geometry problems from Middle European Mathematical Olympiad (MEMO)
with aops links in the names
To be more exact, the Slovak pdf collects all the problems and all solutions that are not contained in the English pdf.
with aops links in the names
To be more exact, the Slovak pdf collects all the problems and all solutions that are not contained in the English pdf.
collected inside aops here
2007 - 2022
Let k be a circle and k1,
k2, k3 and k4 four smaller circles with their
centres O1, O2,O3 and O4 respectively
on k. For i = 1, 2, 3, 4 and k5
= k1 the circles ki and ki+1 meet at Ai
and Bi such that Ai lies on k. The points O1, A1, O2, A2,
O3, A3, O4, A4, lie in that order
on k and are pairwise different.
Prove that B1B2B3B4 is a rectangle.
(Switzerland)
Let ABC be an isosceles triangle with AC = BC. It’s incircle touches AB
and BC at D and E, respectively. A line (different from AE) passes through A
and intersects the incircle at F and G. The lines EF and EG intersect the line
AB at K and L, respectively. Prove that DK = DL.
(Ηungary)
Given an acute-angled triangle ABC, let E be a point situated on the
different side of the line AC than B, and let D be an interior point of the
line segment AE. Suppose that ÐADB = ÐCDE, ÐBAD = ÐECD and ÐACB = ÐEBA. Prove that
B, C and E are collinear.
(Slovenia)
Let ABCD be a convex quadrilateral such that AB and CD are not parallel
and AB = CD. The midpoints of the diagonals AC and BD are E and F. The line EF
meets segments AB and CD at G and H, respectively. Show that ÐAGH = ÐDHG.
(Hungary)
Let ABCD be a parallelogram with ÐBAD = 60o and denote by E the intersection of its diagonals.
The circumcircle of the triangle ACD meets the line BA at K ≠ A, the line BD at
P ≠D and the line BC at L ≠ C. The line EP intersects the circumcircle of the
triangle CEL at points E and M. Prove that the triangles KLM and CAP are
congruent.
(Slovenia)
Suppose that ABCD is a cyclic quadrilateral and CD = DA. Points E and F belong
to the segments AB and BC respectively, and ÐADC = 2 ÐEDF. Segments DK
and DM are height and median of the triangle DEF, respectively. L is the point
symmetric to K with respect to M. Prove that the lines DM and BL are parallel.
(Poland)
We are given a cyclic quadrilateral ABCD with a point E on the diagonal
AC such that AD=AE and CB = CE. Let M be the center of the circumcircle k of the triangle BDE. The circle k intersects the line AC in the points E
and F. Prove that the lines FM, AD, and BC meet at one point.
(Switzerland)
The incircle of the triangle ABC touches the sides BC, CA, and AB in the
points D, E, and F, respectively. Let K be the point symmetric to D with
respect to the incenter. The lines DE and FK intersect at S. Prove that AS is parallel to BC.
(Poland)
Let A, B, C, D, E be points such that ABCD is a cyclic quadrilateral and
ABDE is a parallelogram. The diagonals AC and BD intersect at S and the rays AB
and DC intersect at F. Prove that ÐAFS = ÐECD.
(Croatia)
In a plane the circles K1 and K2 with centers I1
and I2, respectively, intersect in two points A and B. Assume that ÐI1AI2 is obtuse. The tangent to K1 in A
intersects K2 again in C and the tangent to K2 in A
intersects K1 again in D. Let K3 be the circumcircle of
the triangle BCD. Let E be the midpoint of that arc CD of K3 that
contains B. The lines AC and AD intersect K3 again in K and L,
respectively. Prove that the line AE is perpendicular to KL.
(Nik Stopar, Slovenia)
Let ABCDE be a convex pentagon with all five sides equal in length. The
diagonals AD and EC meet in S with ÐASE = 60o. Prove that ABCDE has a pair of parallel sides.
(Michal Szabados, Slovakia)
Let ABC be an acute triangle. Denote by B0 and C0
the feet of the altitudes from vertices B and C, respectively. Let X be a point
inside the triangle ABC such that the line BX is tangent to the circumcircle of
the triangle AXC0 and the line CX is tangent to the circumcircle of
the triangle AXB0. Show that the line AX is perpendicular to BC.
(Michal Rolinek, Josef Tkadlec, Czech Republic)
In a given trapezium ABCD with AB parallel to CD and AB > CD, the
line BD bisects the angle ÐADC. The line
through C parallel to AD meets the segments BD and AB in E and F, respectively.
Let O be the circumcentre of the triangle BEF. Suppose that ÐACO = 60ο. Prove the
equality CF = AF + FO.
(Croatia)
Let K be the midpoint of the side AB of a given triangle ABC. Let L and
M be points on the sides AC and BC, respectively, such that ÐCLK = ÐKMC. Prove that
the perpendiculars to the sides AB, AC, and BC passing through K, L, and M,
respectively, are concurrent.
(Poland)
Let ABCD be a convex quadrilateral with no pair of parallel sides, such
that ÐABC =ÐCDA. Assume that the intersections of the pairs of neighbouring angle
bisectors of ABCD form a convex quadrilateral EFGH. Let K be the intersection
of the diagonals of EFGH. Prove that the lines AB and CD intersect on the
circumcircle of the triangle BKD.
(Croatia)
Let ABC be an isosceles
triangle with AC = BC. Let N be a point inside the triangle such that 2ÐANB = 180ο + ÐACB. Let D be the intersection of the line BN and the line parallel to AN that passes through C. Let P be the intersection of the angle bisectors of the angles CAN and ABN. Show that the lines DP
and AN are
perpendicular.
(Matija Basic,
Croatia)
Let ABC be an acute
triangle. Construct a triangle PQR such
that AB = 2PQ, BC = 2QR, CA = 2RP, and the lines PQ,
QR, and RP pass through the points A, B, and C, respectively. (All six points A, B, C, P, Q, and R are
distinct.)
(Gerd Baron, Austria)
Let K be a point inside
an acute triangle ABC, such
that BC is a common tangent of
the circumcircles of AKB and AKC. Let D be the intersection of the lines CK and AB, and
let E be the intersection of
the lines BK and AC. Let F be the intersection of the line BC and the perpendicular bisector of the segment DE. The circumcircle of ABC and the circle k with centre F and radius FD intersect
at points P and Q. Prove that the segment PQ is a diameter of k.
(Patrik Bak, Slovakia)
Let ABC be a triangle
with AB < AC and incentre I. Let E be the point on the side AC such that AE =
AB. Let G be the point on the line EI such that ÐIBG = ÐCBA and such that E
and G lie on opposite
sides of I. Prove that the line
AI, the perpendicular to AE at E, and the bisector of the angle ÐBGI are concurrent.
(Croatia)
Let ABC be a triangle
with AB < AC. Its incircle with centre I touches the sides BC, CA, and AB in
the points D, E, and F respectively. The angle bisector AI intersects the lines DE
and DF in the points X and Y respectively. Let Z be
the foot of the altitude through A with
respect to BC. Prove that D is the incentre of the triangle XY Z.
(Germany)
Let the incircle k of
the triangle ABC touch its side
BC at D. Let the line AD intersect
k at ≠ D and denote the excentre of ABC opposite to A by
K. Let M and N be the
midpoints of BC and KM respectively. Prove that the
points B, C, N, and L are
concyclic.
(Patrik Bak, Slovakia)
Let ABCD be a cyclic
quadrilateral. Let E be the
intersection of lines parallel to AC and
BD passing through points B and A, respectively. The lines EC and ED intersect
the circumcircle of AEB again
at F and G, respectively. Prove that points C, D, F, and G lie on a circle.
(Patrik Bak, Slovakia)
Let ABC be an acute
triangle with AB ą AC. Prove that there exists a point D with the following property:
whenever two distinct points X and
Y lie in the interior of ABC such that the points B, C, X, and Y lie on a circle and ÐAXB - ÐACB = ÐCY A - Ð CBA holds, the line XY passes through D.
(Patrik Bak, Slovakia)
Let I be the incentre of
triangle ABC with AB > AC and let the line AI intersect
the side BC at D. Suppose that point P lies on the segment BC and satisfies PI = PD. Further, let J be the point obtained by reflecting I over the perpendicular bisector of BC, and let Q be the other intersection of the circumcircles of the
triangles ABC and APD. Prove that $\angle BAQ = \angle CAJ$.
(Patrik Bak, Slovakia)
Let ABC be an
acute-angled triangle with ÐBAC > 45ο and with
circumcentre O. The point P lies in its interior such that the
points A, P, O, B lie on a
circle and BP is perpendicular
to CP. The point Q lies on the segment BP such that AQ is parallel to PO.
Prove that ÐQCB=ÐPCO.
(Patrik Bak, Slovakia)
Let ABC be an
acute-angled triangle with AB ≠
AC, and let O be its circumcentre. The line AO intersects the circumcircle ω of ABC a second time in
point D, and the line BC in point E. The circumcircle of CDE
intersects the line CA a
second time in point P. The
line PE intersects the line AB in point Q. The line through O parallel
to PE intersects the altitude
of the triangle ABC that passes
through A in point F. Prove that FP = FQ.
(Croatia)
Let ABC be a triangle
with AB ≠AC. The points K, L, M are the
midpoints of the sides BC, CA,
AB, respectively. The inscribed
circle of ABC with centre I touches the side BC at point D. The line g,
which passes through the midpoint of segment ID and is perpendicular to IK, intersects the line LM
at point P. Prove that ÐPIA = 90 ο.
(Poland)
Let ABCDE be a convex
pentagon. Let P be the
intersection of the lines CE and
BD. Assume that ÐPAD =ÐACB and ÐCAP = ÐEDA. Prove that the
circumcentres of the triangles ABC and
ADE are collinear with P.
(Patrik Bak, Slovakia)
Let ABC be an
acute-angled triangle with AB >AC and circumcircle Γ. Let M be the midpoint of
the shorter arc BC of Γ, and let D be the
intersection of the rays AC and
BM. Let E ≠ C be the
intersection of the internal bisector of the angle ACB and the circumcircle of the triangle BDC. Let us assume that E
is inside the triangle ABC and
there is an intersection N of
the line DE and the circle Γ such that E is the
midpoint of the segment DN.
Show that N is the midpoint of
the segment IBIC, where IB and IC are the excentres of ABC
opposite to B and C, respectively.
(Croatia)
Let ABC be an
acute-angled triangle with AB ≠
AC, circumcentre O and circumcircle Γ. Let the tangents to Γ through B and C meet each other at D,
and let the line AO intersect BC at E. Denote the
midpoint of BC by M and let AM meet Γ again at N ≠ A. Finally, let F ≠
A be a point on Γ such that A,
M, E and F are
concyclic. Prove that FN bisects
the segment MD.
(Patrik Bak, Slovakia)
MEMO 2018 Individual 3
Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A.$ Let $R$ and $Q$ be the centroids of triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and points $P$ $Q$ $R$ and $D$ are concyclic .Prove that the lines $AP$ $BQ$ and $CR$ are concurrent
Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A.$ Let $R$ and $Q$ be the centroids of triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and points $P$ $Q$ $R$ and $D$ are concyclic .Prove that the lines $AP$ $BQ$ and $CR$ are concurrent
Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A,$ points $B'$ and $C'$ lie on the rays $AB$ and $AC,$ respectively , so that points $B',$ $C'$ and $D$ are collinear and points $B,$ $C,$ $B'$ and $C'$ lie on one circle with center $O.$ Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $ABC,$ then $DHMO$ is a parallelogram.
Let $ABC$ be a triangle . The internal bisector of $ABC$ intersects the side $AC$ at $ L$ and the circumcircle of $ABC$ again at $W \neq B.$ Let $K$ be the perpendicular projection of $L$ onto $AW.$ the circumcircle of $BLC$ intersects line $CK$ again at $P \neq C.$ Lines $BP$ and $AW$ meet at point $T.$ Prove that $$AW=WT.$$
Let $ABC$ be an acute-angled triangle with $AC>BC$ and circumcircle $\omega$. Suppose that $P$ is a point on $\omega$ such that $AP=AC$ and that $P$ is an interior point on the shorter arc $BC$ of $\omega$. Let $Q$ be the intersection point of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA=QR$ and $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$. Prove that the points $P, Q, R$ and $S$ are concyclic
Let $ABC$ be a a right-angled triangle with the right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$. Prove that $X, Y$ and $P$ are collinear.
Let $ABC$ be an acute-angled triangle such that $AB<AC$. Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$. Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$. Finally, let $M$ be the midpoint of the side $AB$. Prove that $\angle APD=\angle MPB$.
Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$. Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.
Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A, D, E$ and $F$ are concyclic.
Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.
Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$. Let $X$ be a point on the ray $AB$ such that $2 \angle CXA=\angle CMA$. Let $Y$ be a point on the ray $AC$ such that $2 \angle AYB=\angle AMB$. The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$, such that the points $P, B, C$, and $Q$ lie in this order on the line $BC$. Prove that $PB=QC$.
by Dominik Burek, Poland
Let $ABCD$ be a parallelogram with $\angle DAB < 90$ Let $E$ be the point on the line $BC$ such that $AE = AB$ and let $F$ be the point on the line $CD$ such that $AF = AD$. The circumcircle of the triangle $CEF$ intersects the line $AE$ again in $P$ and the line $AF$ again in $Q$. Let $X$ be the reflection of $P$ over the line $DE$ and $Y$ the reflection of $Q$ over the line $BF$. Prove that $A, X, Y$ lie on the same line.
Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle CAB = 90$. The medians through $B$ and $C$ meet $\Omega$ again at $D$ and $E$, respectively. The tangent to $\Omega$ at $D$ intersects the line $AC$ at $X$ and the tangent to $\Omega$ at $E$ intersects the line $AB$ at $Y$ . Prove that the line $XY$ is tangent to $\Omega$.
Let $ABCD$ be a convex quadrilateral such that $AC = BD$ and the sides $AB$ and $CD$ are not parallel. Let $P$ be the intersection point of the diagonals $AC$ and $BD$. Points $E$ and $F$ lie, respectively, on segments $BP$ and $AP$ such that $PC=PE$ and $PD=PF$. Prove that the circumcircle of the triangle determined by THE lines $AB, CD, EF$ is tangent to the circumcircle of the triangle $ABP$.
sources:
No comments:
Post a Comment