[Solutions] Hanoi Open Mathematics Competition 2018

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.

From 2018, with the support of the Hanoi People’s Committee, the competition was opened to international teams to participate by invitation only. The 15th Hanoi Open Mathematics Competition 2018 (HOMC), an annual event held by the Hanoi Mathematical Society, will see the participation of international contestants for the first time, announced the Hanoi Department of Education and Training on March 23. 87 international contestants from 10 countries, including China, Thailand, Myanmar, Indonesia, Poland, Hungary, and others will attend the competition.

Junior Individual

1. Let $x$ and $y$ be real numbers satisfying the conditions $x + y = 4$ and $xy = 3$. Compute the value of $(x - y)^2$. 
2. Let $f(x)$ be a polynomial such that $2f(x) + f(2 - x) = 5 + x$ for any real number $x$. Find the value of $f(0) + f(2)$. 
3. There are $3$ unit squares in a row as shown in the figure below. Each side of this figure is painted by one of the three colors: Blue, Green or Red. It is known that for any square, all the three colors are used and no two adjacent sides have the same color. Find the number of possible colorings. 
4. Find the number of distinct real roots of the following equation $$x^2 +\frac{9x^2}{(x + 3)^2} = 40.$$
5. 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. Calculate the length of $BC$ if $HO = 3 MO$. 
6. Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books? 
7. Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^\circ$, $\angle B = 105^\circ$, $\angle C = 90^\circ$ 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 $a + b$? 
8. Let $k$ be a positive integer such that $$1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}.$$ Find the remainder when $k$ is divided by $7$. 
9. 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 $m$. 
10. Let $T=\frac{1}{4}x^{2}-\frac{1}{5}y^{2}+\frac{1}{6}z^{2}$ where $x,y,z$ are real numbers such that $1 \leq x,y,z \leq 4$ and $x-y+z=4$. Find the smallest value of $10T$. 
11. Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $. 
12. Let $ABCD$ be a rectangle with $45^\circ < \angle ADB < 60^\circ$. 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 $\angle MKN = 90^\circ$. 
13. A competition room of HOMC has $m \times n$ students where $m, n$ are integers larger than $2$. Their seats are arranged in $m$ rows and $n$ columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally $27$ handshakes. Find the number of students in the room. 
14. Let $P(x)$ be a polynomial with degree $2017$ such that $$P(k) =\frac{k}{k + 1},\,\forall k = 0, 1, 2, ..., 2017.$$ Calculate $P(2018)$. 
15. Find all pairs of prime numbers $(p,q)$ such that for each pair $(p,q)$, there is a positive integer $m$ satisfying $$\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}.$$

Junior Team

1. Let $a, b$, and $c$ be distinct positive integers such that $a + 2b + 3c < 12$. Which of the following inequalities must be true? 
• $a + b + c < 7$ 
• $a- b + c < 4$ 
• $b + c- a < 3$ 
• $a + b- c <5 $ 
• $5a + 3b + c < 27$ 
2. Let $ABCD$ be a rectangle with $\angle ABD = 15^\circ$, $BD = 6cm$. Compute the area of the rectangle. 
3. Consider all triples $(x,y,p)$ of positive integers, where $p$ is a prime number, such that $$4x^2 + 8y^2 + (2x-3y)p-12xy = 0.$$ Which below number is a perfect square number for every such triple $(x,y, p)$? 
4. How many triangles are there for which the perimeters are equal to $30cm$ and the lengths of sides are integers in centimeters? 
5. Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$ 
6. In the below figure, there is a regular hexagon and three squares whose sides are equal to $4cm$. 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 $a + b$. 
7. For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have? 
8. Let $a,b, c$ be real numbers with $a+b+c = 2018$. Suppose $x, y$, and $z$ are the distinct positive real numbers which are satisfied $a = x^2 - yz - 2018, b = y^2 - zx - 2018$ , and $c = z^2 - xy - 2018$. Compute the value of the following expression $$P = \frac{\sqrt{a^3 + b^3 + c^3 - 3abc}}{x^3 + y^3 + z^3 - 3xyz}$$
9. Each of the thirty squares in the diagram below contains a number $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ of which each number is used exactly three times. The sum of three numbers in three squares on each of the thirteen line segments is equal to $S$. 
10. The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The top level of the tower model has no doorways. The front of the tower model is signified by a disk symbol on the top level. We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements.
    
    • The top level is painted with only one color. 
    • In the second level, the $3$ doorways at the front are painted with the same color which is different from the one used for the center doorway at the back. Besides, any two adjacent doorways, including the pairs at the same corners, are painted with different colors. 
    • For the first level, we apply the same rules as for the second level. 
    a) In how many ways the first level can be painted?.
    b) In how many ways the whole tower model can be painted?

Senior Individual

1. How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle). 
2. What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$? 
3. How many integers $n$ are there those satisfy the following inequality$$n^4 - n^3 - 3n^2 - 3^n - 17 < 0$$
4. Let $$\begin{align}a &= (\sqrt2 +\sqrt3 +\sqrt6)(\sqrt2 +\sqrt3 -\sqrt6)(\sqrt3 +\sqrt6 -\sqrt2)(\sqrt6 +\sqrt2 -\sqrt3), \\ b &= (\sqrt2 +\sqrt3 +\sqrt5)(\sqrt2 +\sqrt3 -\sqrt5)(\sqrt3 +\sqrt5 -\sqrt2)(\sqrt5 +\sqrt2 -\sqrt3).\end{align}$$ Which set does he difference $a - b$ belongs to?. 
5. The center of a circle and nine randomly selected points on this circle are colored in red. Every pair of those points is connected by a line segment, and every point of intersection of two line segments inside the circle is colored in red. What is the largest possible number of red points? 
6. Write down all real numbers $(x, y)$ satisfying two conditions $x^{2018} + y^2 = 2$ and $x^2 + y^{2018} = 2$. 
7. Let $\{u_n\}_ {n\ge 1}$ be given sequence satisfying the conditions: $u_1 = 0$, $u_2 = 1$, $u_{n+1} = u_{n-1} + 2n - 1$ for $n \ge 2$.
   a) Calculate $u_5$.
   b) Calculate $u_{100} + u_{101}$. 
8. Let $P$ be a point inside the square $ABCD$ such that $\angle PAC = \angle PCD = 17^\circ$. Calculate $\angle APB$? 
9. How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point. 
10. There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$.
    a) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$?
    b) What is the number of students from club $A$? 
11. Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$. 
12. 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 $D$. Prove that $DE \parallel BC$, and $AC$ is tangent to the circumcircle of $\vartriangle DEF$. 
13. For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively.
    a) Find all values of n such that $n = P(n)$.
    b) Determine all values of n such that $n = S(n) + P(n)$. 
14. Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$. Consider $$T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}.$$ Determine the largest possible value of $T$. 
15. There are $n$ distinct straight lines on a plane such that every line intersects exactly $12$ others. Determine all the possible values of $n$. 

Senior Team

1. If $x$ and $y$ are positive real numbers such that $$(x + \sqrt{x^2 + 1})(y +\sqrt{y^2 + 1}) = 2018.$$ What is the minimum possible value of $x + y$?. 
2. In triangle $ABC$, $\angle BAC = 60^\circ$, $AB = 3a$ 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 $BI$. 
3. The lines $\ell_1$ and \ell_2 are parallel. The points $A_1,A_2, ...,A_7$ are on $\ell_1$ and the points $B_1,B_2,...,B_8$ are on $\ell_2$. The points are arranged in such a way that the number of internal intersections among the line segments is maximized (example Figure). What is the greatest number of intersection points? 
4. A pyramid of non-negative integers is constructed as follows
   
   • The first row consists of only $0$, 
   • The second row consists of $1$ and $1$, 
   • The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row.
   Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$.
5. Let $f$ be a polynomial such that, for all real number $x$, $$f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019.$$ Compute $f(2018)$. 
6. Three students $A$, $B$ and $C$ are traveling from a location on the National Highway No.$5$ on direction to Hanoi for participating the HOMC $2018$. At beginning, $A$ takes $B$ on the motocycle, and at the same time $C$ rides the bicycle. After one hour and a half, $B$ switches to a bicycle and immediately continues the trip to Hanoi, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the travel on the motocycle to Hanoi with $A$. Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motocycle is $50$ km per hour and of the both bicycles are $10$ km per hour. Find the distance from the starting point to Hanoi. 
7. Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of $2$. What is the maximal possible number written on the blackboard? 
8. Let $ABCD$ be rhombus, with $\angle ABC = 80^\circ$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree. 
9. 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 $C$ 
10. The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level. We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements
    
    • The top level is painted with only one color. 
    • The $3$ doorways at the front on the second level are painted with the same color. 
    • The $3$ doorways at the front on the first level are painted with the same color. 
    Each of the remaining $14$ doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors.
    a) How many ways are there to paint the first level?
    b) How many ways are there to paint the entire tower model? 

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