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Mediterranean 1998 - 2019 (MMC) 26p

geometry problems from Mediterranean Mathematical Competitions 
(also known as MMC, and as Peter O' Halloran Memorial)
with aops links in the names

1998 - 2019

Mediterranean  1998 P1 (GRE)
A square ABCD is inscribed in a circle. If M is a point on the shorter arc AB, prove that MC·MD > 3√3 ·MA·MB.

Mediterranean  1998 P3 (SPA)
In a triangle ABC, I is the incenter and D,E,F are the points of tangency of the incircle with BC,CA,AB, respectively. The bisector of angle BIC meets BC at M, and the line AM intersects EF at P. Prove that DP bisects the angle FDE.

Mediterranean  1999 P4
In a triangle ABC with BC = a, CA = b, AB = c we have <B = 4<A. Show that  ab2c3 = (b2 −a2−ac)((a2−b2)2 −a2c2).

Mediterranean  2000 P2
Suppose that in the exterior of a convex quadrilateral ABCD equilateral triangles XAB,YBC,ZCD,WDA with centroids S1,S2,S3,S4 respectively are constructed. Prove that S1S3 $\bot $  S2S4 if and only if AC $\bot $  BD.

Mediterranean  2000 P4
Let P,Q,R,S be the midpoints of the sides BC,CD,DA,AB of a convex quadrilateral, respectively. Prove that 4 (AP2+BQ2+CR2+DS2) ≤ 5 (AB2+BC2+CD2+DA2).

Mediterranean  2001 P1
Let P and Q be points on a circle k. A chord AC of k passes through the midpoint M of PQ. Consider a trapezoid ABCD inscribed in k with AB // CD. Prove that the intersection point X of AD and BC depends only on k and P,Q.

In an acute-angled triangle $ABC$, $M$ and $N$ are points on the sides $AC$ and $BC$ respectively, and $K$ the midpoint of $MN$. The circumcircles of triangles $ACN$ and $BCM$ meet again at a point $D$. Prove that the line $CD$ contains the circumcenter $O$ of $\triangle ABC$ if and only if $K$ is on the perpendicular bisector of $AB.$

Mediterranean  2003 P2
In a triangle ABC with BC = CA+ ö AB, point P is given on side AB such that BP /PA= 1/ 3. Prove that <CAP = 2<CPA.

Mediterranean  2004 P1
In a triangle ABC, the altitude from A meets the circumcircle again at T. Let O be the circumcenter. The lines OA and OT intersect the side BC at Q and M, respectively. Prove that $\frac{{{S}_{AQC}}}{{{S}_{CMT}}}={{\left( \frac{sinB}{cosC} \right)}^{2}}$

Mediterranean  2004 P4
Let z1, z2, z3 be pairwise distinct complex numbers satisfying |z1| = |z2| = |z3| =1 and$\frac{1}{2+\left| {{z}_{1}}+{{z}_{2}} \right|}+\frac{1}{2+\left| {{z}_{2}}+{{z}_{3}} \right|}+\frac{1}{2+\left| {{z}_{3}}+{{z}_{1}} \right|}=1$. If the points A(z1),B(z2),C(z3) are vertices of an acute-angled triangle, prove that this triangle is equilateral.

Mediterranean  2005 P2
Two circles k and k′ have the common center O and radii r and r′ respectively. A ray Ox meets k at A, while its complementary ray Ox′ meets k′ at B. Another ray Ot meets k at E and k′ at F. Prove that the circles OAE, OBF and the circles with diameters EF and AB all pass through a single point.

Mediterranean  2006 P2 (GRE)
Let P be a point inside a triangle ABC, and A1B2,B1C2,C1A2 be segments through P parallel to AB,BC,CA respectively, where points A1,A2 lie on BC,B1,B2 on CA, and C1,C2 on AB. Prove that Area(A1A2B1B2C1C2) ≥ 2/3 Area(ABC).
by Dimitris Kontogiannis
Mediterranean  2007 P2
The diagonals AC and BD of a convex cyclic quadrilateral ABCD intersect at point E. Given that AB = 39, AE = 45, AD = 60 and BC = 56, determine the length of CD.

Mediterranean  2007 P3
In the triangle ABC, the angle α =<BAC and the side a =BC are given. Assume that $a=\sqrt{rR}$ where r is the inradius and R the circumradius. Compute all possible lengths of sides AB and AC.

Mediterranean 2008 P1 (GRE)
Let ABCDEF be a convex hexagon such that all of its vertices are on a circle. Prove that AD, BE and CF are concurrent if and only if  $\frac{AB}{BC}\cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$

by Dimitris Kontogiannis
Mediterranean 2009 P2
Let ABC be a triangle with 90þ ≠ <A ≠135þ. Let D and E be external points to the triangle ABC such that DAB and EAC are isosceles triangles with right angles at D and E. Let                  F = BE ∩ CD, and let M and N be the midpoints of BC and DE, respectively. Prove that, if three of the points A, F, M, N are collinear, then all four are collinear.

Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that $R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}$ where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$

Mediterranean 2011 P4
Let D be the foot of the internal bisector of the angle <A of the triangle ABC. The straight line which joins the incenters of the triangles ABD and ACD cut AB and AC at M and N, respectively. Show that BN and CM meet on the bisector AD.

Mediterranean 2012 P4
Let O be the circumcenter, R be the circumradius, and k be the circumcircle of a triangle ABC. Let k1 be a circle tangent to the rays AB and AC, and also internally tangent to k. Let k2 be a circle tangent to the rays AB and AC , and also externally tangent to k. Let A1 and A2 denote the respective centers of k1 and k2. Prove that: (OA1 + OA2)2 − A1A22= 4R2.

Mediterranean 2013 P4
ABCD is quadrilateral inscribed in a circle Γ .Lines AB and CD intersect at E and lines AD and BC intersect at F. Prove that the circle with diameter EF and circle Γ are orthogonal.

Mediterranean 2014 P4
In triangle ABC let A, B, Crespectively be the midpoints of the sides BC, CA, AB. Furthermore let L, M, N be the projections of the orthocenter on the three sides BC, CA, AB, and let k denote the nine-point circle. The lines AA, BB, CCintersect k in the points D, E, F. The tangent lines on k in D, E, F intersect the lines MN, LN and LM in the points P, Q, R.
Prove that P, Q and R are collinear.

Mediterranean 2015 P 2
Prove that for each triangle, there exists a vertex, such that with the two sides starting from that vertex and each cevian starting from that vertex, is possible to construct a triangle.

Mediterranean 2016 P1
Let ABC be a triangle. Let D be the intersection point of the angle bisector at A with BC. Let T be the intersection point of the tangent line to the circumcircle of triangle ABC at point A with the line through B and C. Let I be the intersection point of the orthogonal line to AT through point D with the altitude ha of the triangle at point A. Let P be the midpoint of AB, and let O be the circumcenter of triangle ABC. Let M be the intersection point of AB and TI, and let F be the intersection point of PT and AD. Prove: MF and AO are orthogonal to each other.

Mediterranean 2017 P1
Let ABC be an equilateral triangle, and let P be some point in its circumcircle. Determine all positive integers n, for which the value of the sum Sn(P) =|PA|n + |PB| n + |PC| n  is independent of the choice of point P.

Mediterranean 2018 P3
Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that $\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$

Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that $\frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)$

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