SMO 2007 Senior Round 1

Question 1

Find the sum of the digits of the product $\left(1+\dfrac12\right)\left(1+\dfrac13\right)\left(1+\dfrac14\right)\cdots\left(1+\dfrac1{2006}\right)\left(1+\dfrac1{2007}\right)$.

$5$
$6$
$9$
$10$
$13$

A bag contains $x$ green and $y$ red sweets. A sweet is selected at random from the bag and its colour noted. It is then replaced into the bag together with 10 additional sweets of the same colour. A second sweet is next randomly drawn. Find the probability that the second sweet is red.

$\dfrac{y+10}{x+y+10}$
$\dfrac y{x+y+10}$
$\dfrac y{x+y}$
$\dfrac x{x+y}$
$\dfrac{x+y}{x+y+10}$

Question 3

What is the remainder when the number
$$\left(\underbrace{999999\ldots999}_{2008\text{ nines}}\right)^{2007}-\left(\underbrace{333333\ldots333}_{2008\text{ threes}}\right)^{2007}$$
is divided by $11$?

$0$
$2$
$4$
$6$
None of the above

Question 4

$W,X,Y$ and $Z$ are the midpoints of the four sides of parallelogram $ABCD$. $P$ is a point on the line segment $YZ$. What percent of the area of parallelogram $ABCD$ is triangle $PXW$?

Question illustration

$50\%$
$45\%$
$30\%$
$25\%$
$20\%$

Question 5

Four rods are connected together with flexible joints at their ends to make a quadrilateral as shown. Rods $PQ=40$ cm, $RS=20$ cm, $PS=60$ cm and $\angle QPS=\angle RSP=60^\circ$. Find $\angle QRS$.

Question illustration

$100^\circ$
$105^\circ$
$120^\circ$
$135^\circ$
$150^\circ$

Question 6

When 2007 bars of soap are packed into $N$ boxes, where $N$ is a positive integer, there is a remainder of 5. How many possible values of $N$ are there?

$14$
$16$
$18$
$20$
$13$

Question 7

Suppose $a_n$ denotes the last two digits of $7^n$. For example, $a_2=49$, $a_3=43$. Find the value of $a_1+a_2+a_3+\cdots+a_{2007}$.

$50189$
$50199$
$50209$
$50219$
$50229$

Question 8

The diagram below shows a quadrilateral $ABCD$ where $AB=10$, $BC=6$, $CD=8$ and $DA=2$. The diagonals $AC$ and $BD$ intersect at the point $O$ and $\angle COB=45^\circ$. Find the area of the quadrilateral $ABCD$.

Question illustration

$28$
$29$
$30$
$31$
$32$

Question 9

In the following diagram, $ABCD$ is a square with $PA=a$, $PB=2a$ and $PC=3a$. Find $\angle APB$.

Question illustration

$120^\circ$
$130^\circ$
$135^\circ$
$140^\circ$
$145^\circ$

Question 10

What is the largest possible prime value of $n^2-12n+27$, where $n$ ranges over all positive integers?

$91$
$37$
$23$
$17$
$7$

Question 11

Suppose that $\log_2[\log_3(\log_4 a)]=\log_3[\log_4(\log_2 b)]=\log_4[\log_2(\log_3 c)]=0$. Find the value of $a+b+c$.

Question 12

Find the unit digit of $17^{17}\times19^{19}\times23^{23}$.

Question 13

Given that $x+y=12$ and $xy=50$, find the exact value of $x^2+y^2$.

Question 14

Suppose that $(21.4)^a=(0.00214)^b=100$. Find the value of $\dfrac1a-\dfrac1b$.

Question 15

Find the value of $100(\sin253^\circ\sin313^\circ+\sin163^\circ\sin223^\circ)$.

Question 16

The letters of the word MATHEMATICS are rearranged in such a way that the first four letters of the arrangement are all vowels. Find the total number of distinct arrangements that can be formed in this way.

(Note: The vowels of English language are A, E, I, O, U.)

Question 17

Given a set $S=\{1,2,3,\ldots,199,200\}$. The subset $A=\{a,b,c\}$ of $S$ is said to be “nice” if $a+c=2b$. How many “nice” subsets does $S$ have?

(Note: The order of the elements inside the set does not matter. For example, we consider $\{a,b,c\}$ or $\{a,c,b\}$ or $\{c,b,a\}$ to be the same set.)