[Solutions] Hanoi Open Mathematics Competition 2006-2014

Kỳ thi Toán học Hà Nội mở rộng (HOMC) được tổ chức từ năm 2004 do Hội Toán học Hà Nội là đơn vị sáng lập và điều hành. Kỳ thi dành cho học sinh ở hai lứa tuổi: Lứa tuổi Senior (thí sinh lứa tuổi 16 - lớp 10 trung học phổ thông) và lứa tuổi Junior (thí sinh lứa tuổi 14 - lớp 8 trung học cơ sở). Đề thi và bài làm của học sinh được trình bày hoàn toàn bằng Tiếng Anh. Học sinh làm bài thi trong khoảng thời gian 180 phút cho 15 câu hỏi (5 câu trắc nghiệm khách quan, 10 câu tự luận). Kỳ thi HOMC được tổ chức thường niên vào tháng 3 hàng năm.

Hanoi Open Mathematics Competition (HOMC) was first organized in 2004 by the Hanoi Mathematical Society for Junior (Grade 8) and Senior (Grade 10) students. As the original regulation of HOMC, all questions, problems, and contestant’s presentation should be presented in English. From 2013 Hanoi Department of Education and Training became the co- organizer and promoted the competition as nationwide with nearly 1,000 contestants participated every year from 50 cities across the country.

1. The figure $ABCDEF$ is a regular hexagon. Find all points M belonging to the hexagon such that Area of triangle $MAC$ = Area of triangle $MCD$.
2. On the circle $(O)$ of radius $15cm$ are given $2$ points $A, B$. The altitude $OH$ of the triangle $OAB$ intersect $(O)$ at $C$. What is $AC$ if $AB = 16cm$?
3. 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 $BC$.
4. On the circle of radius $30cm$ are given $2$ points A,B with $AB = 16cm$ and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?
5. 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 $\sqrt3$.
6. 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.
7. Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$ 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 $AE$.
8. 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.
9. 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 $\angle FHP = \angle BAC$.
10. The figure $ABCDE$ is a convex pentagon. Find the sum $$\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle CEB$$
11. The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?
12. Consider a right-angle triangle $ABC$ with $A=90^\circ$, $AB=c$ 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.
13. 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 $MN \perp PQ$
14. 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 $\angle PMO=\angle OMN$
15. 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 $(\vartriangle PQT)$.
16. 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 $BC$
17. 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 $$T=(\vartriangle BCA')^2+(\vartriangle CAB')^2+(\vartriangle ABC')^2.$$
18. Prove that $d^2+(a-b)^2<c^2$ where $d$ is diameter of the inscribed circle of $\vartriangle ABC$
19. 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 $AB$. Compute $S_{\vartriangle AMN}$ if $S_{\vartriangle BMD} = 9 cm^2$ and $S_{\vartriangle DNC} = 25 cm^2$.
20. 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 $\dfrac{AM}{BN}$.
21. Consider a right -angle triangle $ABC$ with $A=90^\circ$, $AB=c$ 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.
22. Given a quadrilateral $ABCD$ with $AB = BC =3cm$, $CD = 4cm$, $DA = 8cm$ and $\angle DAB + \angle ABC = 180^\circ$. Calculate the area of the quadrilateral.
23. 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 $AB\cdot AC=2 HB \cdot HC$
24. 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 $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$
25. Let be given a triangle $ABC$ with $\angle A=90^\circ$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5cm$, $HC=8cm$, compute the area of $\triangle ABC$.
26. 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 $OM \perp BS$.
27. 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 $$\min\{(\vartriangle ADF), (\vartriangle BED), (\vartriangle CEF)\}\le \frac14 (cm^2).$$
28. Let $ABC$ be a triangle with $\angle A = 90^\circ$, $\angle B = 60^\circ$ and $BC = 1 cm$. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD$, $ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.
29. 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 $\vartriangle ABC$.
30. 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 $\angle BMC$.
31. 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 $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
32. Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB$, $AC=AE$ and $\angle DAB =\angle EAC =90^\circ$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^\circ$. Prove that the triangle $DEF$ is a right- isosceles triangle.
33. 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 $MON$.
34. Let two circles $C_1$, $4C_2$ 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 $\angle ATB = 90^\circ$.
    a) Prove that all such lines $AB$ are concurrent.
    b) Find the locus of the midpoints of all such segments $AB$.
35. Given a rectangle paper of size $15cm$ $\times$ $20cm$, fold it along a diagonal. Determine the area of the common part of two halfs of the paper?
36. 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 $P$.

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