $ AD$.

1998 Irish Paper1 P2

The distances from a point $ P$ inside an equilateral triangle to the vertices of the triangle are $ 3,4$, and $ 5$. Find the area of the triangle.

A circle $ \Gamma$ is inscribed in a quadrilateral $ ABCD$. If $ \angle A=\angle B=120^{\circ}, \angle D=90^{\circ}$ and $ BC=1$, find, with proof, the length of $ AD$.

The distances from a point $ P$ inside an equilateral triangle to the vertices of the triangle are $ 3,4$, and $ 5$. Find the area of the triangle.

1999 Irish Paper1 P3

If $ AD$ is the altitude, $ BE$ the angle bisector, and $ CF$ the median of a triangle $ ABC$, prove that $ AD,BE,$ and $ CF$ are concurrent if and only if $a^2(a\mp c) = (b^2- c^2)(a \pm c)$

where $ a,b,c$ are the lengths of the sides $ BC,CA,AB$, respectively.

If $ AD$ is the altitude, $ BE$ the angle bisector, and $ CF$ the median of a triangle $ ABC$, prove that $ AD,BE,$ and $ CF$ are concurrent if and only if $a^2(a\mp c) = (b^2- c^2)(a \pm c)$

where $ a,b,c$ are the lengths of the sides $ BC,CA,AB$, respectively.

1999 Irish Paper2 P5

A convex hexagon $ ABCDEF$ satisfies $ AB=BC, CD=DE, EF=FA$ and: $ \angle ABC+\angle CDE+\angle EFA = 360^{\circ}$. Prove that the perpendiculars from $ A,C$ and $ E$ to $ FB,BD$ and $ DF$ respectively are concurrent.

2000 Irish Paper1 P2

Let $ ABCDE$ be a regular pentagon of side length $ 1$. Let $ F$ be the midpoint of $ AB$ and let $ G$ and $ H$ be the points on sides $ CD$ and $ DE$ respectively $ \angle GFD = \angle HFD = 30^{\circ}$. Show that the triangle $ GFH$ is equilateral. A square of side $ a$ is inscribed in $ \triangle GFH$ with one side of the square along $ GH$. Prove that:

$ FG = t = \frac {2 \cos 18^{\circ} \cos^2 36^{\circ}}{\cos 6^{\circ}}$ and $ a = \frac {t \sqrt {3}}{2 + \sqrt {3}}$.

A convex hexagon $ ABCDEF$ satisfies $ AB=BC, CD=DE, EF=FA$ and: $ \angle ABC+\angle CDE+\angle EFA = 360^{\circ}$. Prove that the perpendiculars from $ A,C$ and $ E$ to $ FB,BD$ and $ DF$ respectively are concurrent.

Let $ ABCDE$ be a regular pentagon of side length $ 1$. Let $ F$ be the midpoint of $ AB$ and let $ G$ and $ H$ be the points on sides $ CD$ and $ DE$ respectively $ \angle GFD = \angle HFD = 30^{\circ}$. Show that the triangle $ GFH$ is equilateral. A square of side $ a$ is inscribed in $ \triangle GFH$ with one side of the square along $ GH$. Prove that:

$ FG = t = \frac {2 \cos 18^{\circ} \cos^2 36^{\circ}}{\cos 6^{\circ}}$ and $ a = \frac {t \sqrt {3}}{2 + \sqrt {3}}$.

2000 Irish Paper2 P2

In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that $ R^2=\frac{(ab+cd)(ac+bd)(ad+bc)}{16Q^2}$.

Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.

2001 Irish Paper1 P2

Let $ ABC$ be a triangle with sides $ BC=a, CA=b,AB=c$ and let $ D$ and $ E$ be the midpoints of $ AC$ and $ AB$, respectively. Prove that the medians $ BD$ and $ CE$ are perpendicular to each other if and only if $ b^2+c^2=5a^2$.

In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that $ R^2=\frac{(ab+cd)(ac+bd)(ad+bc)}{16Q^2}$.

Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.

2001 Irish Paper1 P2

Let $ ABC$ be a triangle with sides $ BC=a, CA=b,AB=c$ and let $ D$ and $ E$ be the midpoints of $ AC$ and $ AB$, respectively. Prove that the medians $ BD$ and $ CE$ are perpendicular to each other if and only if $ b^2+c^2=5a^2$.

2001 Irish Paper2 P3

In an acute-angled triangle $ ABC$, $ D$ is the foot of the altitude from $ A$, and $ P$ a point on segment $ AD$. The lines $ BP$ and $ CP$ meet $ AC$ and $ AB$ at $ E$ and $ F$ respectively. Prove that $ AD$ bisects the angle $ EDF$.

2002 Irish Paper1 P1

In a triangle $ ABC$ with $ AB=20, AC=21$ and $ BC=29$, points $ D$ and $ E$ are taken on the segment $ BC$ such that $ BD=8$ and $ EC=9$. Calculate the angle $ \angle DAE$.

In an acute-angled triangle $ ABC$, $ D$ is the foot of the altitude from $ A$, and $ P$ a point on segment $ AD$. The lines $ BP$ and $ CP$ meet $ AC$ and $ AB$ at $ E$ and $ F$ respectively. Prove that $ AD$ bisects the angle $ EDF$.

In a triangle $ ABC$ with $ AB=20, AC=21$ and $ BC=29$, points $ D$ and $ E$ are taken on the segment $ BC$ such that $ BD=8$ and $ EC=9$. Calculate the angle $ \angle DAE$.

2002 Irish Paper2 P5

Let $ ABC$ be a triangle with integer side lengths, and let its incircle touch $ BC$ at $ D$ and $ AC$ at $ E$. If $ |AD^2-BE^2| \le 2$, show that $ AC=BC$.

2003 Irish Paper1 P2

$P$, $Q$, $R$ and $S$ are (distinct) points on a circle. $PS$ is a diameter and $QR$ is parallel to the diameter $PS$. $PR$ and $QS$ meet at $A$. Let $O$ be the centre of the circle and let $B$ be chosen so that the quadrilateral $POAB$ is a parallelogram. Prove that $BQ$ = $BP$ .

Let $ ABC$ be a triangle with integer side lengths, and let its incircle touch $ BC$ at $ D$ and $ AC$ at $ E$. If $ |AD^2-BE^2| \le 2$, show that $ AC=BC$.

$P$, $Q$, $R$ and $S$ are (distinct) points on a circle. $PS$ is a diameter and $QR$ is parallel to the diameter $PS$. $PR$ and $QS$ meet at $A$. Let $O$ be the centre of the circle and let $B$ be chosen so that the quadrilateral $POAB$ is a parallelogram. Prove that $BQ$ = $BP$ .

2003 Irish Paper2 P2

$\ ABCD$ is a quadrilateral. the feet of the perpendicular from $\ D$ to $\ AB, BC$ are $\ P,Q$ respectively, and the feet of the perpendicular from $\ B$ to $\ AD,CD$ are $\ R,S$ respectively. Show that if $\angle PSR= \angle SPQ$, then $\ PR=QS$.

2004 Irish Paper1 P3

$AB$ is a chord of length $6$ of a circle centred at $O$ and of radius $5$. Let $PQRS$ denote the square inscribed in the sector $OAB$ such that $P$ is on the radius $OA$, $S$ is on the radius $OB$ and $Q$ and $R$ are points on the arc of the circle between $A$ and $B$. Find the area of $PQRS$.

$\ ABCD$ is a quadrilateral. the feet of the perpendicular from $\ D$ to $\ AB, BC$ are $\ P,Q$ respectively, and the feet of the perpendicular from $\ B$ to $\ AD,CD$ are $\ R,S$ respectively. Show that if $\angle PSR= \angle SPQ$, then $\ PR=QS$.

2004 Irish Paper1 P3

$AB$ is a chord of length $6$ of a circle centred at $O$ and of radius $5$. Let $PQRS$ denote the square inscribed in the sector $OAB$ such that $P$ is on the radius $OA$, $S$ is on the radius $OB$ and $Q$ and $R$ are points on the arc of the circle between $A$ and $B$. Find the area of $PQRS$.

$A$ and $B$ are distinct points on a circle $T$. $C$ is a point distinct from $B$ such that $|AB|=|AC|$, and such that $BC$ is tangent to $T$ at $B$. Suppose that the bisector of $\angle ABC$ meets $AC$ at a point $D$ inside $T$. Show that $\angle ABC>72^\circ$.

2005 Irish Paper1 P2

Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.

2005 Irish Paper2 P1

Let $ X$ be a point on the side $ AB$ of a triangle $ ABC$, different from $ A$ and $ B$. Let $ P$ and $ Q$ be the incenters of the triangles $ ACX$ and $ BCX$ respectively, and let $ M$ be the midpoint of $ PQ$. Prove that: $ MC>MX$.

2006 Irish Paper1 P2

$P$ and $Q$ are points on the equal sides $AB$ and $AC$ respectively of an isosceles triangle $ABC$ such that $AP=CQ$. Moreover, neither $P$ nor $Q$ is a vertex of $ABC$. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of the triangle $ABC$.

Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.

Prove that the sum of the lengths of the medians of a triangle is at least three quarters of its perimeter.

2005 Irish Paper2 P1

Let $ X$ be a point on the side $ AB$ of a triangle $ ABC$, different from $ A$ and $ B$. Let $ P$ and $ Q$ be the incenters of the triangles $ ACX$ and $ BCX$ respectively, and let $ M$ be the midpoint of $ PQ$. Prove that: $ MC>MX$.

$P$ and $Q$ are points on the equal sides $AB$ and $AC$ respectively of an isosceles triangle $ABC$ such that $AP=CQ$. Moreover, neither $P$ nor $Q$ is a vertex of $ABC$. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of the triangle $ABC$.

2006 Irish Paper2 P2

$ABC$ is a triangle with points $D$, $E$ on $BC$ with $D$ nearer $B$; $F$, $G$ on $AC$, with $F$ nearer $C$; $H$, $K$ on $AB$, with $H$ nearer $A$. Suppose that $AH=AG=1$, $BK=BD=2$, $CE=CF=4$, $\angle B=60^\circ$ and that $D$, $E$, $F$, $G$, $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$.

2007 Irish Paper1 P3

The point $ P$ is a fixed point on a circle and $ Q$ is a fixed point on a line. The point $ R$ is a variable point on the circle such that $ P,Q,$ and $ R$ are not collinear. The circle through $ P,Q,$ and $ R$ meets the line again at $ V$. Show that the line $ VR$ passes through a fixed point.

$ABC$ is a triangle with points $D$, $E$ on $BC$ with $D$ nearer $B$; $F$, $G$ on $AC$, with $F$ nearer $C$; $H$, $K$ on $AB$, with $H$ nearer $A$. Suppose that $AH=AG=1$, $BK=BD=2$, $CE=CF=4$, $\angle B=60^\circ$ and that $D$, $E$, $F$, $G$, $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$.

The point $ P$ is a fixed point on a circle and $ Q$ is a fixed point on a line. The point $ R$ is a variable point on the circle such that $ P,Q,$ and $ R$ are not collinear. The circle through $ P,Q,$ and $ R$ meets the line again at $ V$. Show that the line $ VR$ passes through a fixed point.

2007 Irish Paper2 P3

Let $ ABC$ be a triangle the lengths of whose sides $ BC,CA,AB,$ respectively, are denoted by $ a,b,$ and $ c$. Let the internal bisectors of the angles $ \angle BAC, \angle ABC, \angle BCA,$ respectively, meet the sides $ BC,CA,$ and $ AB$ at $ D,E,$ and $ F$. Denote the lengths of the line segments $ AD,BE,CF$ by $ d,e,$ and $ f$, respectively. Prove that:

$ def=\frac{4abc(a+b+c) \Delta}{(a+b)(b+c)(c+a)}$ where $ \Delta$ stands for the area of the triangle $ ABC$.

2008 Irish Paper1 P5

A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| = |AB|$. Prove that $ |AD|^2 = |AB|.|BC|$ if and only if $ \angle CBD = 30^\circ$.

Let $ ABC$ be a triangle the lengths of whose sides $ BC,CA,AB,$ respectively, are denoted by $ a,b,$ and $ c$. Let the internal bisectors of the angles $ \angle BAC, \angle ABC, \angle BCA,$ respectively, meet the sides $ BC,CA,$ and $ AB$ at $ D,E,$ and $ F$. Denote the lengths of the line segments $ AD,BE,CF$ by $ d,e,$ and $ f$, respectively. Prove that:

$ def=\frac{4abc(a+b+c) \Delta}{(a+b)(b+c)(c+a)}$ where $ \Delta$ stands for the area of the triangle $ ABC$.

2008 Irish Paper1 P5

A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| = |AB|$. Prove that $ |AD|^2 = |AB|.|BC|$ if and only if $ \angle CBD = 30^\circ$.

2008 Irish Paper2 P2

Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| = |PB|$.

2009 Irish Paper1 P2

Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| = |PB|$.

2009 Irish Paper1 P2

Let $ABCD$ be a square. The line segment $AB$ is divided internally at $H$ so that $|AB|\cdot |BH|=|AH|^2$. Let $E$ be the midpoints of $AD$ and $X$ be the midpoint of $AH$. Let $Y$ be a point on $EB$ such that $XY$ is perpendicular to $BE$. Prove that $|XY|=|XH|$.

2009 Irish Paper2 P5

In the triangle $ABC$ we have $|AB|<|AC|$. The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$. Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$

2010 Irish Paper1 P2

Let $ABC$ be a triangle and let $P$ denote the midpoint of the side $BC$. Suppose that there exist two points $M$ and $N$ interior to the side $AB$ and $AC$ respectively, such that $|AD|=|DM|=2|DN|,$ where $D$ is the intersection point of the lines $MN$ and $AP$. Show that $|AC|=|BC|$.

In the triangle $ABC$ we have $|AB|<|AC|$. The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$. Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$

2010 Irish Paper1 P2

Let $ABC$ be a triangle and let $P$ denote the midpoint of the side $BC$. Suppose that there exist two points $M$ and $N$ interior to the side $AB$ and $AC$ respectively, such that $|AD|=|DM|=2|DN|,$ where $D$ is the intersection point of the lines $MN$ and $AP$. Show that $|AC|=|BC|$.

2010 Irish Paper2 P3

In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$. Find the length of $AD$.

2011 Irish Paper1 P2

Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$, respectively, the lengths of the medians which connect $A,B,C$, respectively, with the centers of the corresponding opposite sides.

(a) Prove that $2m_a<b+c$. Deduce that $m_a+m_b+m_c<a+b+c$.

(b) Give an example of

(i) a triangle in which $m_a>\sqrt{bc}$;

(ii) a triangle in which $m_a\le \sqrt{bc}$.

2011 Irish Paper1 P4

The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$, and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$. Prove that

(a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$.

(b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$.

In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$. Find the length of $AD$.

Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$, respectively, the lengths of the medians which connect $A,B,C$, respectively, with the centers of the corresponding opposite sides.

(a) Prove that $2m_a<b+c$. Deduce that $m_a+m_b+m_c<a+b+c$.

(b) Give an example of

(i) a triangle in which $m_a>\sqrt{bc}$;

(ii) a triangle in which $m_a\le \sqrt{bc}$.

2011 Irish Paper1 P4

The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$, and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$. Prove that

(a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$.

(b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$.

2011 Irish Paper2 P3

$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$.

2012 Irish Paper1 P2

$A,B,C$ and $D$ are four points in that order on the circumference of a circle $K$. $AB$ is perpendicular to $BC$ and $BC$ is perpendicular to $CD$. $X$ is a point on the circumference of the circle between $A$ and $D$. $AX$ extended meets $CD$ extended at $E$ and $DX$ extended meets $BA$ extended at $F$. Prove that the circumcircle of triangle $AXF$ is tangent to the circumcircle of triangle $DXE$ and that the common tangent line passes through the center of the circle $K$.

$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$.

2012 Irish Paper1 P2

$A,B,C$ and $D$ are four points in that order on the circumference of a circle $K$. $AB$ is perpendicular to $BC$ and $BC$ is perpendicular to $CD$. $X$ is a point on the circumference of the circle between $A$ and $D$. $AX$ extended meets $CD$ extended at $E$ and $DX$ extended meets $BA$ extended at $F$. Prove that the circumcircle of triangle $AXF$ is tangent to the circumcircle of triangle $DXE$ and that the common tangent line passes through the center of the circle $K$.

Consider a triangle $ABC$ with $|AB|\neq |AC|$. The angle bisector of the angle $CAB$ intersects the circumcircle of $\triangle ABC$ at two points $A$ and $D$. The circle of center $D$ and radius $|DC|$ intersects the line $AC$ at two points $C$ and $B’$. The line $BB’$ intersects the circumcircle of $\triangle ABC$ at $B$ and $E$. Prove that $B’$ is the orthocenter of $\triangle AED$.

2013 Irish Paper1 P3

The altitudes of a triangle $\triangle ABC$ are used to form the sides of a second triangle $\triangle A_1B_1C_1$. The altitudes of $\triangle A_1B_1C_1$ are then used to form the sides of a third triangle $\triangle A_2B_2C_2$. Prove that $\triangle A_2B_2C_2$ is similar to $\triangle ABC$.

2014 Irish Paper1 P3

In the triangle $ABC$, $D$ is the foot of the altitude from $A$ to $BC$, and $M$ is the midpoint of the line segment $BC$. The three angles $ \angle BAD, \angle DAM$ and $ \angle MAC$ are all equal. Find the angles of the triangle $ABC$.

2014 Irish Paper2 P7

The square $ABCD$ is inscribed in a circle with center $O$. Let $E$ be the midpoint of $AD$. The line $CE$ meets the circle again at $F$. The lines $FB$ and $AD$ meet at $H$. Prove $HD = 2AH$.

2015 Irish Paper1 P1

In the triangle $ABC$, the length of the altitude from $A$ to $BC$ is equal to $1$. $D$ is the midpoint of $AC$. What are the possible lengths of $BD$?

2015 Irish Paper1 P4

Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that

(a) $|LH| = |KM|$

(b) the line through $B$ perpendicular to $DE$ bisects $HK$.

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$, two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$. Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$.

A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1} $ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that :

$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$

2013 Irish Paper1 P3

The altitudes of a triangle $\triangle ABC$ are used to form the sides of a second triangle $\triangle A_1B_1C_1$. The altitudes of $\triangle A_1B_1C_1$ are then used to form the sides of a third triangle $\triangle A_2B_2C_2$. Prove that $\triangle A_2B_2C_2$ is similar to $\triangle ABC$.

2013 Irish Paper1 P5

$A, B$ and $C$ are points on the circumference of a circle with centre $O$. Tangents are drawn to the circumcircles of triangles $OAB$ and $OAC$ at $P$ and $Q$ respectively, where $P$ and $Q$ are diametrically opposite $O$. The two tangents intersect at $K$. The line $CA$ meets the circumcircle of $\triangle OAB$ at $A$ and $X$. Prove that $X$ lies on the line $KO$.

$A, B$ and $C$ are points on the circumference of a circle with centre $O$. Tangents are drawn to the circumcircles of triangles $OAB$ and $OAC$ at $P$ and $Q$ respectively, where $P$ and $Q$ are diametrically opposite $O$. The two tangents intersect at $K$. The line $CA$ meets the circumcircle of $\triangle OAB$ at $A$ and $X$. Prove that $X$ lies on the line $KO$.

The three distinct points$ B, C, D$ are collinear with C between B and D. Another point A not on

the line BD is such that $|AB| = |AC| = |CD|.$

Prove that ∠$BAC = 36$ if and only if $1/|CD|-1/|BD|=1/(|CD| + |BD|)$

In the triangle $ABC$, $D$ is the foot of the altitude from $A$ to $BC$, and $M$ is the midpoint of the line segment $BC$. The three angles $ \angle BAD, \angle DAM$ and $ \angle MAC$ are all equal. Find the angles of the triangle $ABC$.

2014 Irish Paper2 P7

The square $ABCD$ is inscribed in a circle with center $O$. Let $E$ be the midpoint of $AD$. The line $CE$ meets the circle again at $F$. The lines $FB$ and $AD$ meet at $H$. Prove $HD = 2AH$.

2015 Irish Paper1 P1

In the triangle $ABC$, the length of the altitude from $A$ to $BC$ is equal to $1$. $D$ is the midpoint of $AC$. What are the possible lengths of $BD$?

Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that

(a) $|LH| = |KM|$

(b) the line through $B$ perpendicular to $DE$ bisects $HK$.

In triangle $\triangle ABC$, the angle $\angle BAC$ is less than $90^o$. The perpendiculars from $C$ on $AB$ and from $B$ on $AC$ intersect the circumcircle of $\triangle ABC$ again at $D$ and $E$ respectively. If $|DE| =|BC|$, find the measure of the angle $\angle BAC$.

In triangle $ABC$ we have $|AB| \ne |AC|$. The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$, respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$.

Let $ABC$ be a triangle with $|AC| \ne |BC|$. Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$, so that $P$ is between $A$ and $B$. Prove that if $M$ is any point on the circle with diameter $PQ$, then $\angle AMP = \angle BMP$.

Triangle $ABC$ has sides $a = |BC| > b = |AC|$. The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$. If $G$ is the centroid of triangle $ABC$, prove that $\angle KGH = 90^o$.

2016 Irish Paper2 P5

Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$.

Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.

Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$.

Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.

$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$

The triangle $ABC$ is right-angled at $A$. Its incentre is $I$, and $H$ is the foot of the perpendicular from $I$ on $AB$. The perpendicular from $H$ on $BC$ meets $BC$ at $E$, and it meets the bisector of $\angle ABC$ at $D$. The perpendicular from $A$ on $BC$ meets $BC$ at $F$. Prove that $\angle EFD = 45^o$

2018 Irish Paper1 P5

Points $A, B$ and $P$ lie on the circumference of a circle $\Omega_1$ such that $\angle APB$ is an obtuse angle. Let $Q$ be the foot of the perpendicular from $P$ on $AB$. A second circle $\Omega_2$ is drawn with centre $P$ and radius $PQ$. The tangents from $A$ and $B$ to $\Omega_2$ intersect $\Omega_1$ at $F$ and $H$ respectively. Prove that $FH$ is tangent to $\Omega_2$.

2018 Irish Paper1 P5

Points $A, B$ and $P$ lie on the circumference of a circle $\Omega_1$ such that $\angle APB$ is an obtuse angle. Let $Q$ be the foot of the perpendicular from $P$ on $AB$. A second circle $\Omega_2$ is drawn with centre $P$ and radius $PQ$. The tangents from $A$ and $B$ to $\Omega_2$ intersect $\Omega_1$ at $F$ and $H$ respectively. Prove that $FH$ is tangent to $\Omega_2$.

Let $M$ be the midpoint of side $BC$ of an equilateral triangle $ABC$. The point $D$ is on $CA$ extended such that $A$ is between $D$ and $C$. The point $E$ is on $AB$ extended such that $B$ is between $A$ and $E$, and $|MD| = |ME|$. The point $F$ is the intersection of $MD$ and $AB$. Prove that $\angle BFM = \angle BME$.

source: www.irmo.ie

source: www.irmo.ie

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