Mathley 80p

Here are posted all 16 rounds of Vietnamese Geometry Contest, Mathley from 2011 - 2012 in English with aops links (16x4=64 problems) . Below are the geometry problems from problem solving column of Mathley 2014-15 (16 more problems)

About Geometry Mathley Contest:
This monthly geometry contest is open to people of all ages with the same interest in plane geometry and motivation to hone their problem-solving skill in geometry. You can submit solutions and propose new problems as well. 

collected in as Geometry Mathley 2011-12 here
all posted in the Geometry Mathley Forum
also Mathley 2014-15

Geometry Mathley 2011 - 2012 
(16x4-problem sets / rounds)

Geometry Mathley 2011-12 1.1
Let $ABCDEF$ be a hexagon having all interior angles equal to $120^o$ each. Let $P,Q,R, S, T, V$ be the midpoints of the sides of the hexagon $ABCDEF$. Prove the inequality $$p(PQRSTV ) \ge \frac{\sqrt3}{2} p(ABCDEF)$$ , where $p(.)$ denotes the perimeter of the polygon.

Nguyễn Tiến Lâm

Geometry Mathley 2011-12 1.2
Let $ABC$ be an acute triangle with its altitudes $BE,CF$. $M$ is the midpoint of $BC$. $N$ is the intersection of $AM$ and $EF. X$ is the projection of $N$ on $BC$. $Y,Z$ are respectively the projections of $X$ onto $AB,AC$. Prove that $N$ is the orthocenter of triangle $AYZ$.

Nguyễn Minh Hà

Geometry Mathley 2011-12 1.3
Let $ABC$ be an acute triangle with incenter $O$, orthocenter $H$, altitude $AD. AO$ meets $BC$ at $E$. Line through $D$ parallel to $OH$ meet $AB,AC$ at $M,N$, respectively. Let $I$ be the midpoint of $AE$, and $DI$ intersect $AB,AC$ at $P,Q$ respectively. $MQ$ meets $NP$ at $T$. Prove that $D,O, T$ are collinear.

Trần Quang Hùng

Geometry Mathley 2011-12 1.4
Given are three circles $(O_1), (O_2), (O_3)$, pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let $(X_1)$ be the circle externally tangent to $(O_1)$ and internally tangent to the circles $(O_2), (O_3),$ circles $(X_2), (X_3)$ are defined in the same manner. Let $(Y_1)$ be the circle internally tangent to $(O_1)$ and externally tangent to the circles $(O_2), (O_3)$, the circles $(Y_2), (Y_3)$ are defined in the same way. Let $(Z_1), (Z_2)$ be two circles internally tangent to all three circles $(O_1), (O_2), (O_3)$. Prove that the four lines $X_1Y_1,  X_2Y_2, X_3Y_3, Z_1Z_2$ are concurrent.

Nguyễn Văn Linh

Geometry Mathley 2011-12 2.1
Let $ABC$ be an equilateral triangle with circumcircle of center $O$ and radius $R$. Point $M$ is exterior to the triangle such that $S_bS_c = S_aS_b+S_aS_c$, where $S_a, S_b, S_c$ are the areas of triangles $MBC,MCA,MAB$ respectively. Prove that $OM \ge R$.

Nguyễn Tiến Lâm

Geometry Mathley 2011-12 2.2
Let $ABC$ be a scalene triangle. A circle $(O)$ passes through $B,C$, intersecting the line segments $BA,CA$ at $F,E$ respectively. The circumcircle of triangle $ABE$ meets the line $CF$ at two points $M,N$ such that $M$ is between $C$ and $F$. The circumcircle of triangle $ACF$ meets the line $BE$ at two points $P,Q$ such that $P$ is betweeen $B$ and $E$. The line through $N$ perpendicular to $AN$ meets $BE$ at $R$, the line through $Q$ perpendicular to $AQ$ meets $CF$ at $S$. Let $U$ be the intersection of $SP$ and $NR, V$ be the intersection of $RM$ and $QS$. Prove that three lines $NQ,UV$ and $RS$ are concurrent.

Trần Quang Hùng

Geometry Mathley 2011-12 2.3
Let $ABC$ be a triagle inscribed in a circle $(O)$. A variable line through the orthocenter $H$ of the triangle meets the circle $(O)$ at two points $P , Q$. Two lines through $P, Q$ that are perpendicular to $AP , AQ$ respectively meet $BC$ at $M, N$ respectively. Prove that the line through $P$ perpendicular to $OM$ and the line through $Q$ perpendicular to $ON$ meet each other at a point on the circle $(O)$.

Nguyễn Văn Linh

Geometry Mathley 2011-12 2.4
Let $ABC$ be a triangle inscribed in a circle of radius $O$. The angle bisectors $AD,BE,CF$ are concurrent at $I$. The points $M,N, P$ are respectively on $EF, FD$, and $DE$ such that $IM, IN, IP$ are perpendicular to $BC,CA,AB$ respectively. Prove that the three lines $AM,BN, CP$ are concurrent at a point on $OI$.

Nguyễn Minh Hà

Geometry Mathley 2011-12 3.1
$AB,AC$ are tangent to a circle $(O)$, $B,C$ are the points of tangency. $Q$ is a point iside the angle $BAC$, on the ray $AQ$, take a point $P$ suc that $OP$ is perpendicular to $AQ$. The line $OP$ meets the circumcircles triangles $BPQ$ and $CPQ$ at $I, J$. Prove that $OI = OJ$.

Hồ Quang Vinh

Geometry Mathley 2011-12 3.2
Given a triangle $ABC$, a line $\delta$ and a constant $k$, distinct from $0$ and $1,M$ a variable point on the line $\delta$. Points $E, F$ are on $MB,MC$ respectively such that $\frac{\overline{ME}}{\overline{MB}} =  \frac{\overline{MF}}{\overline{MC}} = k$. Points $P,Q$ are on $AB,AC$ such that $PE, QF$ are perpendicular to $\delta$. Prove that the line through $M$ perpendicular to $PQ$ has a fixed point.

Nguyễn Minh Hà

Geometry Mathley 2011-12 3.3
A triangle $ABC$ is inscribed in circle $(O)$. $P1, P2$ are two points in the plane of the triangle. $P_1A, P_1B, P_1C$ meet $(O)$ again at $A_1,B_1,C_1$ . $P_2A, P_2B, P_2C$ meet $(O)$ again at $A_2,B_2,C_2$.
a) $A_1A_2, B_1B_2, C_1C_2$ intersect $BC,CA,AB$ at $A_3,B_3,C_3$. Prove that three points $A_3,B_3,C_3$ are collinear.
b) $P$ is a point on the line $P_1P_2. A_1P,B_1P,C_1P$ meet (O) again at $A_4,B_4,C_4$. Prove that three lines $A_2A_4,B_2B_4,C_2C_4$ are concurrent.

Trần Quang Hùng

Geometry Mathley 2011-12 3.4
A triangle $ABC$ is inscribed in the circle $(O,R)$. A circle $(O',R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. $P$ is a point on the circle $(O)$. Rays $PA, PB, PC$ meet $(O')$ at $A_1,B_1,C_1$. Let $A_2B_2C_2$ be the triangle formed by the intersections of the line symmetric to $B_1C_1$ about $BC$, the line symmetric to $C_1A_1$ about $CA$ and the line symmetric to $A_1B_1$ about $AB$. Prove that the circumcircle of $A_2B_2C_2$ is tangent to $(O)$.

Nguyễn Văn Linh

Geometry Mathley 2011-12 4.1
Five points $K_i, i = 1, 2, 3, 4$ and $P$ are chosen arbitrarily on the same circle. Denote by $P(i, j)$ the distance from $P$ to the line passing through $K_i$ and $K_j$ . Prove that $$P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)$$

Bùi Quang Tuấn

Geometry Mathley 2011-12 4.2
Let $ABC$ be a triangle. $(K)$ is an arbitrary circle tangent to the lines $AC,AB$ at $E, F$ respectively. $(K)$ cuts $BC$ at $M,N$ such that $N$ lies between $B$ and $M$. $FM$ intersects $EN$ at $I$. The circumcircles of triangles $IFN$ and $IEM$ meet each other at $J$ distinct from $I$. Prove that $IJ$ passes through $A$ and $KJ$ is perpendicular to $IJ$.

Trần Quang Hùng

Geometry Mathley 2011-12 4.3
Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel  FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that
i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency.
ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$.

Nguyễn Minh Hà

Geometry Mathley 2011-12 4.4
et $ABC$ be a triangle with $E$ being the centre of its Euler circle. Through $E$, construct the lines $PS, MQ, NR$ parallel to $BC,CA,AB$ ($R,Q$ are on the line $BC, N, P$ on the line $AC,M, S$ on the line $AB$). Prove that the four Euler lines of triangles $ABC,AMN,BSR,CPQ$ are concurrent.

Nguyễn Văn Linh

Geometry Mathley 2011-12 5.1
Let $a, b$ be two lines intersecting each other at $O$. Point $M$ is not on either $a$ or $b$. A variable circle $(C)$ passes through $O,M$ intersecting $a, b$ at $A,B$ respectively, distinct from $O$. Prove that the midpoint of $AB$ is on a fixed line.

Hạ Vũ Anh

Geometry Mathley 2011-12 5.2
Let $ABCD$ be a rectangle and $U, V$ two points of its circumcircle. Lines $AU,CV$ intersect at $P$ and lines $BU,DV$ intersect at $Q$, distinct from $P$. Prove that $$\frac{1}{PQ^2} \ge \frac{1}{UV^2} - \frac{1}{AC^2}$$

Michel Bataille

Geometry Mathley 2011-12 5.3
Let $ABC$ be an acute triangle, not being isoceles. Let $\ell_a$ be the line passing through the points of tangency of the escribed circles in the angle $A$ with the lines $AB, AC$ produced. Let $d_a$ be the line through $A$ parallel to the line that joins the incenter $I$ of the triangle $ABC$ and the midpoint of $BC$. Lines $\ell_b, d_b, \ell_c, d_c$ are defined in the same manner. Three lines $\ell_a, \ell_b, \ell_c$ intersect each other and these intersections make a triangle called $MNP$. Prove that the lines $d_a, d_b$ and $d_c$ are concurrent and their point of concurrency lies on the Euler line of the triangle $MNP$.

Lê Phúc Lữ

Geometry Mathley 2011-12 5.4
Let $ABC$ be a triangle inscribed in a circle $(O)$. Let $P$ be an arbitrary point in the plane of triangle $ABC$. Points $A',B',C'$ are the reflections of $P$ about the lines $BC,CA,AB$ respectively. $X$ is the intersection, distinct from $A$, of the circle with diameter $AP$ and the circumcircle of triangle $AB'C'$. Points $Y,Z$ are defined in the same way. Prove that five circles $(O), (AB'C')$, $(BC'A'), (CA'B'), (XY Z)$ have a point in common.

Nguyễn Văn Linh

Geometry Mathley 2011-12 6.1
Show that the circumradius $R$ of a triangle $ABC$ equals the arithmetic mean of the oriented distances from its incenter $I$ and three excenters $I_a,I_b, I_c$ to any tangent $\tau$ to its circumcircle. In other words, if $\delta(P)$ denotes the distance from a point $P$ to $\tau$, then with appropriate choices of signs, we have $$\delta(I) \pm \delta_(I_a) \pm  \delta_(I_b) \pm  \delta_(I_c) = 4R$$

Luis González

Geometry Mathley 2011-12 6.2
Let $ABC$ be an acute triangle, and its altitudes $AX,BY,CZ$ concurrent at $H$. Construct circles $(K_a), (K_b), (K_c)$ circumscribing the triangles $AY Z, BZX, CXY$ . Construct a circle $(K)$ that is internally tangent to all the three circles $(Ka), (K_b), (K_c)$. Prove that $(K)$ is tangent to the circumcircle $(O)$ of the triangle $ABC$.

Đỗ Thanh Sơn

Geometry Mathley 2011-12 6.3
Let $AB$ be an arbitrary chord of the circle $(O)$. Two circles $(X)$ and $(Y )$ are on the same side of the chord $AB$ such that they are both internally tangent to $(O)$ and they are tangent to $AB$ at $C,D$ respectively, $C$ is between $A$ and $D$. Let $H$ be the intersection of $XY$ and $AB, M$ the midpoint of arc $AB$ not containing $X$ and $Y$ . Let $HM$ meet $(O)$ again at $I$. Let $IX, IY$ intersect $AB$ again at $K, J$. Prove that the circumcircle of triangle $IKJ$ is tangent to $(O)$.

Nguyễn Văn Linh

Geometry Mathley 2011-12 6.4
Let $P$ be an arbitrary variable point in the plane of a triangle $ABC. A_1$ is the projection of $P$ onto $BC, A_2$ is the midpoint of line segment $PA_1, A_2P$ meets $BC$ at $A_3, A_4$ is the reflection of $P$ about $A_3$. Prove that $PA_4$ has a fixed point.

Trần Quang Hùng

Geometry Mathley 2011-12 7.1
Let $ABCD$ be a cyclic quadrilateral. Suppose that $E$ is the intersection of $AB$ and $CD, F$ is the intersection of $AD$ and $CB, I$ is the intersection of $AC$ and $BD$. The circumcircles $(FAB), (FCD)$ meet $FI$ at $K, L$. Prove that $EK = EL$

Nguyễn Minh Hà

Geometry Mathley 2011-12 7.2
A non-equilateral triangle $ABC$ is inscribed in a circle $\Gamma$ with centre $O$, radius $R$ and its incircle has centre $I$ and touches $BC,CA,AB$ at $D,E, F$, respectively. A circle with centre $I$ and radius $r$ intersects the rays $[ID), [IE), [IF)$ at $A',B',C'$. Show that the orthocentre $K$ of $\vartriangle A'B'C'$ is on the line $OI$ and that $\frac{IK}{IO}=\frac{r}{R}$

Michel Bataille

Geometry Mathley 2011-12 7.3
Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in  AD$, $P,Q \in  BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral.

Nguyễn Văn Linh

Geometry Mathley 2011-12 7.4
Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$.

Trần Quang Hùng

Geometry Mathley 2011-12 8.1
Let $ABC$ be a triangle and $ABDE, BCFZ, CAKL$ be three similar rectangles constructed externally of the triangle. Let $A'$ be the intersection of $EF$ and $ZK, B'$ the intersection of $KZ$ and $DL$, and $C'$ the intersection of $DL$ and $EF$. Prove that $AA'$ passes through the midpoint of the line segment $B'C'$.

Kostas Vittas

Geometry Mathley 2011-12 8.2
Let $ABC$ be a triangle, $d$ a line passing through $A$ and parallel to $BC$. A point $M$ distinct from $A$ is chosen on $d$. $I$ is the incenter of triangle $ABC, K,L$ are the the points of symmetry of $M$ about $IB, IC$. Let $BK$ meet $CL$ at $N$. Prove that $AN$ is tangent to circumcircle of triangle $ABC$.

Đỗ Thanh Sơn

Geometry Mathley 2011-12 8.3
Let $ABC$ be a scalene triangle, $(O)$ and $H$ be the circumcircle and its orthocenter. A line through $A$ is parallel to $OH$ meets $(O)$ at $K$. A line through $K$ is parallel to $AH$, intersecting $(O)$ again at $L$. A line through $L$ parallel to $OA$ meets $OH$ at $E$. Prove that $B,C,O,E$ are on the same circle.

Trần Quang Hùng

Geometry Mathley 2011-12 8.4
Let $ABC$ a triangle inscribed in a circle $(O)$ with orthocenter $H$. Two lines $d_1$ and $d_2$ are mutually perpendicular at $H$. Let $d_1$ meet $BC,CA,AB$ at $X_1, Y_1,Z_1$ respectively. Let $A_1B_1C_1$ be a triangle formed by the line through $X_1$ perpendicular to $BC$, the line through $Y_1$ perpendicular to CA, the line through $Z_1$ perpendicular perpendicular to $AB$. Triangle $A_2B_2C_2$ is defined in the same manner. Prove that the circumcircles of triangles $A_1B_1C_1$ and $A_2B_2C_2$ touch each other at a point on $(O)$.

Nguyễn Văn Linh

Geometry Mathley 2011-12 9.1
Let $ABC$ be a triangle with $(O), (I)$ being the circumcircle, and incircle respectively. Let $(I)$ touch $BC,CA$, and $AB$ at $A_0, B_0, C_0$ let $BC,CA$, and $AB$ intersect $B_0C_0, C_0A_0, A_0Bv$ at $A_1, B_1$, and $C_1$ respectively. Prove that $OI$ passes through the orthocenter of four triangles $A_0B_0C_0, A_0B_1C_1, B_0C_1A_1,C_0A_1B_1$.

Nguyễn Minh Hà

Geometry Mathley 2011-12 9.2
Let $ABDE, BCFZ$ and $CAKL$ be three arbitrary rectangles constructed outside a triangle $ABC$. Let $EF$ meet $ZK$ at $M$, and $N$ be the intersection of the lines through $F,Z$ perpendicular to $FL,ZD$. Prove that $A,M,N$ are collinear.

Kostas Vittas

Geometry Mathley 2011-12 9.3
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$.
(a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$.
(b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear.

Nguyễn Văn Linh

Geometry Mathley 2011-12 9.4
Let $ABC$ be a triangle inscribed in a circle $(O)$, and $M$ be some point on the perpendicular bisector of $BC$. Let $I_1, I_2$ be the incenters of triangles $MAB,MAC$. Prove that the incenters of triangles $A_II_1I_2$ are on a fixed line when $M$ varies on the perpendicular bisector.

Trần Quang Hùng

Geometry Mathley 2011-12 10.1
Let $ABC$ be a triangle with two angles $B,C$ not having the same measure, $I$ be its incircle, $(O)$ its circumcircle. Circle $(O_b)$ touches $BA,BC$ and is internally tangent to $(O)$ at $B_1$. Circle $(O_c)$ touches $CA,CB$ and is internally tangent to $(O)$ at $C_1$. Let $S$ be the intersection of $BC$ and $B_1C_1$. Prove that $\angle AIS = 90^o$.

Nguyễn Minh Hà

Geometry Mathley 2011-12 10.2
Let $ABC$ be an acute triangle, not isoceles triangle and $(O), (I)$ be its circumcircle and incircle respectively. Let $A_1$ be the the intersection of the radical axis of $(O), (I)$ and the line $BC$. Let $A_2$ be the point of tangency (not on $BC$) of the tangent from $A_1$ to $(I)$. Points $B_1,B_2,C_1,C_2$ are defined in the same manner. Prove that
(a) the lines $AA_2,BB_2,CC_2$ are concurrent.
(b) the radical centers circles through triangles $BCA_2, CAB_2$ and $ABC_2$ are all on the line $OI$.

Lê Phúc Lữ

Geometry Mathley 2011-12 10.3
Let $ABC$ be a triangle inscribed in a circle $(O)$. d is the tangent at $A$ of $(O), P$ is an arbitrary point in the plane. $D,E, F$ are the projections of $P$ on $BC,CA,AB$. Let $DE,DF$ intersect the line $d$ at $M,N$ respectively. The circumcircle of triangle $DEF$ meets $CA,AB$ at $K,L$ distinct from $E, F$. Prove that $KN$ meets $LM$ at a point on the circumcircle of triangle $DEF$.

Trần Quang Hùng

Geometry Mathley 2011-12 10.4
Let $A_1A_2A_3...A_n$ be a bicentric polygon with $n$ sides. Denote by $I_i$ the incenter of triangle $A_{i-1}A_iA_{i+1}, A_{i(i+1)}$ the intersection of $A_iA_{i+2}$ and $A_{i-1}A_{i+1},I_{i(i+1)}$ is the incenter of triangle $A_iA_{i(i+1)}A_{i+1}$ ($i = 1, n$). Prove that there exist $2n$ points $I_1, I_2, ..., I_n, I_{12}, I_{23}, ...., I_{n1}$ on the same circle.

Nguyễn Văn Linh

Geometry Mathley 2011-12 11.1
Let $ABCDEF$ be a hexagon with sides $AB,CD,EF$ being equal to $m$ units, sides $BC,DE, FA$ being equal to $n$ units. The diagonals $AD,BE,CF$ have lengths $x, y$, and $z$ units. Prove the inequality $$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}  \ge \frac{3}{(m+ n)^2}$$

Nguyễn Văn Quý

Geometry Mathley 2011-12 11.2
Let $ABC$ be a triangle inscribed in the circle $(O)$. Tangents at $B,C$ of the circles $(O)$ meet at $T$ . Let $M,N$ be the points on the rays $BT,CT$ respectively such that $BM = BC = CN$. The line through $M$ and $N$ intersects $CA,AB$ at $E, F$ respectively; $BE$ meets $CT$ at $P, CF$ intersects $BT$ at $Q$. Prove that $AP = AQ$.

Trần Quang Hùng

Geometry Mathley 2011-12 11.3
Let $ABC$ be a triangle such that $AB = AC$ and let $M$ be a point interior to the triangle. If $BM$ meets $AC$ at $D$. show that $\frac{DM}{DA}=\frac{AM}{AB}$ if and only if $\angle AMB = 2\angle ABC$.

Michel Bataille

Geometry Mathley 2011-12 11.4
Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency.

Nguyễn Văn Linh

Geometry Mathley 2011-12 12.1
Let $ABC$ be an acute triangle with orthocenter $H$, and $P$ a point interior to the triangle. Points $D,E,F$ are the reflections of $P$ about $BC,CA,AB$. If $Q$ is the intersection of $HD$ and $EF$, prove that the ratio $HQ/HD$ is independent of the choice of $P$.

Luis González

Geometry Mathley 2011-12 12.2
Let $K$ be the midpoint of a fixed line segment $AB$, two circles $(O)$ and $(O')$ with variable radius each such that the straight line $OO'$ is throughK and $K$ is inside $(O)$, the two circles meet at $A$ and $C$, center $O'$ is on the circumference of $(O)$ and $O$ is interior to $(O')$. Assume that $M$ is the midpoint of $AC, H$ the projection of $C$ onto the perpendicular bisector of segment $AB$. Let $I$ be a variable point on the arc $AC$ of circle $(O')$ that is inside $(O), I$ is not on the line $OO'$ . Let $J$ be the reflection of $I$ about $O$. The tangent of $(O')$ at $I$ meets $AC$ at $N$. Circle $(O'JN)$ meets $IJ$ at $P$, distinct from $J$, circle $(OMP)$ intersects $MI$ at $Q$ distinct from $M$. Prove that
(a) the intersection of $PQ$ and $O'I$ is on the circumference of $(O)$.
(b) there exist a location of $I$ such that the line segment $AI$ meets $(O)$ at $R$ and the straight line $BI$ meets $(O')$ at $S$, then the lines $AS$ and $KR$ meets at a point on the circumference of $(O)$.
(c) the intersection $G$ of lines $KC$ and $HB$ moves on a fixed line.

Lê Phúc Lữ

Geometry Mathley 2011-12 12.3
Points $E,F$ are chosen on the sides $CA,AB$ of triangle $ABC$. Let $(K)$ be the circumcircle of triangle $AEF$. The tangents at $E, F$ of $(K)$ intersect at $T$ . Prove that
(a) $T$ is on $BC$ if and only if $BE$ meets $CF$ at a point on the circle $(K)$,
(b) $EF, PQ,BC$ are concurrent given that $BE$ meets $FT$ at $M, CF$ meets $ET$ at $N, AM$ and $AN$ intersects $(K)$ at $P,Q$ distinct from $A$.

Trần Quang Hùng

Geometry Mathley 2011-12 12.4
Quadrilateral$ABCD$ has two diagonals $AC,BD$ that are mutually perpendicular. Let $M$ be the Miquel point of the complete quadrilateral formed by lines $AB,BC,CD,DA$. Suppose that $L$ is the intersection of two circles $(MAC)$ and $(MBD)$. Prove that the circumcenters of triangles $LAB,LBC,LCD,LDA$ are on the same circle called $\omega$ and that three circles $(MAC), (MBD), \omega$ are pairwise orthogonal.

Nguyễn Văn Linh

Geometry Mathley 2011-12 13.1
Let $ABC$ be a triangle with no right angle, $E$ on the line $BC$ such that $\angle AEB = \angle BAC$ and $\Delta_A$ the perpendicular to $BC$ at $E$. Let the circle $\gamma$ with diameter $BC$ intersect $BA$ again at $D$. For each point $M$ on $\gamma$ ($M$ is distinct from $B$), the line $BM$ meets $\Delta_A$ at $M'$ and the line $AM$ meets $\gamma$ again at $M''$.
(a) Show that $p(A) = AM' \times DM''$ is independent of the chosen $M$.
(b) Keeping $B,C$ fixed, and let $A$ vary. Show that $\frac{p(A)}{d(A,\Delta_A)}$ is independent of $A$.

Michel Bataille

Geometry Mathley 2011-12 13.2
In a triangle $ABC$, the nine-point circle $(N)$ is tangent to the incircle $(I)$ and three excircles $(I_a), (I_b), (I_c)$ at the Feuerbach points $F, F_a, F_b, F_c$. Tangents of $(N)$ at $F, F_a, F_b, F_c$ bound a quadrangle $PQRS$. Show that the Euler line of $ABC$ is a Newton line of $PQRS$.

Luis González

Geometry Mathley 2011-12 13.3
Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle.

Đỗ Thanh Sơn

Geometry Mathley 2011-12 13.4
Let $P$ be an arbitrary point in the plane of triangle $ABC$. Lines $PA, PB, PC$ meets the perpendicular bisectors of $BC,CA,AB$ at $O_a,O_b,O_c$ respectively. Let $(O_a)$ be the circle with center $O_a$ passing through two points $B,C$, two circles $(O_b), (O_c)$ are defined in the same manner. Two circles $(O_b), (O_c)$ meets at $A_1$, distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Let $Q$ be an arbitrary point in the plane of $ABC$ and $QB,QC$ meets $(O_c)$ and $(O_b)$ at $A_2,A_3$ distinct from $B,C$. Similarly, we have points $B_2,B_3,C_2,C_3$. Let $(K_a), (K_b), (K_c)$ be the circumcircles of triangles $A_1A_2A_3, B_1B_2B_3, C_1C_2C_3$. Prove that
(a) three circles $(K_a), (K_b), (K_c)$ have a common point.
(b) two triangles $K_aK_bK_c, ABC$ are similar.

Trần Quang Hùng

Geometry Mathley 2011-12 14.1
A circle $(K)$ is through the vertices $B, C$ of the triangle $ABC$ and intersects its sides $CA, AB$ respectively at $E, F$ distinct from $C, B$. Line segment $BE$ meets $CF$ at $G$. Let $M, N$ be the symmetric points of $A$ about $F, E$ respectively. Let $P, Q$ be the reflections of $C, B$ about $AG$. Prove that the circumcircles of triangles $BPM , CQN$ have radii of the same length.

Trần Quang Hùng

Geometry Mathley 2011-12 14.2
The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$.

Đỗ Thanh Sơn

Geometry Mathley 2011-12 14.3
Let $ABC$ be a triangle inscribed in circle $(I)$ that is tangent to the sides $BC,CA,AB$ at points $D,E, F$ respectively. Assume that $L$ is the intersection of $BE$ and $CF,G$ is the centroid of triangle $DEF,K$ is the symmetric point of $L$ about $G$. If $DK$ meets $EF$ at $P, Q$ is on $EF$ such that $QF = PE$, prove that $\angle DGE + \angle FGQ = 180^o$.

Nguyễn Minh Hà

Geometry Mathley 2011-12 14.4
Two triangles $ABC$ and $PQR$ have the same circumcircles. Let $E_a, E_b, E_c$ be the centers of the Euler circles of triangles $PBC, QCA, RAB$. Assume that $d_a$ is a line through $Ea$ parallel to $AP$, $d_b, d_c$ are defined in the same manner. Prove that three lines $d_a, d_b, d_c$ are concurrent.

Nguyễn Tiến Lâm, Trần Quang Hùng

Geometry Mathley 2011-12 15.1
Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$.

Nguyễn Minh Hà

Geometry Mathley 2011-12 15.2
Let $O$ be the centre of the circumcircle of triangle $ABC$. Point $D$ is on the side $BC$. Let $(K)$ be the circumcircle of $ABD$. $(K)$ meets $AO$ at $E$ that is distinct from $A$.
(a) Prove that $B,K,O,E$ are on the same circle that is called $(L)$.
(b) $(L)$ intersects $AB$ at $F$ distinct $B$. Point $G$ is on $(L)$ such that $EG \parallel OF$. $GK$ meets $AD$ at $S, SO$ meets $BC$ at $T$ . Prove that $O,E, T,C$ are on the same circle.

Trần Quang Hùng

Geometry Mathley 2011-12 15.3
Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$

Đỗ Thanh Sơn

Geometry Mathley 2011-12 15.4
Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$.

Kostas Vittas

Geometry Mathley 2011-12 16.1
Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $M$, $N$, $P$, $Q$ be the reflections of $E$ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $\triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent.

Trần Quang Hùng

Geometry Mathley 2011-12 16.2
Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel  AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$.

Đỗ Thanh Sơn

Geometry Mathley 2011-12 16.3
The incircle $(I)$ of a triangle $ABC$ touches $BC,CA,AB$ at $D,E, F$. Let $ID, IE, IF$ intersect $EF, FD,DE$ at $X,Y,Z$, respectively. The lines $\ell_a, \ell_b, \ell_c$ through $A,B,C$ respectively and are perpendicular to $YZ,ZX,XY$ .
Prove that $\ell_a, \ell_b, \ell_c$ are concurrent at a point that is on the line segment joining $I$ and the centroid of triangle $ABC$ .

Nguyễn Minh Hà

Geometry Mathley 2011-12 16.4
A triangle $ABC$ is inscribed in the circle $(O)$, and has incircle $(I)$. The circles with diameter $IA$ meets $(O)$ at $A_1$ distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Line $B_1C_1$ meets $BC$ at $A_2$, and points $B_2,C_2$ are defined in the same manner. Prove that $O$ is the orthocenter of triangle $A_2B_2C_2$.

Trần Minh Ngọc

2014 - 2015

[3 problem sets in 2014 and 1 in 2015, missing numbers are not geometry ones]

Mathley 2014.1 p1

Let $AD, BE, CF$ be segments whose midpoints are on the same line $\ell$. The points $X, Y, Z$ lie on the lines $EF, FD, DE$ respectively such that $AX \parallel BY \parallel  CZ \parallel  \ell$. Prove that $X, Y, Z$ are collinear.

Tran Quang Hung

Mathley 2014.1 p6

Let the inscribed circle $(I)$ of the triangle $ABC$, touches $CA, AB$ at $E, F$. $P$ moves along $EF$, $PB$ cuts $CA$ at $M, MI$ cuts the line, through $C$ perpendicular to $AC$, at $N$. Prove that the line through $N$ is perpendicular to $PC$ crosses a fixed point as $P$ moves.

Tran Quang Hung

Mathley 2014.1 p7

The circles $\gamma$ and $\delta$ are internally tangent to the circle $\omega$ at $A$ and $B$. From $A$, draw two tangent lines $\ell_1, \ell_2$ to $\delta$, . From $B$ draw two tangent lines $t_1, t_2$ to $\gamma$ . Let $\ell_1$ intersect $t_1$ at $X$ and $\ell_2$ intersect $t_2$ at $Y$ . Prove that the quadrilateral $AX BY$ cyclic.

Nguyen Van Linh

Mathley 2014.2 p2

Let $ABC$ be a triangle with a circumcircle $(K)$. A circle touching the sides $AB,AC$ is internally tangent to $(K)$ at $K_a$; two other points $K_b,K_c$ are defined in the same manner. Prove that the area of triangle $K_aK_bK_c$ does not exceed that of triangle $ABC$.

Nguyen Minh Ha

Mathley 2014.2 p3

In a triangle $ABC$, $D$ is the reflection of $A$ about the sideline $BC$. A circle $(K)$ with diameter $AD$ meets $DB,DC$ at $M,N$ which are distinct from $D$. Let $E,F$ be the midpoint of $CA,AB$. The circumcircles of $KEM,KFN$ meet each other again at $L$, distinct from $K$. Let $KL$ meets $EF$ at $X$; points $Y,Z$ are defined in the same manner. Prove that three lines $AX,BY,CZ$ are concurrent.

Tran Quang Hung

Mathley 2014.2 p4

Let $(O)$ be the circumcircle of triangle $ABC$, and $P$ a point on the arc $BC$ not containing $A$. $(Q)$ is the $A$-mixtilinear circle of triangle $ABC$, and $(K), (L)$ are the $P$-mixtilinear circles of triangle $PAB, PAC$ respectively. Prove that there is a line tangent to all the three circles $(Q), (K)$ and $(L)$.

Nguyen Van Linh

Mathley 2014.2 p5

A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Another circle $(I)$ is tangent to the diagonals $AC, BD$ at $M, N$ respectively. Suppose that $MN$ meets $AB,CD$ at $P, Q$ respectively. The circumcircle of triangle $IMN$ meets the circumcircles of $IAB, ICD$ at $K, L$ respectively, which are distinct from $I$. Prove that the lines $PK, QL$, and $OI$ are concurrent.

Tran Minh Ngoc

Mathley 2014.3 p3

Let the incircle $\gamma$ of triangle $ABC$ be tangent to $BA, BC$ at $D, E$, respectively. A tangent $t$ to $\gamma$ , distinct from the sidelines, intersects the line $AB$ at $M$. If lines $CM, DE$ meet at$K$, prove that lines $AK,BC$ and $t$ are parallel or concurrent.

Michel Bataille , France

Mathley 2014.3 p4

Let $ABC$ be an acute triangle with $E, F$ being the reflections of $B,C$ about the line $AC, AB$ respectively. Point $D$ is the intersection of $BF$ and $CE$. If $K$ is the circumcircle of triangle $DEF$, prove that $AK$ is perpendicular to $BC$.

Nguyen Minh Ha

Mathley 2014.3 p5

Triangle $ABC$ has incircle $(I)$ and $P,Q$ are two points in the plane of the triangle. Let $QA,QB,QC$ meet $BA,CA,AB$ respectively at $D,E,F$. The tangent at $D$, other than $BC$, of the circle $(I)$ meets $PA$ at $X$. The points $Y$ and $Z$ are defined in the same manner. The tangent at $X$, other than $XD$, of the circle $(I)$ meets $(I)$ at $U$. The points $V,W$ are defined in the same way. Prove that three lines $(AU,BV,CW)$ are concurrent.

Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.

Mathley 2014.3 p6

A quadrilateral is called bicentric if it has both an incircle and a circumcircle. $ABCD$ is a bicentric quadrilateral with $(O)$ being its circumcircle. Let $E, F$ be the intersections of $AB$ and $CD, AD$ and $BC$ respectively. Prove that there is a circle with center $O$ tangent to all of the circumcircles of the four triangles $EAD, EBC, FAB, FCD$.

Nguyen Van Linh

Mathley 2014.3 p8

Two circles $(U)$ and $(V)$ intersect at $A,B$. A line d meets $(U), (V)$ at $P, Q$ and $R,S$ respectively. Let $t_P, t_Q, t_R,t_S$ be the tangents at $P,Q,R, S$ of the two circles. Another circle $(W)$ passes through through $A, B$. Prove that if the circumcircle of triangle that is formed by the intersections of $t_P,t_R, AB$ is tangent to $(W)$ then the circumcircle of triangle formed by $t_Q, t_S, AB$ is also tangent to $(W)$.

Tran Minh Ngoc

Mathley 2015 p1

Let $ABC$ be an acute triangle inscribed in a circle $(O)$ that is fixed, and two of the vertices $B$, $C$ are fixed while vertex $A$ varies on the circumference of the circle. Let $I$ be the center of the incircle, and $AD$ the angle bisector. Let $K$, $L$ be the circumcenters of $CAD$, $ABD$. A line through $O$ parallel to $DL$, $DK$ intersects the line that is through $I$ perpendicular to $IB$, $IC$ at $M$, $N$ respectively. Prove that $MN$ is tangent to a fixed circle when $A$ varies on the circle $(O)$.

Tran Quang Hung

Mathley 2015 p2

A quadrilateral $ABCD$ is inscribed in a circle and its two diagonals $AC,BD$ meet at $G$. Let $M$ be the center of $CD, E,F$ be the points on $BC, AD$ respectively such that $ME \parallel AC$ and $MF  \parallel  BD$. Point $H$ is the projection of $G$ onto $CD$. The circumcircle of $MEF$ meets $CD$ at $N$ distinct from $M$. Prove that $MN = MH$

Tran Quang Hung, Nguyen Le Phuoc, Thanh Xuan, Hanoi

Mathley 2015 p3

A point $P$ is interior to the triangle $ABC$ such that $AP \perp BC$. Let $E, F$ be the projections of $CA, AB$. Suppose that the tangents at $E, F$ of the circumcircle of triangle $AEF$ meets at a point on $BC$. Prove that $P$ is the orthocenter of triangle $ABC$.

Do Thanh Son

Mathley 2015 p4

Points $E, F$ are in the plane of triangle $ABC$ so that triangles $ABE$ and $ACF$ are the opposite directed, and the two triangles are isosceles in that $BE = AE, AF = CF$. Let $H, K$ be the orthocenter of triangle $ABE, ACF$ respectively. Points $M, N$ are the intersections of $BE$ and $CF, CK$ and $CH$. Prove that $MN$ passes through the center of the circumcircle of triangle $ABC$.

Nguyen Minh Ha

source:  www.hexagon.edu.vn/mathley.html