geometry problems from Benelux Mathematical Olympiads (BxMO)
with aops links in the names
with aops links in the names
(Belgium, Netherlands, Luxembourg)
collected inside aops here
2009 - 2022
Benelux 2009 P4
Given trapezoid ABCD with parallel sides AB and CD, let E be a point on line BC outside segment BC, such that segment AE intersects segment CD. Assume that there exists a point F inside segment AD such that <EAD = <CBF. Denote by I the point of intersection of CD and EF, and by J the point of intersection of AB and EF. Let K be the midpoint of segment EF, and assume that K is different from I and J. Prove that K belongs to the circumcircle of ∆ABI if and only if K belongs to the circumcircle of ∆CDJ.
Given trapezoid ABCD with parallel sides AB and CD, let E be a point on line BC outside segment BC, such that segment AE intersects segment CD. Assume that there exists a point F inside segment AD such that <EAD = <CBF. Denote by I the point of intersection of CD and EF, and by J the point of intersection of AB and EF. Let K be the midpoint of segment EF, and assume that K is different from I and J. Prove that K belongs to the circumcircle of ∆ABI if and only if K belongs to the circumcircle of ∆CDJ.
On a line l there are three different points A, B and P in that order.
Let a be the line through A perpendicular to l, and let b be the line through B
perpendicular to l. A line through P, not coinciding with l, intersects a in Q
and b in R. The line through A perpendicular to BQ intersects BQ in L and BR in
T. The line through B perpendicular to AR intersects AR in K and AQ in S.
(a) Prove that P, T, S are collinear.
(b) Prove that P, K, L are collinear.
Let ABC be a triangle
with incentre I. The angle
bisectors AI, BI and CI meet [BC], [CA] and [AB] at D, E and F, respectively. The perpendicular bisector of [AD] intersects the lines BI and CI at M and N, respectively. Show that A, I, M and N lie on a circle.
In triangle ABC the midpoint of BC is called M. Let P be a variable
interior point of the triangle such that <CPM = <PAB. Let Γ be the circumcircle of triangle ABP. The line MP intersects Γ a second time in Q. Define R as the reflection of P in the tangent to Γ in B. Prove that the length |QR| is independent of the position of P
inside the triangle.
Let ∆ABC be a triangle with circumcircle Γ, and let I be the center of the incircle of ∆ABC. The lines AI, BI and
CI intersect Γ in D ≠ A, E ≠ B and F ≠ C. The tangent lines to Γ in F, D and E intersect the lines AI, BI and CI in R, S and T,
respectively. Prove that
|AR|·|BS|·|CT| = |ID|·|IE|·|IF| .
Let ABCD be a square. Consider a variable point P inside the square for
which <BAP ≥ 60þ. Let Q
be the intersection of the line AD and the perpendicular to BP in P. Let R be
the intersection of the line BQ and the perpendicular to BP from C.
(a) Prove that |BP| ≥ |BR|.
(b) For which point(s) P does the inequality in (a) become an equality?
Let ABC be an acute
triangle with circumcentre O.
Let ΓΒ be the circle through A and
B that is tangent to AC, and let ΓC be the circle through A and
C that is tangent to AB. An arbitrary line through A intersects ΓΒ again in X and
intersects ΓC again in Y . Prove that
|OX| = |OY |.
A circle ω passes through the two vertices B and C of a triangle ABC. Furthermore,
ω intersects segment AC in D ≠ C and segment AB in E ≠ B. On the ray from
B through D lies a point K such that |BK| = |AC|, and on the ray from C through
E lies a point L such that |CL|=|AB|. Show that the circumcentre O of triangle
AKL lies on ω.
In the convex quadrilateral ABCD we have <B= <C and <D = 90þ. Suppose that |AB|=2|CD|. Prove that the angle bisector of <ACB is perpendicular to CD.
Benelux 2018 P3
Let $ABC$ be a triangle with orthocentre $H$, and let $D$, $E$, and $F$ denote the respective midpoints of line segments $AB$, $AC$, and $AH$. The reflections of $B$ and $C$ in $F$ are $P$ and $Q$, respectively.
a) Show that lines $PE$ and $QD$ intersect on the circumcircle of triangle $ABC$.
b) Prove that lines $PD$ and $QE$ intersect on line segment $AH$.
Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $Z$ (with $A\neq Z$). Let $B$ be the centre of $\Gamma_1$ and let $C$ be the centre of $\Gamma_2$. The exterior angle bisector of $\angle{BAC}$ intersects $\Gamma_1$ again at $X$ and $\Gamma_2$ again at $Y$. Prove that the interior angle bisector of $\angle{BZC}$ passes through the circumcenter of $\triangle{XYZ}$.
Let $ABC$ be a triangle. The circle $\omega_A$ through $A$ is tangent to line $BC$ at $B$. The circle $\omega_C$ through $C$ is tangent to line $AB$ at $B$. Let $\omega_A$ and $\omega_C$ meet again at $D$. Let $M$ be the midpoint of line segment $[BC]$, and let $E$ be the intersection of lines $MD$ and $AC$. Show that $E$ lies on $\omega_A$.
A cyclic quadrilateral $ABXC$ has circumcentre $O$. Let $D$ be a point on line $BX$ such that
$AD = BD$. Let $E$ be a point on line $CX$ such that $AE = CE$. Prove that the circumcentre of
triangle $\triangle DEX$ lies on the perpendicular bisector of $OA$.
Let $ABC$ be a scalene acute triangle. Let $B_1$ be the point on ray $[AC$ such that $|AB_1|=|BB_1|$. Let $C_1$ be the point on ray $[AB$ such that $|AC_1|=|CC_1|$. Let $B_2$ and $C_2$ be the points on line $BC$ such that $|AB_2|=|CB_2|$ and $|BC_2|=|AC_2|$. Prove that $B_1$, $C_1$, $B_2$, $C_2$ are concyclic.
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