## The Two Special Sets of Numbers

Posted: May 3, 2017 in Mathematics

Sum of Zeroth Powers

$1^0 + 6^0 + 7^0 + 23^0 + 24^0 + 30^0 + 38^0 + 47^0 + 54^0 + 55^0 = 10$

$2^0 + 3^0 + 10^0 + 19^0 + 27^0 + 33^0 + 34^0 + 50^0 + 51^0 + 56^0 = 10$

Sum of First Powers

$1^1 + 6^1 + 7^1 + 23^1 + 24^1 + 30^1 + 38^1 + 47^1 + 54^1 + 55^1 = 285$

$2^1 + 3^1 + 10^1 + 19^1 + 27^1 + 33^1 + 34^1 + 50^1 + 51^1 + 56^1 = 285$

Sum of Second Powers

$1^2 + 6^2 + 7^2 + 23^2 + 24^2 + 30^2 + 38^2 + 47^2 + 54^2 + 55^2 = 11685$

$2^2 + 3^2 + 10^2 + 19^2 + 27^2 + 33^2 + 34^2 + 50^2 + 51^2 + 56^2 = 11685$

Sum of Third Powers

$1^3 + 6^3 + 7^3 + 23^3 + 24^3 + 30^3 + 38^3 + 47^3 + 54^3 + 55^3 = 536085$

$2^3 + 3^3 + 10^3 + 19^3 + 27^3 + 33^3 + 34^3 + 50^3 + 51^3 + 56^3 = 536085$

Sum of Fourth Powers

$1^4 + 6^4 + 7^4 + 23^4 + 24^4 + 30^4 + 38^4 + 47^4 + 54^4 + 55^4 = 26043813$

$2^4 + 3^4 + 10^4 + 19^4 + 27^4 + 33^4 + 34^4 + 50^4 + 51^4 + 56^4 = 26043813$

Sum of Fifth Powers

$1^5 + 6^5 + 7^5 + 23^5 + 24^5 + 30^5 + 38^5 + 47^5 + 54^5 + 55^5 = 1309753125$

$2^5 + 3^5 + 10^5 + 19^5 + 27^5 + 33^5 + 34^5 + 50^5 + 51^5 + 56^5 = 1309753125$

Sum of Sixth Powers

$1^6 + 6^6 + 7^6 + 23^6 + 24^6 + 30^6 + 38^6 + 47^6 + 54^6 + 55^6 = 67334006805$

$2^6 + 3^6 + 10^6 + 19^6 + 27^6 + 33^6 + 34^6 + 50^6 + 51^6 + 56^6 = 67334006805$

Sum of Seventh Powers

$1^7 + 6^7 + 7^7 + 23^7 + 24^7 + 30^7 + 38^7 + 47^7 + 54^7 + 55^7 = 3512261547765$

$2^7 + 3^7 + 10^7 + 19^7 + 27^7 + 33^7 + 34^7 + 50^7 + 51^7 + 56^7 = 3512261547765$

Sum of Eighth Powers

$1^8 + 6^8 + 7^8 + 23^8 + 24^8 + 30^8 + 38^8 + 47^8 + 54^8 + 55^8 = 185039471773893$

$2^8 + 3^8 + 10^8 + 19^8 + 27^8 + 33^8 + 34^8 + 50^8 + 51^8 + 56^8 = 185039471773893$

## 2187 in Star Wars

Posted: April 15, 2017 in Mathematics

Last night the Pearl was playing the Star Wars movie episode 4. When Princess Leia was captured in the Death Star, she was held in the cell number 2187. Cell “2187” sounds very familiar. The black dude who is casting as Finn in the Force Awaken also has the code name “FN-2187”. I don’t know if the number 2187 has any special meaning to the Star Wars story or the screenwriter, but 2187 reminds me of tutoring full time 10 years ago on topic like Geometric Sequence because $3^7=2187$.

There are also some interesting facts about the number 2187. If you write the number in backward, you get 7812. When you add the two numbers, you get 9999. When you split the number 2187 into two 2-digit numbers 21 and 87 and then multiple them, you get 1827. And also notice that $2187 = 3^7 = 3^3 \cdot 3^4 = 27 \cdot 81$.

## Stanford Summer Institute Math Problems

Posted: April 9, 2017 in Mathematics

(1)  Suppose $x, y \in\mathbb{Z}$.

$x^2-9y^2=(x-3y)(x+3y)=2017=1\cdot2017=2017\cdot1=(-1)(-2017)=(-2017)(-1)$

Case $(x-3y)(x+3y) = 1\cdot2017 \Rightarrow (x,y)=(1009, 336)$

Case $(x-3y)(x+3y) = 2017\cdot1 \Rightarrow (x,y)=(1009, -336)$

Case $(x-3y)(x+3y) = (-1)(-2017) \Rightarrow (x,y)=(-1009, -336)$

Case $(x-3y)(x+3y) = (-2017)(-1) \Rightarrow (x,y)=(-1009, 336)$

(2)  Since there are more 2’s than 5’s in 2017! and $2\cdot5=10$, all we need to do is to count the number of 5’s in the prime factorization of 2017!.

$\displaystyle \left\lfloor\frac{2017}{5}\right\rfloor + \left\lfloor\frac{2017}{25}\right\rfloor + \left\lfloor\frac{2017}{125}\right\rfloor + \left\lfloor\frac{2017}{625}\right\rfloor = 403 + 80 + 16 + 3 = 502$

(3)  Suppose that $n \in\mathbb{Z}$.

$n^2+9n+14 = m^2$, for some $m\in\mathbb{Z}$

By completing the square,

$\displaystyle \left(n+\frac{9}{2}\right)^2 - \frac{25}{4} = m^2$

$\displaystyle \left(n+\frac{9}{2}-m\right)\left(n+\frac{9}{2}+m\right) = \frac{25}{4}$

$(2n+9-2m)(2n+9+2m) = 25 = 1\cdot25 = 25\cdot1 = 5\cdot5 \\ = (-1)(-25) = (-25)(-1) = (-5)(-5)$

Case $1\cdot25 \text{ and } 25\cdot1 \Rightarrow 2n+9=\dfrac{1+25}{2}=13 \Rightarrow n=2$

Case $5 \cdot 5 \Rightarrow 2n+9=\dfrac{5+5}{2}=5 \Rightarrow n=-2$

Case $(-1)(-25) \text{ and } (-25)(-1) \Rightarrow 2n+9=-\dfrac{1+25}{2}=-13 \Rightarrow n=-11$

Case $(-5)(-5) \Rightarrow 2n+9=-\dfrac{5+5}{2}=-5 \Rightarrow n=-7$

If we define zero being a perfect square, then $n=2, -2, -11, -7$. Otherwise $n=2, -11$.

(4)  $ax^4+bx^3+cx^2+dx+e=0$

$x=\sqrt3+\sqrt5$

$x^2=8-2\sqrt{15}$

$x^3=18\sqrt3-14\sqrt5$

$x^4=124-32\sqrt{15}$

$a(124-32\sqrt{15}) + b(18\sqrt3-14\sqrt5) + c(8-2\sqrt{15}) + d(\sqrt3-\sqrt5) + e = 0$

$\sqrt{15}: \quad -32a-2c=0 \Rightarrow c=-16a \Rightarrow \dfrac{c}{a} = -16$

$\sqrt5: \quad -14b-d=0$

$\sqrt3: \quad 18b+d=0$

$\therefore b=d=0$

constant term: $\displaystyle 124a+8c+e=0 \Rightarrow 124+\frac{8b}{a}+\frac{c}{a}=0$

$\displaystyle \therefore \frac{e}{a} = -\frac{8c}{a}-124 = 128-124=4$

In particular,

$\displaystyle x^4+\frac{c}{a}x^2+\frac{e}{a}=0 \Rightarrow x^4-16x^2+4=0$

In general, for any integer n,

$a=n, b=0, c=-16n, d=0, e=4n$

(5)  Let $r$ be the radius of the smallest semi-circle.

$2\sqrt2 r + 4\sqrt2 r = 6\sqrt2 \quad \therefore \; r=1$

Note that the perimeter of semi-circle is $\pi r$,

$\text{perimeter of the curve} = \pi + 2\pi + 4\pi = 7\pi$

(6)  Let the side length of B, C, D, E, F, G, H, and I be b, c, d, e, f, g, h, and i respectively.

$g = f+1$

$b = f+2$

$c = f+3$

$d = f+3+e$

$i = f+3+2e$

$h = f+g = 2f+1$

$b+c+d=i+h \Rightarrow 3f+8+e = 3f+4+2e \Rightarrow e=4$

$d+i=b+g+h \Rightarrow 2f+18=4f+4 \Rightarrow f=7$

$\therefore \text{Area} = (d+i)(i+h) = 32\cdot33 = 1056$

(7)  By Fermat’s little theorem, $3^6 \equiv 1 \mod 7$

$\displaystyle 8\sum_{k=51}^{100}10^k + 10^{50}a + 9\sum_{k=0}^{49}10^k \pmod{7} \equiv \sum_{k=0}^{49}3^{k+51} + 3^{2}a + 2\sum_{k=0}^{49}3^k$

$\displaystyle \equiv -\sum_{k=0}^{49}3^k + 2a + 2\sum_{k=0}^{49}3^k \equiv 2a+\sum_{k=0}^{49}3^k \equiv 2a+\frac{3^{50}-1}{3-1} \equiv 0$

$\displaystyle 4a + 3^{50}-1 \equiv 0 \pmod{7}$

$\displaystyle 4a + 1 \equiv 0 \pmod7$

$\displaystyle a \equiv 5 \pmod7$

$\displaystyle \therefore a = 5$

(8)  By Pythagoras theorem,

$x^2+13 = 4^2 = 16 \therefore x=\sqrt3$

(9)  By Euclidean Algorithm,

$11n+7 = (7n+1)(1)+4n+6$

$7n+1 = (4n+6)(1) + 3n-5$

$4n+6 = (3n-5)(1) + n+11$

$3n-5 = (n+11)(3) - 38$

hence $(11n+7, 7n+1) = (n+11, -38)$

$(n+11, -38) > 1 \quad\text{if}\quad 2|n+11 \text{ or } 19|n+11$

$n+11\equiv 0 \mod 2 \Rightarrow n\equiv1 \mod 2 \quad\therefore n=2k+1$

$n+11\equiv 0 \mod 19 \Rightarrow n \equiv 8 \mod 19 \quad\therefore n=19k+8$

(10)  Divide the numbers from 1 to 100 into congruence classes mod 11.

[1] : 1, 12, … , 100; There are 10 members so we can choose at most 5 members, since we don’t want any pair of numbers with difference of 11. (Any adjacent members in the class have difference of 11)

[2] : 2, 13, … , 90; There are 9 members so we can choose at most 5 members.

[3] : 3, 14, … , 91; There are 9 members so we can choose at most 5 members.

… … , similarly for classes such as [4] to [11], each of them have 9 members. Therefore we can at most pick $5\times 11 = 55$ numbers without any pair in the set having difference of 11.

## DSE Math 2016 Paper 1 ANS

Posted: March 27, 2017 in Mathematics

Exam week’s coming, so here it is.

###### Section A (35 marks)

(1)  Simplify $\displaystyle\frac{(x^8 y^7)^2}{x^5 y^{-6}}$ and express your answer with positive indices.  (3 marks)

Solution

$\displaystyle\frac{(x^8 y^7)^2}{x^5 y^{-6}} = \frac{x^{16}y^{14}}{x^5 y^{-6}} = x^{11}y^{20}$

(2)  Make $x$ the subject of the formula $Ax=(4x+B)C$.  (3 marks)

Solution

$Ax=4Cx+BC$

$Ax-4Cx=BC$

$\therefore x = \dfrac{BC}{A-4C}$

(3)  Simplify $\dfrac{2}{4x-5}+\dfrac{3}{1-6x}$.  (3 marks)

Solution

$\displaystyle \frac{2(1-6x)+3(4x-5)}{(4x-5)(1-6x)} = \frac{2-12x+12x-15}{(4x-5)(1-6x)} = \frac{13}{(4x-5)(6x-1)}$

(4)  Factorize

(a)  $5m-10n$.

(b)  $m^2+mn-6n^2$.

(c)  $m^2+mn-6n^2-5m+10n$.  (4 marks)

Solution

(a)  $5(m-2n)$

(b)  $(m+3n)(m-2n)$

(c)  $(m+3n)(m-2n)-5(m-2n) = (m-2n)(m+3n-5)$

(5)  In a recreation club, there are 180 members and the number of male members is 40% more than the number of female members. Find the difference of the number of male members and the number of female members.  (4 marks)

Solution

Let $f$ be the number of female members.

$1.4f + f = 180 \quad \Rightarrow \quad f = \dfrac{180}{2.4} = 75$

$\therefore \text{difference} = 1.4f-f=0.4f=0.4(75)=30$

(6)  Consider the compound inequality

$x+6 < 6(x+11) \text{ or } x\leq -5 \qquad (*)$

(a)  Solve (*).

(b)  Write down the greatest negative integer satisfying (*).  (4 marks)

Solution

(a)  $x+6<6x+66 \text{ or } x\leq -5 \\ \Leftrightarrow 5x>-60 \text{ or } x\leq -5 \\ \Leftrightarrow x>-12 \text{ or } x\leq -5 \\ \Leftrightarrow x\in\mathbb{R}$

(b)  $-1$

(7)  In a polar coordinate system, $O$ is the pole. The polar coordinate of the points $A$ and $B$ are $(12, 75^\circ)$ and $(12, 135^\circ)$ respectively.

(a)  Find $\angle AOB$.

(b)  Find the perimeter of $\triangle AOB$.

(c)  Write down the number of folds of rotational symmetry of $\triangle AOB$.  (4 marks)

Solution

(a)  $\angle AOB = 135^\circ - 75^\circ = 60^\circ$

(b)  Obviously the triangle is isosceles and hence $\alpha = \beta = 60^\circ$ and hence $\triangle AOB$ is equilateral. Therefore the perimeter is 36.

(c)  3

(8)  It is given that $f(x)$ is the sum of two parts, one part varies as $x$ and the other part varies as $x^2$. Suppose that $f(3)=48$ and $f(9)=198$.

(a)  Find $f(x)$.

(b)  Solve the equation $f(x)=90$.  (5 marks)

Solution

(a)  Let $f(x) = ax^2+bx$ for some non-zero real numbers $a$ and $b$.

$f(3) = 9a+3b = 48 \Leftrightarrow 3a+b=16 \qquad (1)$

$f(9) = 81a+9b = 198 \Leftrightarrow 9a+b=22 \qquad (2)$

(2) – (1) gives $6a=6 \Rightarrow a=1 \Rightarrow b=13$

$\therefore f(x) = x^2+13x$

(b)  $x^2+13x-90=0 \Rightarrow (x+18)(x-5)=0 \Rightarrow x=-18 \text{ or } x=5$

(9)  The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.

(a)  Find $x$, $y$, and $z$.

(b)  If a plant is randomly selected from the garden, find the probability that the height of the selected plant is less than 1.25 m but not less than 0.65 m.

Solution

(a)  $a=2, \therefore x=2+4=6, y=37-15=22, z=37+3=40$

(b)  $\dfrac{22-6}{40}=\dfrac{16}{40}=\dfrac{2}{5}$

(10)  The coordinates of the points $A$ and $B$ are $(5, 7)$ and $(13,1)$ respectively. Let $P$ be a moving point in the rectangular coordinate plane such that $P$ is equidistant from $A$ and $B$. Denote the locus of $P$ by $\Gamma$.

(a)  Find the equation of $\Gamma$.  (2 marks)

(b)  $\Gamma$ intersects the x-axis and y-axis at $H$ and $K$ respectively. Denote the origin by $O$. Let $C$ be the circle which passes through $O$, $H$, and $K$. Someone claims that the circumference of $C$ exceeds 30. Is the claim correct? Explain your answer.  (3 marks)

Solution

(a)  $(x-5)^2+(y-7)^2=(x-13)^2+(y-1)^2$

$\Rightarrow -10x+25-14y+49=-26x+169-2y+1$

$\Rightarrow 16x=12y+96$

$\Rightarrow 4x-3y-24=0$

(b)  Let $y=0$, then $x=6$, hence $H(6, 0)$

Let $x=0$, then $y=-8$, hence $K(0, -8)$

Since $\triangle HKO$ is a right triangle where $\angle O = 90^\circ$, hence $HK$ is the diameter of the circle $C$.

$\text{Circumference} = \pi d = \pi\sqrt{6^2+8^2} = 10\pi > 30$

Therefore, the claim is correct.

(11)  An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is $12 \text{ cm}$. Peter pours $444\pi\text{ cm}^2$ of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is $16\text{ cm}$.

(a)  Express the final volume of milk in the vessel in terms of $\pi$.  (3 marks)

(b)  Peter claims that the final area of the wet curved surface of the vessel is at least $800\text{ cm}^2$. Do you agree? Explain your answer.  (3 marks)

Solution

(a)  Let $x$ be the original volume of milk.

$\displaystyle\frac{444\pi+x}{x}=\left(\frac{16}{12}\right)^3=\left(\frac{4}{3}\right)^3$

$27(444\pi+x)=64x$

$27\cdot444\pi=37x$

$x=27\cdot4\cdot3\pi = 81\cdot4\pi=324\pi$

$\therefore \text{Final volume} = 444\pi+324\pi = 768\pi\text{ cm}^3$

(b)  $\dfrac{\pi r^2 (16)}{3} = 768 \pi \Rightarrow r^2=144 \Rightarrow r=12\text{ cm}$

$\text{Curved surface area} = \pi r s = \pi(12)\sqrt{12^2+16^2} = 240\pi \approx 754 \text{ cm}^2 < 800\text{ cm}^2$

Therefore, disagree.

(12)  The bar chart below shows the distribution of the ages of the children in a group, where $a>11$ and $4. The median of the ages of the children in the group is 7.5.

(a)  Find $a$ and $b$.  (3 marks)

(b)  Four more children now join the group. It is found that the ages of these four children are all different and the range of the ages of the children in the group remains unchanged. Find

(i)  the greatest possible median of the ages of the children in the group.

(ii)  the least possible mean of the ages of the children in the group.  (4 marks)

Solution

(a)  Since the median age is 7.5, hence $11+a=11+b+4 \Rightarrow a=b+4$

Now, $a>11 \text{ and } 47 \text{ and } 4.

Since $a,b\in\mathbb{Z}^+ \therefore b=8\text{ or } 9$

$\therefore (a,b) = (12,8)\text{ or }(13,9)$

(b) (i)  To maximize the median, we add age 7, 8, 9, 10. Therefore the new median will be 8.

(b) (ii)  To minimize the mean, we add age 6, 7, 8, 9.

Case $(a, b)=(12,8)$,

$\text{mean} = \dfrac{12\cdot6+13\cdot7+12\cdot8+9\cdot9+4\cdot10}{12+13+12+9+4} = \dfrac{380}{50}=\dfrac{38}{5}=7.6$

Case $(a, b)=(13,9)$,

$\text{mean} = \dfrac{12\cdot6+14\cdot7+12\cdot8+10\cdot9+4\cdot10}{12+14+12+10+4} = \dfrac{396}{52}=\dfrac{99}{13}=7.\overline{615384}$

(13)  In figure 1, $ABC$ is a triangle. $D$, $E$, and $M$ are points lying on $BC$ such that $BD=CE$, $\angle ADC = \angle AEB$ and $DM=EM$.

(a)  Prove that $\triangle ACD\cong\triangle ABE$.  (2 marks)

(b) Suppose that $AD=15\text{ cm}$, $BD=7\text{ cm}$ and $DE=18\text{ cm}$.

(i)  Find $AM$.

(ii)  Is $\triangle ABE$ a right-angled triangle? Explain your answer.  (5 marks)

Solution

(a)  $\angle ADC = \angle AEB \quad \text{(given)}$

$AD=AE \quad \text{(sides opp. equal }\angle\text{s)}$

$BD=CE \quad\text{(given)}$

$BE = BD+DE = CE+DE = CD\quad (BD=CE)$

$\therefore \triangle ACD\cong\triangle ABE \quad\text{(SAS)}$

(b) (i)  $AD=AE=15 cm\quad \text{(given)}$

Hence, $\triangle ADE$ is isosceles.

$DM=EM\quad\text{(given)}$

$\therefore AM \perp DE\quad\text{(property of isos }\triangle \text{)}$

$\therefore AM=\sqrt{15^2-9^2} = 12\text{ cm} \quad\text{(Pyth. thm)}$

(b) (ii) $DM=EM=9\text{ cm}$

By Pyth. thm, $AB=\sqrt{12^2+(7+9)^2} = 20\text{ cm}$

$AB^2+AE^2 = 20^2+15^2 = 25^2 = AE^2$

Therefore, $\triangle ABE$ is right-angled.

(14)  Let $p(x)=6x^4+7x^3+ax^2+bx+c$, where $a$, $b$, and $c$ are constants. When $p(x)$ is divided by $x+2$ and when $p(x)$ is divided by $x-2$, the two remainders are equal. It is given that $p(x)=(lx^2+5x+8)(2x^2+mx+n)$, where $l$, $m$, and $n$ are constants.

(a)  Find $l$, $m$, and $n$.  (5 marks)

(b)  How many real roots does the equation $p(x)=0$ have? Explain your answer.  (5 marks)

Solution

(a)  $p(-2)=p(2)$

$6(-2)^4+7(-2)^3+a(-2)^2+b(-2)+c = 6(2)^4+7(2)^3+a(2)^2+b(2)+c$

$-56-2b = 56+2b$

$4b = -112$

$b=-28$

$\therefore p(x)=6x^4+7x^3+ax^2-28x+c$

$(lx^2+5x+8)(2x^2+mx+n) = (2l)x^4+(lm+10)x^3+(ln+5m+16)x^2+(5n+8m)x+8n$

By matching the coefficient,

$2l=6 \Rightarrow l=3$

$lm+10=7 \Rightarrow m=-1$

$5n+8m=-28 \Rightarrow n=-4$

(b)  $p(x)=(lx^2+5x+8)(2x^2+mx+n)=(3x^2+5x+8)(2x^2-x-4)=0$

$\therefore (3x^2+5x+8)=0\text{ or }(2x^2-x-4)=0$

For $3x^2+5x+8=0, \quad \Delta=25-4\cdot3\cdot8 = -71\quad \therefore$ no real root.

For $2x^2-x-4=0, \quad \Delta=1+4\cdot2\cdot4 = 33\quad \therefore$ two real roots.

Therefore, $p(x)=0$ has two real roots.

###### Section B (35 marks)

(15)  If 4 boys and 5 girls randomly form a queue, find the probability that no boys are next to each other in the queue.  (3 marks)

Solution

Consider this : _ G _ G _ G _ G _ G _

There are 6 places to put 4 B’s so there are 6P4 ways to place the boys. Moreover, there are 5! ways to permute those 5 G’s. Therefore,

$\displaystyle \text{P(no boys are next to each other)}=\frac{_6P_4 \cdot 5!}{9!} = \frac{6\cdot5\cdot4\cdot3}{9\cdot8\cdot7\cdot6} = \frac{5}{42}$

(16)  In a test, the mean of the distribution of the scores of a class of students is 61 marks. The standard scores of Albert and Mary are -2.6 and 1.4 respectively. Albert gets 22 marks. A student claims that the range of the distribution is at most 59 marks. Is the claim correct? Explain your answer.  (3 marks)

Solution

$Z=\dfrac{X-\mu}{\sigma} \quad \Rightarrow\quad -2.6 = \dfrac{22-61}{\sigma} \quad\Rightarrow\quad \sigma = 15$

So, the standard deviation is 15. Now, let’s calculate what Mary gets on the test.

$X=Z\sigma+\mu = 1.4\cdot 15 + 61 = 82$

$82-22 = 60$, which is greater than 59. Therefore the claim is false.

(17)  The 1st term and the 38th term of an arithmetic sequence are 666 and 555 respectively. Find

(a)  the common difference of the sequence.  (2 marks)

(b)  the greatest value of $n$ such that the sum of the first $n$ term of the sequence is positive.  (3 marks)

Solution

(a)  $t_{38} =t_1 + 37d$

$\therefore d=\dfrac{t_{38}-t_1}{37} = \dfrac{555-666}{37} = \dfrac{-111}{37}=-3$

(b)  $S_n = \dfrac{(2a+(n-1)d)n}{2} > 0$

$\displaystyle \frac{(2(666)+(n-1)(-3))n}{2}>0$

Since $n$ is positive integer, we can divide both side by $n$ without switching the sign.

$2(666)-3(n-1)>0$

$n-1 < 2(222)$

$n < 445$

Therefore, the greatest such $n$ is 444.

(18)  Let $f(x) = \dfrac{-1}{3}x^2 + 12x - 121$.

(a)  Using the method of completing the square, find the coordinates of the vertex of the graph of $y = f(x)$.  (2 marks)

(b)  The graph of $y=g(x)$ is obtained by translating the graph of $y=f(x)$ vertically. If the graph of $y=g(x)$ touches the x-axis, find $g(x)$.  (2 marks)

(c)  Under a transformation, $f(x)$ is changed to $latex \dfrac{-1}{3}x^2 – 12x – 121$. Describe the geometric meaning of the transformation.  (2 marks)

Solution

(a)  $\displaystyle f(x) = \frac{-1}{3}x^2 + 12x - 121$

$= -\frac13 (x^2-36x+18^2-18^2) - 121$

$= -\frac13(x-18)^2+\frac{18^2}{3}-121$

$= -\frac13(x-18)^2 -13$

Therefore, the vertex is $(18, -13)$.

(b)  Basically translate the graph of $f(x)$ 13 units upward.

$\therefore g(x) = -\frac13(x-18)^2$

(c)  Since $\displaystyle\frac{-1}{3}x^2-12x-121=\frac{-1}{3}(-x)^2+12(-x)-121 = f(-x)$

Therefore, the transformation is reflecting the graph of f(x) horizontally along the y-axis.

(19)  Figure 2 shows a geometric model $ABCD$ in the form of tetrahedron. It is given that $\angle BAD=86^\circ, \angle CBD=43^\circ, AB=10\text{ cm}, AC=6\text{ cm}, BC=8\text{ cm} \text{ and } BD=15\text{ cm}$.

(a)  Find $\angle ABD$ and $CD$.  (4 marks)

(b)  A craftsman claims that the angle between $AB$ and the face $BCD$ is $\angle ABC$. Do you agree? Explain your answer.  (2 marks)

Solution

(a)  By Sine Law, $\displaystyle \frac{\sin\angle ADB}{AB}=\frac{\sin\angle BAD}{BD}$

$\displaystyle \frac{\sin\angle ADB}{10}=\frac{\sin 86^\circ}{15}$

$\therefore \angle ADB = \sin^{-1} \dfrac{10\sin 86^\circ}{15}$

$\therefore \angle ABD = 180^\circ - 86^\circ - \sin^{-1} \dfrac{2\sin 86^\circ}{3} = 52.31439868 \approx 52.3^\circ \text{ (3 s.f.)}$

By Cosine Law, $CD^2=AC^2+AD^2-2AC\cdot AD \cos\angle CBD$.

$\therefore CD = \sqrt{8^2+15^2-2\cdot8\cdot15\cdot\cos 43^\circ} = 10.65246974 \approx 10.7 \text{ cm (3 s.f.)}$

(b)  By Sine law, $\displaystyle \frac{AD}{\sin\angle ABD}=\frac{BD}{\sin\angle BAD}$

$\therefore AD=\dfrac{15\sin\angle ABD}{\sin 86^\circ} \approx 11.8996447\ldots$

$AC^2+CD^2=6^2+10.65246974^2=149.4751116\neq AD^2=141.6015440$

Therefore, do not agree.

(20)  $\triangle OPQ$ is an obtuse-angled triangle. Denote the in-centre and the circumcentre of $\triangle OPQ$ by $I$ and $J$ respectively. It is given that $P$, $I$, and $J$ are collinear.

(a)  Prove that $OP=PQ$.  (3 marks)

(b)  A rectangular coordinate system is introduced so that the coordinates of $O$ and $Q$ are $(0,0)$ and $(40,30)$ respectively while the y-coordinate of $P$ is 19. Let $C$ be the circle which passes through $O$, $P$, and $Q$.

(i)  Find the equation of $C$.

(ii)  Let $L_1$ and $L_2$ be two tangents to $C$ such that the slope of each tangent is $\dfrac34$ and the y-intercept of $L_1$ is greater than that of $L_2$. $L_1$ cuts the x-axis and y-axis at $S$ and $T$ respectively while $L_2$ cuts the x-axis and the y-axis at $U$ and $V$ respectively. Someone claims that the area of the trapezium $STUV$ exceeds 17000. Is the claim correct? Explain your answer.  (9 marks)

Solution

(a)  Since $P$, $I$, and $J$ are collinear, we can draw a line passing through all three points and meet the line segment $OQ$ at $K$. Since $I$ is incenter, $PI$ is angle bisector of $\angle OPQ$, hence $\angle OPK = \angle QPK$. Since $J$ is circumcenter, then $PK$ is perpendicular bisector of $OQ$ and so $\angle PKO = \angle PKQ = 90^\circ$. $PK = PK$ obviously by common sides. Hence, $\triangle PKO \cong \triangle PKQ$ by ASA. Therefore the corresponding sides of the two congruent triangles are congruent and so $OP=PQ$.

(b) (i)  Let $P=(x, 19)$.  Since $OP = PQ$,

$\sqrt{x^2+19^2} = \sqrt{(x-40)^2+(19-30)^2}$

$x^2+361=x^2-80x+1600+121$

$80x=1360$

$x=17$

Let circle $C: x^2+y^2+Dx+Ey+F=0$

Substitute $O(0,0)$ into circle C yields $F=0$. Hence,

$P(17, 19): 17^2+19^2+17D+19E=0 \Rightarrow 650+17D+19E \qquad (1)$

$Q(40, 30): 40^2+30^2+40D+30E=0 \Rightarrow 250+4D+3E=0 \qquad (2)$

3(1)-19(2):

$1950+51D-4750-76D=0$

$\therefore D=-112$

$\therefore E=66$

Therefore, the equation of circle C is $x^2+y^2-112x+66y=0$

(b) (ii)  Since $L_1$ and $L_2$ have slope of $\dfrac34$, They both have the equation of the form: $y=\dfrac34x+b$

They are both tangents to the circle hence:

$x^2+y^2-112x+66y =0 \quad\text{and}\quad y=\dfrac34x+b$

$x^2+\left(\frac34x+b\right)^2-112x+66\left(\frac34x+b\right)=0$

$x^2+\frac{9}{16}x^2+\frac{3b}{2}x+b^2-112x+\frac{99}{2}x+66b=0$

$16x^2+9x^2+24bx+16b^2-1792x+792x+1056b=0$

$25x^2+(24b-1000)x+16b^2+1056b=0$

Since they are tangents to the circle,

$\Delta = (24b-1000)^2-4(25)(16b^2+1056b)=0$

$576b^2-48000b+1000000-1600b^2-105600b=0$

$1024b^2+153600b-1000000=0$

$16b^2+2400b-15625=0$

$(4b+625)(4b-25)=0$

$b=-\dfrac{625}{4} \text{ or } b=\dfrac{25}{4}$

$\displaystyle\therefore \quad L_1: y=\frac34x+\frac{25}{4} \quad\text{and}\quad L_2: y =\frac34x-\frac{625}{4}$

$L_1:$ Let $y=0 \Rightarrow x=-\dfrac{25}{3} \Rightarrow S\left(-\dfrac{25}{3},0\right) \Rightarrow T\left(0,\dfrac{25}{4}\right)$

$L_2:$ Let $y=0 \Rightarrow x=\dfrac{625}{3} \Rightarrow U\left(\dfrac{625}{3},0\right) \Rightarrow V\left(0,-\dfrac{625}{4}\right)$

$\displaystyle \therefore ST=\sqrt{\left(\frac{25}{3}\right)^2+\left(\frac{25}{4}\right)^2}=25\sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{25}{12}$

$\displaystyle \therefore UV=\sqrt{\left(\frac{625}{3}\right)^2+\left(\frac{625}{4}\right)^2}=625\sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{3125}{12}$

$C: x^2+y^2-112x+66y=0$

$\Rightarrow x^2-112x+56^2+y^2+66y+33^2=56^2+33^2=4225$

$\Rightarrow (x-56)^2+(y+33)^2=65^2 \Rightarrow r=65 \Rightarrow d=130$

$\displaystyle\therefore \text{Area} = \frac{(ST+UV)d}{2} = \left(\frac{125}{12}+\frac{3125}{12}\right)65 =17604.1\overline{6}>17000$

Therefore, the claim is correct.

## 777, 365, and 21

Posted: March 26, 2017 in Mathematics

The number “689” has been the laughingstock of the HK chief executive for the past 4 years. “689” basically means 冇柒用, so in response, this year’s chief executive will receive “777” votes just to emphasis that she is very 好有柒用. It’s not one 7 or two 7s, but three 7s in a row, how nice!

Actually 777 is quite an interesting number. In gambling, it is a lucky number to most people because “777” is a jackpot number. In computer science, “777” means you have been granted read, write, and executable permission on a file (in UNIX or UNIX-like operating system such as Linux, BSD, Mac OSX, etc). To most HKers, 777, 365, and 21 mean the number of vote of today’s HK chief executive election. To me, I have another interpretation.

$777 = 7 \cdot 111 = 7 \cdot 3 \cdot 37$

$365 = 5 \cdot 73$

$21 =3 \cdot 7$

$\therefore 777 \cdot 365 \cdot 21 = 7 \cdot 3 \cdot 37 \cdot 5 \cdot 73 \cdot 3 \cdot 7$

Math is beautiful, isn’t it?

###### Remarks

The joke has made into wikipedia LOL

## Fibonacci Number & Golden Ratio

Posted: March 22, 2017 in Mathematics

Today I saw this funny picture of Donald Trump and then I know why everybody says Golden Ratio can be found all over nature. Yes it is living among us. You can’t escape from it, not even from the president of America.

So, what is Golden Ratio?

In Euclid’s Elements, Book VI Definition 2, Euclid wrote (in Greek of course):

Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν

Or in English,

A straight-line is said to have been cut in extreme and mean ratio when as the whole is to the greater segment so the greater (segment is) to the lesser.

Or in plain English, it basically says that when you divide a line segment into two uneven pieces, the ratio of the whole line segment to the longer piece equals the ratio of the longer piece to the shorter piece. And this ratio is called the Golden Ratio (often denote as $\phi$).

$\displaystyle\frac{a+b}{a} = \frac{a}{b} = \phi \quad\Leftrightarrow\quad 1+\frac{1}{\phi}=\phi \quad\Leftrightarrow\quad \phi^2=\phi+1$

After solving the quadratic equation, you will get:

$\displaystyle\phi = \frac{1+\sqrt5}{2} \approx 1.61803398874989\ldots$

Now, let’s consider the Fibonacci sequence.

$F_n = 1,1,2,3,5,8,13,21,34,55,89,\ldots$

and examine some nth power of the golden ratio.

$\phi^1 = \phi$

$\phi^2 = \phi+1$

$\phi^3 = \phi^2+\phi = 2\phi + 1$

$\phi^4 = 2\phi^2+\phi = 2(\phi+1) + \phi = 3\phi+2$

$\phi^5 = 3\phi^2+2\phi = 3(\phi+1) + 2\phi = 5\phi + 3$

$\phi^6 = 5\phi^2+3\phi = 5(\phi + 1) + 3\phi = 8\phi + 5$

This suggests that

$x^2=x+1 \quad \Rightarrow \quad x^n = F_n x + F_{n-1} \quad\forall n\in\mathbb{Z}^+$

Let’s define $F_0 = 0$, then

$x^1 = F_1 x + F_0 = x$

$x^2 = F_2 x + F_1 = x+1$

Assume $x^k = F_k x + F_{k-1}$, then

$x^{k+1} = x^k\cdot x = F_k x^2 + F_{k-1} x \\= F_k(x+1)+F_{k-1}x = (F_k+F_{k-1})x+F_k \\ = F_{k+1} x + F_k$

Hence, by math induction, $x^n = F_n x + F_{n-1} \quad \forall n\in\mathbb{Z}^+ \text{ where } x^2=x+1$.

Now, go back to the equation $x^2 = x+1$, the two roots are

$\displaystyle\phi = \frac{1+\sqrt5}{2} \quad\text{and}\quad \Phi=\frac{1-\sqrt5}{2}$

Hence,

$\phi^n = F_n\phi + F_{n-1} \qquad \text{(1)}$

$\Phi^n = F_n\Phi + F_{n-1} \qquad \text{(2)}$

(1) – (2) yields

$\displaystyle F_n = \frac{\phi^n - \Phi^n}{\phi - \Phi} = \frac{\phi^n - \Phi^n} {\sqrt5}$

Or

$\displaystyle F_n = \frac{1}{\sqrt5}\left(\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n\right)$

###### Remarks

Since

$\displaystyle\Phi = 1-\phi \quad\text{and}\quad \phi = 1+\frac{1}{\phi}$

then

$\displaystyle F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt5} = \frac{\phi^n-(-\frac{1}{\phi})^n}{\sqrt5}$

$\displaystyle\lim_{n\to\infty}\frac{F_{n+1}}{F_n} = \lim_{n\to\infty} \frac{\phi^{n+1}-(-\frac{1}{\phi})^{n+1}}{\sqrt5} \cdot \frac{\sqrt5}{\phi^n-(-\frac{1}{\phi})^n}$

As $\displaystyle n\to\infty, \quad -\frac{1}{\phi^{n+1}} \text{ and } -\frac{1}{\phi^n} \to 0$

$\displaystyle \therefore \lim_{n\to\infty} \frac{F_{n+1}}{F_n} = \phi$

As $n\to\infty, \quad \displaystyle -\frac{1}{\phi^n} \to 0$

$\displaystyle \therefore F_n \approx \frac{\phi^n}{\sqrt5}$ as n is large.

The last approximation comes in handy when you want to find the nth Fibonacci number, let’s say the 30th Fibonacci number.

$\displaystyle F_{30} \approx \frac{\phi^{30}}{\sqrt5} \approx 832040.00000024\ldots$

So, you can compute the 30th Fibonacci number to be 832040 in seconds with any normal scientific calculator.

## Einstein’s Five House Riddle

Posted: March 15, 2017 in Mathematics

This riddle is claimed to be written by Einstein when he was a boy. He claimed that only 2% of the people can solve it. Are you that 2% who can solve it?

There are five houses in five different colors in a row. In each house lives a person with a different nationality. The five owners drink a certain type of beverage, smoke a certain brand of cigar, and keep a certain pet. No owners have the same pet, smoke the same brand of cigar, or drink the same beverage.

The question is: Who owns the fish?

Hints:

1. The Brit lives in the red house.
2. The Swede keeps dogs as pets.
3. The Dane drinks tea.
4. The green house is on the left of the white house.
5. The green house’s owner drinks coffee.
6. The owner who smokes Pall Mall rears birds.
7. The owner of the yellow house smokes Dunhill.
8. The owner living in the center house drinks milk.
9. The Norwegian lives in the first house.
10. The owner who smokes blends lives next to the one who keeps cats.
11. The owner who keeps horses lives next to the man who smokes Dunhill.
12. The owner who smokes BlueMaster drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next to the blue house.
15. The man who smokes blend has a neighbor who drinks water.