Spain 1984 - 2019 52p

geometry problems from Spanish Mathematical Olympiads
with aops links in the names

1984 - 2019

At a position $O$ of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments $AB$ and $CD$ attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length $\ell$.

Let $A$ and $A' $ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$ be arcs of these circles of the same length $x$. Find the locus of the midpoint of segment $BB'$ when $x$ varies:
(a) if the arcs have the same direction,
(b) if the arcs have opposite directions.

Let $f : P\to P$ be a bijective map from a plane $P$ to itself such that:
(i) $f (r)$ is a line for every line $r$,
(ii) $f (r) $ is parallel to $r$ for every line $r$.
What possible transformations can $f$ be?

Let $OX$ and $OY$ be non-collinear rays. Through a point $A$ on $OX$, draw two lines $r_1$ and $r_2$ that are antiparallel with respect to $\angle XOY$. Let $r_1$ cut $OY$ at $M$ and $r_2$ cut $OY$ at $N$. (Thus, $\angle OAM = \angle ONA$). The bisectors of $ \angle AMY$ and $\angle ANY$ meet at $P$. Determine the location of $P$.

Let $a, b, c$ be the side lengths of a scalene triangle and let $O_a, O_b and O_c$ be three concentric circles with radii $a, b$ and $c$ respectively.
(a) Howmany equilateral triangles with different areas can be constructed such that the lines containing the sides are tangent to the circles?
(b) Find the possible areas of such triangles.

In a triangle $ABC, D$ lies on $AB, E$ lies on $AC$ and $ \angle ABE = 30^o, \angle EBC = 50^o, \angle ACD = 20^o$, $\angle DCB = 60^o$. Find $\angle EDC$.

Points $A' ,B' ,C'$ on the respective sides $BC,CA,AB$ of triangle $ABC$ satisfy $\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k$. The lines $AA' ,BB' ,CC' $ form a triangle $A_1B_1C_1$ (possibly degenerate). Given $k$ and the area $S$ of $\triangle ABC$, compute the area of $\triangle A_1B_1C_1$.

On the sides $BC,CA$ and $AB$ of a triangle $ABC$ of area $S$ are taken points $A' ,B' ,C'$ respectively such that $AC' /AB = BA' /BC = CB' /CA = p$, where $0 < p < 1$ is variable.
(a) Find the area of triangle $A' B' C'$ in terms of $ p$.
(b) Find the value of $p$ which minimizes this area.
(c) Find the locus of the intersection point $P$ of the lines through $A' $ and $C'$ parallel to $AB$ and $AC$ respectively.

The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$.

Given two circles of radii $r$ and $r'$ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths $\ell$.

Given a triangle $ABC$, show how to construct the point $P$ such that $\angle PAB= \angle PBC= \angle PCA$. Express this angle in terms of $\angle A,\angle B,\angle C$ using trigonometric functions.

Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.

Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$. Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$, the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$.  Find the locus of the projection of $W$ on the plane O$xy$. Also find the locus of points $W$.

1994 Spanish P4
In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

1995 Spanish P3
A line through the centroid G of the triangle ABC intersects the side AB at P and the side AC at Q Show that $\frac{PB}{PA} \cdot \frac{QC}{QA} \leq \frac{1}{4}$.

1995 Spanish P6
Let $C$ be a variable interior point of a fixed segment $AB$. Equilateral triangles $ACB' $ and $CBA'$  are constructed on the same side and $ABC' $ on the other side of the line $AB$.
(a) Prove that the lines $AA' ,BB'$ , and $CC'$ meet at some point $P$.
(b) Find the locus of $P$ as $C$ varies.
(c) Prove that the centers $A'' ,B'' ,C''$  of the three triangles form an equilateral triangle.
(d) Prove that $A'' ,B'',C''$ , and $P$ lie on a circle.

1996 Spanish P2
Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles.

1996 Spanish P6
A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.

1997 Spanish P5
Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.

1998 Spanish P1
A unit square $ABCD$ with centre $O$ is rotated about $O$ by an angle $\alpha$. Compute the common area of the two squares.

Let $ABC$ be a triangle. Points $D$ and $E$ are taken on the line $BC$ such that $AD$ and $AE$ are parallel to the respective tangents to the circumcircle at $C$ and $B$. Prove that
$\frac{BE}{CD}=\left(\frac{AB}{AC}\right)^2 $

The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that:
a) $g_a,g_b,g_c \ge  \frac{2}{3}r$
b) $g_a+g_b+g_c \ge 3r$

Two circles $C_1$ and $C_2$ with the respective radii $r_1$ and $r_2$ intersect in $A$ and $B.$ A variable line $r$ through $B$ meets $C_1$ and $C_2$ again at $P_r$ and $Q_r$ respectively. Prove that there exists a point $M,$ depending only on $C_1$ and $C_2,$ such that the perpendicular bisector of each segment $P_rQ_r$ passes through $M.$

Let $P$ be a point on the interior of triangle $ABC$, such that the triangle $ABP$ satisfies $AP = BP$. On each of the other sides of $ABC$, build triangles $BQC$ and $CRA$ exteriorly, both similar to triangle $ABP$ satisfying: $BQ = QC$ and $CR = RA.$  Prove that the point $P,Q,C,$ and $R$ are collinear or are the vertices of a parallelogram.

A quadrilateral $ABCD$ is inscribed in a circle of radius 1 whose diameter is $AB$. If the quadrilateral $ABCD$ has an incircle, prove that $CD \leq 2 \sqrt{5} - 2$.

In the triangle $ABC$, $A'$ is the foot of the altitude to $A$, and $H$ is the orthocenter.
a) Given a positive real number $k = \frac{AA'}{HA'}$ , find the relationship between the angles $B$ and $C$, as a function of $k$.
b) If $B$ and $C$ are fixed, find the locus of the vertice $A$ for any value of $k$.

The altitudes of the triangle ${ABC}$ meet in the point ${H}$. You know that ${AB = CH}$. Determine the value of the angle $\widehat{BCA}$

How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths $1, 2, 3, 4, 5$ and $6,$ in any order?

${ABCD}$ is a quadrilateral, ${P}$ and ${Q}$ are midpoints of the diagonals ${BD}$ and ${AC}$, respectively. The lines parallel to the diagonals originating from ${P}$ and ${Q}$ intersect in the point ${O}$. If we join the four midpoints of the sides, ${X}$, ${Y}$, ${Z}$, and ${T}$, to ${O}$, we form four quadrilaterals: ${OXBY}$, ${OYCZ}$, ${OZDT}$, and ${OTAX}$. Prove that the four newly formed quadrilaterals have the same areas.

Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is:  ${A/5 = B/10 = C/13}$.

We will say that a triangle is multiplicative if the product of the heights of two of its sides is equal to the length of the third side. Given $ABC \dots XYZ$ is a regular polygon with $n$ sides of length $1$. The $n-3$ diagonals that go out from vertex $A$ divide the triangle $ZAB$ in $n-2$ smaller triangles. Prove that each one of these triangles is multiplicative.

In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that:
a) $0^o \le  A \le 60^o$.
b) The height relative to side $a$ is three times the inradius $r$.
c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.

$ABC$ is an isosceles triangle with $AB = AC$. Let $P$ be any point of a circle tangent to the sides $AB$ in $B$ and to AC in C. Denote $a$, $b$ and $c$ to the distances from $P$ to the sides $BC, AC$ and $AB$ respectively. Prove that: $a^2=bc$

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?

2007 Spanish P3
$O$ is the circumcenter of triangle $ABC$. The bisector from $A$ intersects the opposite side in point $P$. Prove that the following is satisfied: $AP^2 + OA^2 - OP^2 = bc.$

2007 Spanish P6
Given a halfcircle of diameter $AB = 2R$, consider a chord $CD$ of length $c$. Let $E$ be the intersection of $AC$ with $BD$ and $F$ the inersection of $AD$ with $BC$. Prove that the segment $EF$ has a constant length and direction
when varying the chord $CD$ about the halfcircle.

2008 Spanish P5
Given a circle, two fixed points $A$ and $B$ and a variable point $P$, all of them on the circle, and a line $r$, $PA$ and $PB$ intersect $r$ at $C$ and $D$, respectively. Find two fixed points on $r$, $M$ and $N$, such that $CM\cdot DN$ is constant for all $P$.

2009 Spanish P2
Let $ ABC$ be an acute triangle with the incircle $ C(I,r)$ and the circumcircle $ C(O,R)$ . Denote
$ D\in BC$ for which $ AD\perp BC$ and $ AD = h_a$ . Prove that $ DI^2 = (2R - h_a)(h_a - 2r)$ .

2009 Spanish P6
Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$  ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q =\overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.

2010 Spanish P3
Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which $ EG+3HF\ge kd+(1-k)s $
where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?

2010 Spanish P5
In a triangle $ABC$, let $P$ be a point on the bisector of $\angle BAC$ and let $A',B'$ and $C'$ be points on lines $BC,CA$ and $AB$ respectively such that $PA'$ is perpendicular to $BC,PB'\perp AC$, and $PC'\perp AB$. Prove that $PA'$ and $B'C'$ intersect on the median $AM$, where $M$ is the midpoint of $BC$.

2011 Spanish P3
Let $A$, $B$, $C$, $D$ be four points in space not all lying on the same plane. The segments $AB$, $BC$, $CD$, and $DA$ are tangent to the same sphere. Prove that their four points of tangency are coplanar.

2011 Spanish P5
In triangle $ABC$, $\angle B=2\angle C$ and $\angle A>90^\circ$. Let $D$ be the point on the line $AB$ such that $CD$ is perpendicular to $AC$, and let $M$ be the midpoint of $BC$. Prove that $\angle AMB=\angle DMC$.

2012 Spanish P6
Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$, $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$. The circle $\omega$ be tangent to the side $BC$ at the point $D$. The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$. The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.

2013 Spanish P6
Let $ABCD$ a convex quadrilateral where: $|AB|+|CD|=\sqrt{2} |AC|$ and $|BC|+|DA|=\sqrt{2} |BD|$ . What form does the quadrilateral have?

2014 Spanish P3
Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.

2015 Spanish P2
In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$. Prove that:
a) The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $4R^2-4Rr-2r^2$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively.
b) The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $AL=\sqrt{ AB.AC}$ where $I$ is the centre of the inscribed circle of $ABC$.

2015 Spanish P6
Let $ABC$ be a triangle. $M$, and $N$ points on $BC$, such that $BM=CN$, with $M$ in the interior of $BN$. Let $P$ and $Q$ be points in $AN$ and $AM$ respectively such that $\angle PMC= \angle MAB$, and $\angle QNB= \angle NAC$. Prove that $ \angle QBC= \angle PCB$.

2016 Spanish P3
In the circumscircle of a triangle $ABC$, let $A_1$ be the point diametrically opposed to the vertex $A$. Let $A'$ the intersection point of $AA'$ and $BC$. The perpendicular to the line $AA'$ from $A'$ meets the sides $AB$ and $AC$ at $M$ and $N$, respectively. Prove that the points $A,M,A_1$ and $N$ lie on a circle which has the center on the height from $A$ of the triangle $ABC$.

2017 Spanish P6
In the triangle $ABC$, the respective mid points of the sides $BC$, $AB$ and $AC$ are $D$, $E$ and $F$. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$. Also, let $O$ be the intersection point of $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR=AD$.

2018 Spanish P3
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$. Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.

2019 Spanish P6
In the scalene triangle $ABC$, the bisector of angle A cuts side $BC$ at point $D$. The tangent lines to the circumscribed circunferences of triangles $ABD$ and $ACD$ on point D, cut lines $AC$ and $AB$ on points $E$ and $F$ respectively. Let $G$ be the intersection point of lines $BE$ and $CF$. Prove that angles $EDG$ and $ADF$ are equal.


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