SMO 2007 Junior Round 1

Question 1

Among the four statements on integers below,
- “If $a\lt b$ then $a^2\lt b^2$”;
- “$a^2\gt 0$ is always true”;
- “$-a\lt 0$ is always true”;
- “If $ac^2\lt bc^2$ then $a\lt b$”,
how many of them are correct?

Question 2

Which of the following numbers is odd for any integer values of $k$?

$2007+k^3$
$2007+7k$
$2007+2k^2$
$2007+2007k$
$2007k$

Question 3

In a school, all 300 Secondary 3 students study either Geography, Biology or both Geography and Biology. If 80% study Geography and 50% study Biology, how many students study both Geography and biology?

$30$
$60$
$80$
$90$
$150$

Question 4

An unbiased six-sided dice is numbered 1 to 6. The dice is thrown twice and the two scores added. Which of the following events has the highest probability of occurrence?

The total score is a prime number
The total score is a multiple of 4
The total score is a perfect square
The total score is 7
The total score is a factor of 12

Question 5

The cardboard below can be cut out and folded to make a cube. Which face will then be opposite the face marked $A$?

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Question 6

How many triangles can you find in the following figure?

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$7$
$10$
$12$
$16$
$20$

Question 7

Suppose $x_1,x_2$ and $x_3$ are roots of $(11-x)^3+(13-x)^3=(24-2x)^3$. What is the sum of $x_1+x_2+x_3$?

$30$
$36$
$40$
$42$
$44$

Question 8

In the following right-angled triangle $ABC$, $AC=BC=1$ and $DEF$ is an arc of a circle with center $A$. Suppose the shaded areas $BDE$ and $CEF$ are equal and $AD=\dfrac{x}{\sqrt\pi}$. What is the value of $x$?

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Question 9

Suppose
$$\frac1x=\frac2{y+z}=\frac3{z+x}=\frac{x^2-y-z}{x+y+z}.$$
What is the value of $\dfrac{z-y}{x}$?

Question 10

Suppose $x^2-13x+1=0$. What is the last digit of $x^4+x^{-4}$?

Question 11

In a triangle $ABC$, it is given that $AB=1$ cm, $BC=2007$ cm and $AC=a$ cm, where $a$ is an integer. Determine the value of $a$.

Question 12

Find the value (in the simplest form) of $\sqrt{21+12\sqrt3}-\sqrt{21-12\sqrt3}$.

Question 13

Find the value of $\dfrac{2007^2+2008^2-1993^2-1992^2}{4}$.

Question 14

Find the greatest integer $N$ such that $N\le\sqrt{2007^2-20070+31}$.

Question 15

Suppose that $x$ and $y$ are non-zero real numbers such that
$$\frac x3=y^2\quad\text{and}\quad\frac x9=9y.$$
Find the value of $x+y$.

Question 16

Evaluate the sum $\dfrac{2007}{1\times2}+\dfrac{2007}{2\times3}+\cdots+\dfrac{2007}{2006\times2007}$.

Question 17

Find the sum of the digits of the product $\underbrace{111111111\ldots111}_{2007\text{ ones}}\times2007$.

Question 18

The diagram shows two identical squares, $ABCD$ and $PQRS$, overlapping each other in such a way that their edges are parallel, and a circle of radius $(2-\sqrt2)$ cm covered within these squares. Find the length of the square $ABCD$.

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Question 19

When 2007 bars of soap are packed into $N$ boxes of equal size, where $N$ is an integer strictly between 200 and 300, there are extra 5 bars remaining. Find $N$.

Question 20

Suppose that $a+x^2=2006$, $b+x^2=2007$ and $c+x^2=2008$ and $abc=3$. Find the value of
$$\frac a{bc}+\frac b{ca}+\frac c{ab}-\frac1a-\frac1b-\frac1c.$$