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Question bank SMO 2013

Exam papers

110 questions

Question 1

If $a=8^{53}$, $b=16^{41}$ and $c=64^{27}$, which inequality is true?
  1. $a>b>c$
  2. $c>b>a$
  3. $b>a>c$
  4. $b>c>a$
  5. $c>a>b$

Question 2

Real numbers $a,b,c$ satisfy $|a-b|=1$, $|b-c|=1$, $|c-a|=2$ and $abc=60$. Find
$$\frac a{bc}+\frac b{ca}+\frac c{ab}-\frac1a-\frac1b-\frac1c.$$
  1. $1/30$
  2. $1/20$
  3. $1/10$
  4. $1/4$
  5. None of the above.

Question 3

If a complex number $x$ satisfies $x^2+x+1=0$, find $x^{49}+x^{50}+x^{51}+x^{52}+x^{53}$.
  1. -1
  2. $-1/2$
  3. 0
  4. $1/2$
  5. 1

Question 4

In triangle $ABC$, $\angle ACB=36^\circ$, and the interior angle bisectors of $\angle CAB,\angle ABC$ meet at $P$. Find $\angle APB$.

Question illustration
  1. $72^\circ$
  2. $108^\circ$
  3. $126^\circ$
  4. $136^\circ$
  5. None of the above.

Question 5

Find the number of integer pairs $(x,y)$ satisfying $xy-3x+5y=0$.
  1. 1
  2. 2
  3. 4
  4. 8
  5. 16

Question 6

Five young ladies sit around a circular table. Miss Ong sits between Miss Lim and Miss Mak. Ellie sits between Cindy and Miss Nai. Miss Lim sits between Ellie and Amy. Beatrice has Miss Poh on her left and Miss Mak on her right. What is Daisy's surname?
  1. Lim
  2. Mak
  3. Nai
  4. Ong
  5. Poh

Question 7

In triangle $ABC$, $D$ is the midpoint of $AC$, and $E\in CB$ satisfies $CE=2EB$. The lines $AE,BD$ meet at $F$. If $[AFB]=1$ square unit, find $[ABC]$.

Question illustration
  1. 3
  2. $10/3$
  3. $11/3$
  4. 4
  5. 5

Question 8

Square $ABCD$ has side $8$ cm. Point $M$ lies on $CB$ with $CM=2$ cm. If $N$ varies on diagonal $DB$, find the least value of $CN+MN$.

Question illustration
  1. 8
  2. $6\sqrt2$
  3. 10
  4. $8\sqrt2$
  5. 12

Question 9

Rectangle $ABCD$ has diagonals meeting at $O$. Point $E\in AB$ is such that $CE$ bisects $\angle BCD$. If $\angle ACE=15^\circ$, find $\angle BOE$.

Question illustration
  1. $60^\circ$
  2. $65^\circ$
  3. $70^\circ$
  4. $75^\circ$
  5. $80^\circ$

Question 10

Let $S$ be the smallest positive multiple of $15$ having exactly $3k$ digits: $k$ zeros, $k$ threes and $k$ eights. Find the remainder when $S$ is divided by $11$.
  1. 0
  2. 3
  3. 5
  4. 6
  5. 8

Question 12

If the graphs $y=x^2+2ax+6b$ and $y=x^2+2bx+6a$ meet at only one point in the $xy$-plane, what is its $x$-coordinate?

Question 13

Find the number of multiples of $11$ in $99,100,101,102,\ldots,20130$.

Question 14

In the figure, $B,A,D$; $B,C,E$; $A,C,F$; and $D,E,F$ are collinear triples. Given $BA=BC$, $AD=AF$ and $EB=ED$, find $x$, where $\angle BED=x^\circ$.

Question illustration

Question 15

If $a=1.69$, $b=1.73$ and $c=0.48$, find
$$\frac1{a^2-ac-ab+bc}+\frac2{b^2-ab-bc+ac}+\frac1{c^2-ac-bc+ab}.$$

Question 16

Let $x_1,x_2$ be the roots of $(x-2)^2=3(x+5)$. Find $x_1x_2+x_1^2+x_2^2$.

Question 17

Let $ABCD$ be a square, with points $X,Y$ as shown, and $XY=6$, $AX=8$, $AY=10$. Its area is $m/n$ square units, where $m,n$ are coprime positive integers. Find $m+n$.

Question illustration

Question 18

Real numbers $x,y$ satisfy $5x^2+y^2-4xy+24\le10x-1$. Find $x^2+y^2$.

Question 19

A painting job takes Team A alone $2.5$ hours, or Team B alone $75$ minutes. Team A completes a fraction $m/n$ of the job, then Team B immediately takes over. The whole job takes $1.5$ hours. If $m,n$ are coprime positive integers, find $m+n$.

Question 20

Real numbers $a,b,c$ satisfy $\frac{ab}{a+b}=\frac13$, $\frac{bc}{b+c}=\frac14$, $\frac{ca}{c+a}=\frac15$. Find $\frac{24abc}{ab+bc+ca}$.