SMO 2018 Junior Round 1

Which of the five numbers $\frac 9{11}$, $\frac{19}{22}$, $\frac{29}{33}$, $\frac{39}{44}$ and $\frac{49}{55}$ has the largest value?

Find the value of $$\begin{aligned}&\left(\frac 12+\frac 13+\frac 14+\cdots+\frac 1{2018}\right)\left(1+\frac 12+\frac 13+\frac 14+\cdots+\frac 1{2017}\right)\\&-\left(1+\frac 12+\frac 13+\frac 14+\cdots+\frac 1{2018}\right)\left(\frac 12+\frac 13+\frac 14+\cdots+\frac 1{2017}\right).\end{aligned}$$

Given that $ABCD$ is a right-angled trapezium with $AB=BC$, $\angle ABC=\angle BCD=90^\circ$ and $E$ is a point on $BC$ such that $AE=AD$. If $AD=10$ and $BE=6$, find the length of $DE$.



Original choices: A. $\sqrt 2$; B. $\sqrt 3$; C. $2$; D. $2\sqrt 2$; E. $2\sqrt 3$.

In a strange island, there are only two types of inhabitants: truth-tellers who only tell the truth and liars who only tell lies. One day, you meet two such inhabitants A and B. A said: “At least one of us is a liar.” B kept silent. Which of the following must be true?

Suppose $ax+5=0$ is satisfied by some x, where $-5\lt x\lt 5$. Which one of the following conditions describes completely the range of a?

There are 24 identical containers, each with a capacity of 12 litres. The containers contain some amount of water. If half of the containers are half full, a quarter of the containers are a quarter full, one-sixth of the containers are one-sixth full and one-twelfth of the containers are one-twelfth full, find the total amount of water (in litres) that are in the containers.

A quadruple $(a,b,c,d)$ of positive integers is balanced if the mean, median and mode of a,b,c,d are equal. How many balanced quadruples $(a,b,c,d)$ of positive integers are there that satisfy $a\le b\le c\le d$ and $a+b+c+d=44$?

In the following diagram, $ED$ is a diameter of the circle. $EC$, $CO$ and $OD$ are respective diameters of the three semi-circles. If $EC=12$, $CO=4$, $OD=16$ and the area of the shaded region is $k\pi$, what is the value of k?

If $m^2-5 m-300=0$, find the value of $$m^3-325 m+1503.$$

In Round 1 of a game, three numbers 2,3 and 5 are written around a circle. In Round 2, the sum of every two adjacent numbers with respect to the circle is written between them. The following diagram shows the numbers after Rounds 1 and 2. Find the sum of all the numbers written around the circle after Round 5.

In the following diagram, if $AD=2$, $AC=6$, $\angle BAC=\angle BCA=\angle DEC=\angle CDE$, find the value of $AB^2$.

A business man bought p identical mobile phones at a total cost of m dollars, where p is a prime and m is a positive integer. He sold 2 of the mobile phones each at half of the cost price to a charitable organisation and sold the rest of the mobile phones in his shop at a profit of 200 dollars per phone. If his total profit from selling all these mobile phones was 1800 dollars, find the least possible value of m.

Consider the $x$-$y$ plane, where we adopt the convention that the positive x-direction is towards the right and the positive y-direction is upwards. An ant starts from the origin and crawls in the following manner:2 units left, followed by 3 units up, followed by 4 units right, followed by 2 units down. The ant then continues to move in the same pattern until it has moved a total of 2018 units. If the ant ends on the coordinate $(m,n)$, what is the value of $m+n$?

A three-digit positive integer has the following properties: (i) if its tens and ones digits are swapped, the integer would increase by 36; (ii) if its hundreds and ones digits are swapped, the integer would decrease by 495. Find the three-digit integer.

Let $ABCD$ be a rectangular sheet of paper with $AB=12$ and $BC=24$. If we fold the sheet of paper along the diagonal $AC$, there will be a overlapping region as shown in the diagram. Find the area of this overlapping region.

If $a-b=3+\sqrt 8$ and $b-c=3-\sqrt 8$, find the value of $$a^2+b^2+c^2-ab-bc-ac.$$

A teacher organised an outing for her students. The outing costs T dollars, regardless of how many students participated. This cost will be shared equally among the participants. If all her students were to participate, each student would pay a whole number of dollars. After the teacher announced how much each student would pay, there were 4 students who decided to withdraw. Without these four students, each of the remaining students would still pay a whole number of dollars. However, on the day of the outing, another 2 students were sick and absent. As these 2 sick students did not participate, each student who participated in the outing had to fork out an additional 3 dollars. If the teacher has between 24 and 40 students (inclusive), find T, the total cost of the outing in dollars.

In the following diagram, $\triangle AOB$ is an isosceles right-angled triangle with $OA=OB=40$. Point D lies on the side OB such that $OD:DB=3:1$. Point E is a point on AD such that OE is perpendicular to AD and OE extended meets AB at C. If $[OAC]$ denotes the area of $\triangle OAC$, find the value of $7[OAC]$.

Find the value of $$\sqrt{9^2+19}+\sqrt{19^2+39}+\sqrt{29^2+59}+\sqrt{39^2+79}+\cdots+\sqrt{639^2+1279}.$$

If x is a positive real number that satisfies $x^2=\sqrt 3$, evaluate $$\frac 1{x-1}-\frac 1{x+1}-\frac 2{x^2+1}.$$