SMO 2009 Junior Round 1

Question 1

Let $C_1,C_2$ be distinct circles of radius $7$ cm in the same plane, tangent to each other. How many circles of radius $26$ cm in this plane are tangent to both $C_1$ and $C_2$?

$2$
$4$
$6$
$8$
None of the above.

Question 2

In the diagram, quadrant $OAD$ has radius $4$ and quadrant $OBC$ has radius $8$. Given $\angle COD=30^\circ$, find the area of shaded region $ABCD$.

Question illustration

$12\pi$
$13\pi$
$15\pi$
$16\pi$
None of the above.

Question 3

Let $k$ be a real number. Find the maximum value of $k$ such that the following inequality holds: $$\sqrt{x-2}+\sqrt{7-x}\ge k.$$

$\sqrt5$
$3$
$\sqrt2+\sqrt3$
$\sqrt{10}$
$2\sqrt3$

Question 4

Three circles of radius $20$ are arranged with their respective centres $A,B,C$ in a row, as shown. If line $WZ$ is tangent to the third circle, find $XY$.

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$30$
$32$
$34$
$36$
$38$

Question 5

Given negative integers $x,y$ satisfying $y=\dfrac{10x}{10-x}$, find the maximum value of $y$.

$-10$
$-9$
$-6$
$-5$
None of the above.

Question 6

The sequence satisfies $a_n=a_{n-1}+n^2$ and $a_0=2009$. Find $a_{50}$.

$42434$
$42925$
$44934$
$45029$
$45359$

Question 7

Coins of the same size are arranged on a very large table (the infinite plane) so that each coin touches six other coins. Find the percentage of the plane covered by the coins.

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$\frac{20\pi}{\sqrt3}\%$
$\frac{50\pi}{\sqrt3}\%$
$16\sqrt3\pi\%$
$17\sqrt3\pi\%$
$18\sqrt3\pi\%$

Question 8

Given real numbers $x,y$ with $x+xy+y=2+3\sqrt2$ and $x^2+y^2=6$, find $|x+y+1|$.

$1+\sqrt3$
$2-\sqrt3$
$2+\sqrt3$
$3-\sqrt2$
$3+\sqrt2$

Question 9

Given $y=(x-16)(x-14)(x+14)(x+16)$, find the minimum value of $y$.

$-896$
$-897$
$-898$
$-899$
$-900$

Question 10

How many positive integer solutions $(a,b,c,d)$ satisfy $\frac1a+\frac1b+\frac1c+\frac1d=1$ with $a\lt b\lt c\lt d$?

$6$
$7$
$8$
$9$
$10$

Two models of LCD television are on sale: a “20 inch” standard model and a “20 inch” widescreen model. The ratio of screen length to height is $4:3$ for the standard model and $16:9$ for the widescreen model. Screen size is measured by the diagonal, so both have diagonal $20$ inches. If the ratio of the standard screen area to the widescreen area is $A:300$, find $A$.

Question 12

The diagram shows an overlapping pentagon (regions $A,B$) and rectangle (regions $B,C$). Region $B$ is $3/16$ of the pentagon and $2/9$ of the rectangle. If the ratio of the area of region $A$ to that of region $C$ is $m/n$ in lowest terms, find $m+n$.

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Question 13

2009 students take a test consisting of ten true-or-false questions. Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers.

Question 14

The number of ways to arrange $5$ boys and $6$ girls in a row, allowing girls next to girls but no two boys next to each other, is $6!\,k$. Find $k$.

Question 15

Triangle $ABC$ has $\angle BAC=90^\circ$. Squares are constructed on $AB$ and $BC$ as shown. Square $ABDE$ has area $8\text{ cm}^2$, and square $BCFG$ has area $26\text{ cm}^2$. Find the area of triangle $DBG$ in square centimetres.

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