Al lives in Alton and Ben lives 12 km away in Benburg. They want to go to Centreville which is 30 km from Alton and 20 km from Benburg. Ben asks Al to take a taxi from Alton to Benburg to pick him up so they can go to Centreville together. The cost of the taxi is 1000 rupiahs per km. Ben pays the part of the fare resulting from the extra distance caused by this detour, and shares equally the remaining cost with Al. How much does Ben save from sharing the taxi fare with Al?
Alice and Brian each have some number of cows. Alice says to Brian: “If I add three times the number of cows you have to mine, then I am satisfied.” Brian replies: “If I add five times the number of cows you have to mine, then I am satisfied.” If the number of cows which makes them satisfied is the same, what is the minimum value of this number?
Initially, a robot faces north. Whenever it stops moving, it automatically faces north. It is programmed to do the following:
(1) Turn $30^\circ$ to the right, move $1$ km forward and stop.
(2) Turn $90^\circ$ to the right, move $1$ km forward and stop.
(3) Turn $150^\circ$ to the right, move $1$ km forward and stop.
(4) Turn $210^\circ$ to the right, move $1$ km forward and stop.
(5) Turn $270^\circ$ to the right, move $1$ km forward and stop.
(6) Turn $330^\circ$ to the right, move $1$ km forward and stop.
What is the distance between the initial and final positions of the robot?
Holly earns 67510 rupiahs per hour while Molly earns 32490 rupiahs per hour. Together, they earned a total of 267510 rupiahs. If Holly worked the number of hours as Molly did and Molly worked the number of hours as Holly did, their combined earnings would have been 232490 rupiahs. How many hours did Holly and Molly worked individually?
Let $ABCD$ be a rectangle such that $AB=25$ cm and $BC=30$ cm. If $M$ is a point on $AD$ such that $\frac{AM}{AD}=\frac13$ and $N$ is a point on diagonal $AC$ such that $\frac{AN}{AC}=\frac35$, what is the area, in cm², of triangle $BMN$?
Let $A,B,C,D$ be four different positive integers such that the sum of any two is divisible by $2$ and the sum of any three is divisible by $3$. What is the minimum value of the sum of $A+B+C+D$?
Let $ABCD$ be a square with side length $10$ cm. Points $E,F,G,H$ are on sides $AB,BC,CD,DA$, respectively, such that $EG$ is parallel to $AD$ and $FH$ is parallel to $AB$. If $P$ is a point on $AE$ such that $PE=2$ cm and $Q$ is a point on $DH$ such that $HQ=3$ cm, what is the area, in cm², of quadrilateral $PFGQ$?
Each of the numbers $1,2,3,4,5,6,7,8,9$ is to be placed into a different square exactly once in the expression below. $$\square+\frac1{\square+\frac1\square}+\square+\frac1{\square+\frac1\square}+\square+\frac1{\square+\frac1\square}.$$ What is the maximum value of this expression?
In triangle $ABC$, it is known that $AD$ and $BE$ are altitudes and $AP$ and $BQ$ are angle bisectors at $A$ and $B$, where $P$ and $Q$ lies on $CD$ and $CE$, respectively. If $\angle PAD=6^\circ$ and $\angle QBE=18^\circ$, what is the angle measure, in degrees, of $\angle BCA$?
A 2014-digit number is the smallest positive integer such that when it is multiplied by 3, every digit of the product is even. How many times does the digit 3 appear in the original number?
There are $18$ people in a party. Each of them shakes hands with at least one other person, and no two people shake hands more than once. If $X$ shakes hands with $Y$, then $X$ does not shake hands with anyone who shakes hands with $Y$. If $X$ does not shake hands with $Y$, then $X$ shakes hands with everyone who shakes hands with $Y$. What is the maximum and minimum number of handshakes that will take place?
A $112$-meter by $75$-meter farm has been divided into $13$ square fields, as shown in the diagram below. The field at the bottom right corner has side length $33$ meters. There is a straight path which serves as a boundary for several fields, and cut across two other fields along the dotted lines. What is the side length, in meters, of each field?
Each of the $36$ cells in a $6\times6$ table is to be filled with numbers $0,1$ or $2$. The sum of the six numbers in each row and each column is then computed. How do you fill the cells to get the maximum number of different values among the $12$ sums?
In the Indonesian village Sevenia, each bill is worth a number of rupiahs equal to a power of $7$, namely, $7^0=1$, $7^2=49$, $7^3=343$, $7^4=2401$, $7^5=16807$, $7^6=117649$, $7^7=823543$ and so on. A millionaire has some bills worth a total of exactly $1000000$ rupiahs, but at most $8$ bills of any kind. No two millionaires have exactly the same combination of bills. What is the maximum number of millionaires in Sevenia?
A magic triangle consists of several circles in a triangular configuration and are filled with consecutive integers starting from $0$. The integer written in a circle that is supported by two other circles below it is the remainder when the sum of the integers in the supporting circles is divided by the number of circles of the triangle. Moreover, the integer in the bottom leftmost circle should be always less than the integer in the bottom rightmost circle. The diagram below shows an example of a triangle with three circles such that the only division $(1+2)\div3$ leaves a remainder of $0$.
(a) Write down all magic triangles with six circles. (2 marks)
(b) Write down all magic triangles with ten circles. (4 marks)
