SMO 2009 Senior Round 1

Question 1

Suppose that $\pi$ is a plane and $A,B$ are two points on the plane $\pi$. If the distance between $A$ and $B$ is $33$ cm, how many lines are there in the plane such that the distance between each line and $A$ is $7$ cm and the distance between each line and $B$ is $26$ cm respectively?

$1$
$2$
$3$
$4$
Infinitely many

Question 2

Let $y=(17-x)(19-x)(19+x)(17+x)$, where $x$ is a real number. Find the smallest possible value of $y$.

$-1296$
$-1295$
$-1294$
$-1293$
$-1292$

Question 3

If two real numbers $a,b$ are randomly chosen from the interval $(0,1)$, find the probability that the equation $x^2-\sqrt a\,x+b=0$ has real roots.

$\frac18$
$\frac5{16}$
$\frac3{16}$
$\frac14$
$\frac13$

Question 4

If $x,y$ are real numbers for which $|x|+x+5y=2$ and $|y|-y+x=7$, find $x+y$.

$-3$
$-1$
$1$
$3$
$5$

Question 5

In a triangle $ABC$, $\sin A=\dfrac35$ and $\cos B=\dfrac5{13}$. Find $\cos C$.

$\frac{56}{65}$ or $\frac{16}{65}$
$\frac{56}{65}$
$\frac{16}{65}$
$-\frac{56}{65}$
$\frac{56}{65}$ or $-\frac{16}{65}$

Question 6

The area of triangle $ABC$ is $40\text{ cm}^2$. Points $D,E,F$ are on sides $AB,BC,CA$ respectively, as shown. If $AD=3$ cm, $DB=5$ cm, and the area of triangle $ABE$ is equal to the area of quadrilateral $DBEF$, find the area of triangle $AEC$ in $\text{cm}^2$.

Question illustration

$11$
$12$
$13$
$14$
$15$

Question 7

Find the value of $\displaystyle\frac3{1!+2!+3!}+\frac4{2!+3!+4!}+\cdots+\frac{22}{20!+21!+22!}$.

$1-\frac1{24!}$
$\frac12-\frac1{23!}$
$\frac12-\frac1{22!}$
$1-\frac1{22!}$
$\frac12-\frac1{24!}$

Question 8

There are eight envelopes numbered $1$ to $8$. Find the number of ways in which $4$ identical red buttons and $4$ identical blue buttons can be put in the envelopes such that each envelope contains exactly one button, and the sum of the numbers on the envelopes containing the red buttons is more than the sum of the numbers on the envelopes containing the blue buttons.

$35$
$34$
$32$
$31$
$62$

Question 9

Determine the number of acute-angled triangles (i.e. all angles are less than $90^\circ$) in which all angles (in degrees) are positive integers and the largest angle is three times the smallest angle.

Question 10

Let $ABCD$ be a quadrilateral inscribed in a circle with diameter $AC$, and let $E$ be the foot of the perpendicular from $D$ onto $AB$, as shown. If $AD=DC$ and the area of quadrilateral $ABCD$ is $24\text{ cm}^2$, find $DE$ in cm.

Question illustration

$3\sqrt2$
$2\sqrt6$
$2\sqrt7$
$4\sqrt2$
$6$

Question 11

Find the number of positive divisors of $\bigl(2008^3+(3\times2008\times2009)+1\bigr)^2$.

Question 12

Suppose that $a,b,c$ are real numbers greater than $1$. Find the value of
$$\frac1{1+\log_{a^2b}(c/a)}+\frac1{1+\log_{b^2c}(a/b)}+\frac1{1+\log_{c^2a}(b/c)}.$$

Question 13

Find the remainder when $(1!\times1)+(2!\times2)+(3!\times3)+\cdots+(286!\times286)$ is divided by $2009$.

Question 14

Find the value of $(25+10\sqrt5)^{1/3}+(25-10\sqrt5)^{1/3}$.

Question 15

Let $a=\dfrac{1+\sqrt{2009}}2$. Find $(a^3-503a-500)^{10}$.

Question 16

In the figure below, $ABC$ is a triangle and $D$ is a point on side $BC$. Point $E$ is on side $AB$ such that $DE$ is the angle bisector of $\angle ADB$, and point $F$ is on side $AC$ such that $DF$ is the angle bisector of $\angle ADC$. Find
$$\frac{AE}{EB}\cdot\frac{BD}{DC}\cdot\frac{CF}{FA}.$$
Question illustration

Question 17

Find $(\cot25^\circ-1)(\cot24^\circ-1)(\cot23^\circ-1)(\cot22^\circ-1)(\cot21^\circ-1)(\cot20^\circ-1)$.

Question 18

Find the number of $2$-element subsets $\{a,b\}$ of $\{1,2,3,\ldots,99,100\}$ such that $ab+a+b$ is a multiple of $7$.

Question 19

Let $x$ be a real number such that $x^2-15x+1=0$. Find $x^4+\dfrac1{x^4}$.

Question 20

In the figure below, $ABC$ is a triangle with $AB=10$ cm and $BC=40$ cm. Points $D,E$ lie on side $AC$ and point $F$ on side $BC$ such that $EF\parallel AB$ and $DF\parallel EB$. Given that $BE$ is an angle bisector of $\angle ABC$ and $AD=13.5$ cm, find $CD$ in cm.

Question illustration