Hanoi Open MC 2006-18 (HOMC) (Vietnam) 71p

geometry problems from Hanoi Open Mathematics Competitions (HOMC) , the proof questions,
with aops links in the names

HOMC 2019 individual 

aops post collection 

2006 - 2018

2006 HOMC Junior Q6 Senior Q5
The figure $ABCDEF$ is a regular hexagon. Find all points M belonging to the hexagon such that Area of triangle $MAC$ = Area of triangle $MCD$.

2006 HOMC Junior Q7
On the circle $(O)$ of radius $15$ cm are given $2$ points $A, B$. The altitude $OH$ of the triangle $OAB$ intersect $(O)$ at $C$. What is $AC$ if $AB = 16$ cm?

2006 HOMC Junior Q8 Seniors Q7
In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in  EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.

2006 HOMC Senior Q7
On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?


2007 HOMC Junior Q7
Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side $4$. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.

2007 HOMC Junior Q9
A triangle is said to be the Heron triangle if it has integer sides and integer area. In a Heron triangle, the sides $a, b,c$ satisfy the equation $b = a(a - c)$. Prove that the triangle is isosceles.

2007 HOMC Junior Q13
Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle  MAB$= area of $\vartriangle MAC$

2007 HOMC Senior Q5
Suppose that $A,B,C,D$ are points on a circle, $AB$ is the diameter, $CD$ is perpendicular to $AB$ and meets $AB$ and meets $AB$ at $E , AB$ and $CD$ are integers and $AE - EB=\sqrt{3}$. Find $AE$.

2007 HOMC Senior Q8
Let $ABC$ be an equilateral triangle. For a point $M$ inside $\vartriangle ABC$, let $D,E,F$ be the feet of the perpendiculars from $M$ onto $BC,CA,AB$, respectively. Find the locus of all such points $M$ for which $\angle FDE$ is a right angle.

2007 HOMC Senior Q13
Let $ABC$ be an acute-angle triangle with $BC >CA$. Let $O, H$ and $F$ be the circumcenter, orthocentre and the foot of its altitude $CH,$ respectively. Suppose that the perpendicular to $OF$ at $F$ meet the side $CA$ at $P$. Prove $\angle FHP = \angle BAC$.

2008 HOMC Junior Q7
The figure $ABCDE$ is a convex pentagon. Find the sum $\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle  CEB$?

2008 HOMC Junior Q8
The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

2008 HOMC Junior Q9 Senior Q10 (also  2011 Junior Q10)
Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

2008 HOMC Senior Q8
Consider a convex quadrilateral $ABCD$. Let $O$ be the intersection of $AC$ and $BD$,$M, N$ be the centroid of $\vartriangle AOB$ and $\vartriangle COD$ and $P, Q$ be orthocenter of $\vartriangle BOC$ and $\vartriangle DOA$, respectively. Prove that $MN \perp PQ$

2008 HOMC Senior Q9 (also)
Consider a triangle $ABC$. For every point M $\in BC$ ,we define $N \in  CA$ and $P \in  AB$ such that $APMN$ is a parallelogram. Let $O$ be the intersection of $BN$ and $CP$. Find $M \in  BC$ such that $\angle PMO=\angle OMN$


2009 HOMC Junior Q9
Let be given $\vartriangle ABC$ with area $(\vartriangle ABC) = 60$ cm$^2$. Let $R,S$ lie in $BC$ such that $BR = RS = SC$ and $P,Q$ be midpoints of $AB$ and $AC$, respectively. Suppose that $PS$ intersects $QR$ at $T$. Evaluate area $(\vartriangle PQT)$.

2009 HOMC Junior Q10
Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$. Find the distance between the midpoints of $AD$ and $BC$

2009 HOMC Senior Q9 (also)
Give an acute-angled triangle $ABC$ with area $S$, let points $A',B',C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards facing semicirle drawn on $BX$ as diameter.Points $B',C'$ are located similarly. Evaluate the sum  $T=($area $\vartriangle BCA')^2+($area  $\vartriangle CAB')^2+($area $\vartriangle ABC')^2$.

2009 HOMC Senior Q10 (also)
Prove that $d^2+(a-b)^2<c^2$ ,where $d$ is diameter of the inscribed circle of $\vartriangle ABC$


2010 HOMC Junior Q9
Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?

2010 HOMC Senior Q7 (also) (also 2012 HOMC Senior Q9)
Let $P$ be the common point of $3$ internal bisectors of a given ABC. The line passing through P and perpendicular to $CP$ intersects $AC$ and $BC$ at $M$ and $N$, respectively. If $AP=3$ cm, $BP=4$ cm, compute the value of $\frac{AM}{BN}$.


2011 HOMC Junior Q10 Senior Q11
Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

2011 HOMC Junior Q11
Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$. Calculate the area of the quadrilateral.

2011 HOMC Senior Q10 (also)
Two bisectors $BD$ and $CE$ of the triangle $ABC$ interect at $O$.Suppose that $BD \cdot CE=2BO\cdot  OC$.Denote by $H$ the point in $BC$ such that $OH \perp BC$.Prove that $AB\cdot  AC=2 HB \cdot HC$


2012 HOMC Junior Q8
Given a triangle $ABC$ and $2$ point $K \in AB, \; N \in BC$ such that $BK=2AK, \; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$. Compute $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$

2012 HOMC Junior Q14
Let be given a triangle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5$ cm, $HC=8$ cm, compute the area of $\triangle ABC$.

2012 HOMC Senior Q12
In an isosceles triangle ABC with the base AB given a point M \in BC. Let O be the center of its circumscribed circle and S be the center of the inscribed circle in \vartriangle ABC and
SM // AC: Prove that OM \perp BS.


2013 HOMC Junior Q6
Let $ABC$ be a triangle with area $1$ (cm$^2$). Points $D,E$ and $F$ lie on the sides $AB, BC$ and CA, respectively. Prove that $min\{$area of $\vartriangle ADF,$ area of $\vartriangle BED,$ area of $\vartriangle CEF\}\le \frac14$ (cm$^2$).

2013 HOMC Junior Q7
Let $ABC$ be a triangle with $\angle  A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.

2013 HOMC Junior Q8 Senior Q8
Let $ABCDE$ be a convex pentagon and area of $\vartriangle ABC =$ area of $\vartriangle BCD =$ area of $\vartriangle CDE=$ area of $\vartriangle DEA =$ area of $\vartriangle EAB$. Given that area of $\vartriangle ABCDE = 2$. Evaluate the area of area of $\vartriangle ABC$.

2013 HOMC Senior Q7
Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$. Draw an equilateral triangle $ACD$ where $D \ne B$. Let the point $N$ inside $\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\angle BMC$.


2014 HOMC Junior Q6
Let $a,b,c$ be the length sides of a given triangle and $x,y,z$ be the sides length of bisectrises, respectively. Prove the following inequality $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

2014 HOMC Junior Q8
Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.

2014 HOMC Junior Q11 Senior Q6
Let $S$ be area of the given parallelogram $ABCD$ and the points $E,F$ belong to $BC$ and $AD$, respectively, such that $BC = 3BE, 3AD = 4AF$. Let $O$ be the intersection of $AE$ and $BF$. Each straightline of $AE$ and $BF$ meets that of $CD$ at points $M$ and $N$, respectively. Determine area of triangle $MON$.

2014 HOMC Senior Q7
Let two circles $C_1,C_2$ with different radius be externally tangent at a point $T$.
Let $A$ be on $C_1$ and $B$ be on $C_2$, with $A,B \ne  T$ such that $\angle ATB = 90^o$.
(a) Prove that all such lines $AB$ are concurrent.
(b) Find the locus of the midpoints of all such segments $AB$.

2014 HOMC Senior Q12
Given a rectangle paper of size $15$ cm $\times$  $20$ cm, fold it along a diagonal.
Determine the area of the common part of two halfs of the paper?

2014 HOMC Senior Q14
Let $\omega$ be a circle with centre $O$, and let $\ell$ be a line that does not intersect $\omega$. Let $P$ be an arbitrary point on $\ell$. Let $A,B$ denote the tangent points of the tangent lines from $P$. Prove that $AB$ passes through a point being independent of choosing $P$.

2015 HOMC Junior Q4 Senior Q2
A regular hexagon and an equilateral triangle have equal perimeter.
If the area of the triangle is $4\sqrt3$ square units, what is the area of the hexagon is?

2015 HOMC Junior Q10 Senior Q10
A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is $9$ cm$^2$. Determine the length of sides of the triangle.

2015 HOMC Junior Q11 Senior Q11
Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I , K , H$ be feet of perpendiculars from $B , O , C$ to $AD$, respectively. Prove that $AD \times BI \times CH \le  AC \times BD \times OK$.

2015 HOMC Junior Q12
Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$.

2015 HOMC Senior Q12
Give an isosceles triangle $ABC$ at $A$. Draw ray $Cx$ being perpendicular to $CA, BE$ perpendicular to $Cx$ ($E \in Cx$).Let $M$ be the midpoint of $BE$, and $D$ be the intersection point of $AM$ and $Cx$. Prove that $BD \perp  BC$.


2016 HOMC Junior Q10
Let $h_a, h_b, h_c$ and $r$ be the lengths of altitudes and radius of the inscribed circle of $\vartriangle ABC$, respectively. Prove that $h_a + 4h_b + 9h_c > 36r$.

2016 HOMC Junior Q11
Let be given a triangle $ABC$, and let $I$ be the middle point of $BC$. The straight line $d$ passing $I$ intersects $AB,AC$ at $M,N$ , respectively. The straight line $d'$ ($\ne d$) passing $I$ intersects $AB, AC$ at $Q, P$ , respectively. Suppose $M, P$ are on the same side of $BC$ and $MP , NQ$ intersect $BC$ at $E$ and $F$, respectively. Prove that $IE = I F$.

2016 HOMC Junior Q12
In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.

2016 HOMC Junior Q13
Let $H$ be orthocenter of the triangle $ABC$. Let $d_1, d_2$ be lines perpendicular to each-another at $H$. The line $d_1$ intersects $AB, AC$ at $D, E$ and the line d_2 intersects $B C$ at $F$. Prove that $H$ is the midpoint of segment $DE$if and only if $F$ is the midpoint of segment $BC$.

2016 HOMC Senior Q11
Let $I$ be the incenter of triangle $ABC$ and $\omega$ be its circumcircle. Let the line $AI$ intersect $\omega$ at point $D \ne A$. Let $F$ and $E$ be points on side $BC$ and arc $BDC$ respectively such that $\angle BAF = \angle CAE < \frac12 \angle BAC$ . Let $X$ be the second point of intersection of line $EI$ with $\omega$  and $T$ be the point of intersection of segment $DX$ with line $AF$ . Prove that $TF \cdot AD = ID \cdot AT$ .

2016 HOMC Senior Q12
Let $A$ be a point inside the acute angle $xOy$. An arbitrary circle $\omega$ passes through $O, A$, intersecting $Ox$ and $Oy$ at the second intersection $B$ and $C$, respectively. Let $M$ be the midpoint of $BC$. Prove that $M$ is always on a fixed line (when $\omega$ changes, but always goes through $O$ and $A$).


2017 HOMC Junior Q9
Prove that the equilateral triangle of area $1$ can be covered by five arbitrary equilateral triangles having the total area $2$.

2017 HOMC Junior Q11
Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.

2017 HOMC Junior Q14
Given trapezoid $ABCD$ with bases $AB \parallel  CD$ ($AB < CD$). Let $O$ be the intersection of $AC$ and $BD$. Two straight lines from $D$ and $C$ are perpendicular to $AC$ and $BD$ intersect at $E$ , i.e. $CE \perp  BD$ and $DE \perp  AC$ . By analogy, $AF \perp  BD$ and $BF \perp  AC$ . Are three points $E , O, F$ located on the same line?

2017 HOMC Junior Q15
Show that an arbitrary quadrilateral can be divided into nine isosceles triangles.

2017 HOMC Senior Q11
Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle.
Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

2017 HOMC Senior Q12
Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ?

2017 HOMC Senior Q13
Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB|  - |P B|  \ge d$, and $|AC | - |P C |  \ge d$. Is the following inequality true $|AM | - |P M |  \ge  d$, for any position of $M \in  BC$?

2017 HOMC Senior Q15
Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$ .



2018 HOMC Junior Individual Q5
Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$, respectively. If $HO = 3 MO$, then find the length of $BC$ .

2018 HOMC Junior Individual Q7
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?

2018 HOMC Junior Individual Q9

There are three polygons and the area of each one is $3$. They are drawn inside a square of area $6$. Find the greatest value of $m$ such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than $m$.

2018 HOMC Junior Individual Q12

Let ABCD be a rectangle with $45^o < \angle ADB < 60^o$. The diagonals $AC$ and$BD$ intersect at $O$. A line passing through $O$ and perpendicular to $BD$ meets $AD$ and $CD$ at $M$ and $N$ respectively. Let $K$ be a point on side $BC$ such that $MK \parallel AC$. Show that $\angle MKN = 90^o$.

2018 HOMC Junior Team Q2

Let $ABCD$ be a rectangle with $\angle ABD = 15^o, BD = 6$ cm. Compute the area of the rectangle.

2018 HOMC Junior Team Q4

How many triangles are there for which the perimeters are equal to $30$ cm and the lengths of sides are integers in centimeters?

2018 HOMC Junior Team Q6

In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$.

2018 HOMC Senior Individual Q1
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).

2018 HOMC Senior Individual Q2
What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$?

2018 HOMC Senior Individual Q8
Let $P$ be a point inside the square $ABCD$ such that $\angle PAC = \angle PCD = 17^o$ (see Figure 1). Calculate $\angle APB$?

2018 HOMC Senior Individual Q12
Let $ABC$ be an acute triangle with $AB < AC$, and let $BE$ and $CF$ be the altitudes. Let the median $AM$ intersect $BE$ at point $P$, and let line $CP$ intersect $AB$ at point $D$ (see Figure 2). Prove that $DE \parallel BC$, and $AC$ is tangent to the circumcircle of $\vartriangle DEF$.

2018 HOMC Senior Team Q2 
In triangle $ABC,\angle BAC = 60^o, AB = 3a$ and $AC = 4a, (a > 0)$. Let $M$ be point on the segment $AB$ such that $AM =\frac13 AB, N$ be point on the side $AC$ such that $AN =\frac12AC$. Let $I$ be midpoint of $MN$. Determine the length of $BI$.

A. $\frac{a\sqrt2}{19}$ B. $\frac{2a}{\sqrt{19}}$ C. $\frac{19a\sqrt{19}}{2}$ D. $\frac{19a}{\sqrt2}$ E. $\frac{a\sqrt{19}}{2}$

2018 HOMC Senior Team Q8

Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.

2018 HOMC Senior Team Q9

Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively.
(a) Prove that $AO \perp  EF$.
(b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar.
(c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$