Question bank SMO 2018

Consider the integer $\overline{30 x 070 y 03}$ where $x,y$ are unknown digits. Find all possible values of $x,y$ so that the given integer is a multiple of $37$.

In $\triangle ABC$, $AB=AC=14\sqrt 2$, $D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$. Suppose $\triangle CDE$ is similar to $\triangle ABC$. Find the length of $BD$.

One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum of the squares of its $4$ smallest factors.

You are given some equilateral triangles and squares, all with side length $1$, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?

Consider a regular cube with side length $2$. Let $A$ and $B$ be two vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1,A_2,\ldots,A_k$ so that $A_1=A$, $A_k=B$ and for any $i=1,\ldots,k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value of $k$.

Let $O$ be a point inside a triangle $ABC$ such that $\angle BOC=90^\circ$ and $\angle BAO=\angle BCO$. Prove that $\angle OMN=90^\circ$, where $M,N$ are the midpoints of $AC$ and $BC$ respectively.

Let $n$ be a positive integer. Show that there exists an integer $m$ such that $$2018 m^2+20182017 m+2017$$ is divisible by $2^n$.

Each of the squares in a $2\times 2018$ grid of squares is to be coloured black or white such that in any $2\times 2$ block, at least one of the 4 squares is white. Let $P$ be the number of ways of colouring the grid. Find the largest $k$ so that $3^k$ divides $P$.

Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c$, $$P(a,b,c)=0\quad\Longleftrightarrow\quad a=b=c.$$ Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $n,m$, $$m^r\mid P(n,n+m,n+2 m).$$

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.