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SMO 2016 Junior Round 1

Question 1

We know that $\mathrm{I}^2=\overline{\mathrm{ME}}$ and $\mathrm{I}^3=\overline{\mathrm{YOU}}$. If each distinct letter represents a unique digit such that the arithmetic holds, what is the value of $\mathrm{E}$?
  1. $4$
  2. $5$
  3. $6$
  4. $8$
  5. $9$

Question 2

Given three integers, we form another three integers by adding the mean of any two of them to the third. If the three new integers formed are 33, 35 and 40, find the mean of the original three integers.
  1. $18$
  2. $27$
  3. $30$
  4. $36$
  5. None of the above

Question 3

Let $ABCD$ be a rectangle, $E$ be a point on $BC$ such that $BE=2EC$, and $F$ be a point on $AE$ such that $AF=3FE$. If the area of $ABCD$ is $1200$, what is the area of the quadrilateral $DFEC$?
  1. $100$
  2. $300$
  3. $350$
  4. $400$
  5. None of the above

Question 4

The diagram shows a square $ABCD$ inscribed in a semicircle with centre $O$ and another square $PQRS$ inscribed in the entire circle with the same centre. If the area of $ABCD$ is $16$, find the area of $PQRS$.

Question illustration
  1. $20$
  2. $24$
  3. $32$
  4. $40$
  5. $48$

Question 5

$ABCD$ is a parallelogram. $E$ is a point on the side $AB$ such that the ratio of the area of the quadrilateral $AECD$ to the area of the triangle $ABC$ is $7:5$. The ratio of $AE:EB$ is

Question illustration
  1. $2:3$
  2. $2:7$
  3. $3:4$
  4. $3:5$
  5. $5:7$

Question 6

In the diagram $D$ is the midpoint of $AB$, $EF=2CF$ and $AF$ is parallel to $DE$. Given the area of triangle $GFC$ is $4$, find the area of triangle $ADG$.

Question illustration
  1. $32$
  2. $36$
  3. $40$
  4. $64$
  5. $76$

Question 7

Which of the following five numbers has the greatest value?
  1. $(10-\pi)(10+\pi)$
  2. $(11-\pi)(9+\pi)$
  3. $(12-\pi)(8+\pi)$
  4. $(13-\pi)(7+\pi)$
  5. $(14-\pi)(6+\pi)$

Question 8

Two real numbers $u$ and $v$ satisfy the following equations respectively: $$2015u^2+2016u+1=0;\qquad v^2+2016v+2015=0.$$ If $uv\ne1$, find the value of $\frac uv$.
  1. $-\frac{2016}{2015}$
  2. $\frac1{2015}$
  3. $\frac1{2016}$
  4. $\frac{2016}{2015}$
  5. $1$

Question 9

Find the minimum value of the real valued function $x+2016-\sqrt{2x-3}$.
  1. $2015$
  2. $2016$
  3. $2017$
  4. $2018$
  5. $2019$

Question 10

The number $123456789101112\ldots$ is formed by writing the whole numbers $1,2,3,\ldots$ until there are 201 digits in the number. Find the remainder when this number is divided by 9.
  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. $5$

Question 11

If the sum and product of two positive real numbers are both equal to 13, find the sum of the squares of these two numbers.

Question 12

Let $x$ be a positive integer. If the highest common factor of $x$ and $168$ is $12$ and the highest common factor of $x$ and $270$ is $18$, find the smallest possible value of $x$.

Question 13

In the diagram, $AB$, $AC$, $CD$ and $DB$ are respectively the diameters of the circles $c_1,c_2,c_3$ and $c_4$. If the circumference of $c_1$ is $2016$, what is the sum of the circumferences of all four circles?

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Question 14

Find the integer closest to $10\sqrt{0.7}+10\sqrt{2.8}$.

Question 15

In the figure below, each distinct letter represents a unique digit such that the arithmetic sum holds. What is the five digit number represented by $\mathrm{SIXTY}$? The displayed sum is $\overline{\mathrm{FORTY}}+\overline{\mathrm{TEN}}+\overline{\mathrm{TEN}}=\overline{\mathrm{SIXTY}}$.

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Question 16

In the diagram, $ABCD$ is a rectangle in which $AB=34$ and $BC=12$. $P$ and $Q$ are points on $AB$ and $CD$ respectively such that $\angle CPQ$ and $\angle PQA$ are right angles. Find the sum of the two possible lengths of side $AP$.

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Question 17

Let $a,b$ and $c$ be positive integers such that $$ab+ac=144,\qquad ab+bc=209,\qquad ac+bc=221.$$ Find the value of $a^2+b^2+c^2$.

Question 18

$P$ and $Q$ are two regular polygons with respectively $n$ and $m$ number of sides. The ratio of the interior angles of $P$ and $Q$ is $4:3$. If $n\gt m$, how many possible pairs of $(n,m)$ are there?

Question 19

It is given that $x$ and $y$ are positive integers such that $x\gt y$ and $$\sqrt{x}+\sqrt{y}=\sqrt{2000}.$$ How many different possible values can $x$ can take?

Question 20

The diagram shows a $12$ by $9$ rectangle which is cut by a pair of parallel line segments into three parts with the equal areas. If $h$ denotes the distance between the two parallel lines, find the value of $30h^2$.

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