Ireland 1988 - 2018 (IRMO) 67p

geometry problems from Irish Mathematical Olympiads (IRMO)
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 $AY$.

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 $L$.

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.

1990 Irish Paper1 P1
A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$

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
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$.

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)$.

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.

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.

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.

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.

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}}$.

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$.

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$.

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 $BQ$ = $BP$ .

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 $PQRS$.
$A$ 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$.

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$.

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 $ABC$.

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.

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$.

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
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 $|AC|=|BC|$.

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 $\mathcal{C}_2$.

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 $K$.

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  $\triangle ABC$.

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 $KO$.

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 $1/|CD|-1/|BD|=1/(|CD| + |BD|)$

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 $HK$.

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.

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

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 $\Omega_2$.

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 $\angle BFM = \angle BME$.


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