IMSO 2014 — Math Division — Short Answer

Calculate $9+99+999+9999+99999+999999+9999999+99999999+999999999$.

The income of a taxi driver is the sum of his regular salary and some tips. If the tips are $\frac54$ of his regular salary, what is the fraction of his total income which comes from his tips?

The base of a larger triangle is two times the altitude of a smaller triangle, while the altitude of the larger triangle is three times the base of the smaller triangle. What is the ratio of the area of the larger triangle to the area of the smaller triangle?

In an election, only $80\%$ of people planned to vote. However, only $85\%$ of those who planned to vote actually voted. What is the percentage of people who actually casted their vote?

Campant A offers an internet service with an initial payment of 300000 rupiahs and a monthly fee of 72000 rupiahs while Company B offers the same internet service with no initial payment but has a monthly fee of 90000 rupiahs. If Johnny prefers Company A, what is the minimum number of months he should subscribe in Company A so that he will pay less than the subscription with Company B?

Twelve identical squares are put together in a $6\times2$ configuration to form a rectangle. If the perimeter of each square is $6$ cm, what is the perimeter, in cm, of the rectangle?

The sum of two 2-digit numbers is also a 2-digit number. What is the maximum value of the product of those three 2-digit numbers?

How many positive integers less than 2014 are there such that the sum of its digits is a multiple of 5?

The diagram below shows two different $3\times4$ pieces of paper, parts of which are shaded. If a student combines both diagrams on a single $3\times4$ piece of paper, what fraction of the paper is now shaded?

If Alice and Brian have a total of 377 cards, Brian and Colin have a total of 685 cards and Colin and Alice have a total of 546 cards, then how many cards does Brian have?

Write down six positive integers whose sum is 100 such that each integer contains at least one digit 6.

Answer entry (editorial guidance): Enter the numbers in ascending order, separated by commas, without units. Include repeated numbers as separate entries.

Let $ABCD$ be a square piece of paper such that $M$ and $N$ are the midpoints of $AB$ and $CD$, respectively. Let $P$ be a point on $AM$ such that if it is folded along $DP$, then $A$ lands on point $Q$ on line segment $MN$. What is the angle measure, in degrees, of $\angle ADP$?

The sum of five different positive integers is 364. If the largest integer is 95, what is the maximum possible value of the smallest among these five integers?

Alice gives $\frac14$ of all her apples to Brian and then gives $\frac13$ of the remaining apples to Colin. If Alice’s remaining apples are worth $13500$ rupiahs, what is the total worth of the apples that Colin received from Alice?

Let $ABCD$ be a rectangle such that $AD=8$ cm and $CD=12$ cm. If point $P$ is on $CD$ such that $DP=AD$ and point $Q$ is on $AD$ such that $DQ=CP$, then what is the area, in cm², of quadrilateral $ABPQ$?

Each pair of five positive integers is added so that there are ten sums: $110,112,113,114,115,116,117,118,120,121$. What is the largest among these five integers?

What is the sum of all multiples of 6 which exactly have ten positive divisors each?

What is the smallest positive integer which leaves a remainder of 3, 4 and 5 when divided by 5, 7 and 9 respectively?

What is the remainder when $1\times2\times3\times\cdots\times14\times15$ is divided by $1+2+3+\cdots+14+15$?

We want to draw some number of straight lines on a chessboard such that the lines must pass through an interior point of all the cells of the board. At least how many lines do we need to draw on a $3\times3$ chessboard?