IMSO 2017 — Mathematics — Short Answer

Find the smallest positive integer $n$ such that the three numbers: $n-96$, $n$ and $n+96$ are all positive prime numbers.

The diagram below shows five circles touching each other at points $B$, $C$, $D$ and $E$. Points $A$ and $B$ are two different points on the leftmost circle. Points $E$ and $F$ are two different points on the rightmost circle. An ant wants to crawl from $A$ to $F$ along part of the circumference of the circles. If the ant is only allowed to crawl along any parts of the circumference of the circles at most once, how many possible paths can the ant take to crawl from $A$ to $F$?

Alex spent a total of \$119 in 7 days. Each day, he spent \$4 more than the previous day. How much did Alex spend on the 6th day?

A sequence of numbers is added in the following way:
For example, for the sequence $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,\ldots\}$, all the numbers in the sequence are first written as decimals as shown below and then added up together.

$$\begin{array}{r}0.1\\0.02\\0.003\\0.0004\\0.00005\\0.000006\\0.0000007\\0.00000008\\0.000000009\\0.0000000010\\0.00000000011\\0.000000000012\\+\quad\vdots\\\hline0.123456790123\ldots\end{array}$$

What is the result when the sequence $\{9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117,\ldots\}$ is added up in the same way?

A positive integer is lucky if each digit in its base-ten representation, starting from the leftmost digit, is not more than the digit on its right. For example, 79, 335 and 679 are lucky numbers but 41, 523 and 786 are not. A super-lucky number is a lucky number with 8 in its ones place and whose square is also lucky. Find the smallest super-lucky number.

Harry, Larry and Parry were each given some marbles on Monday.

On Tuesday, Harry gave Larry and Parry some marbles so that Larry and Parry each ended up with 4 times their original number of marbles.

On Wednesday, Larry gave Harry and Parry some marbles so that Harry and Parry each ended up with 3 times the number of marbles they had on Tuesday.

On Thursday, Parry gave Harry and Larry some marbles so that Harry and Larry each ended up with twice the number of marbles they had on Wednesday.

If Harry, Larry and Parry ended up with 48 marbles each on Thursday, how many marbles did Parry have on Monday?

Form 9-digit numbers using each of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once. Let $A$ and $B$ be two such numbers with $B=8A$. Find the sum of the digits of the number $A+B$.

Straight lines are used to divide a square into identical regions as shown in the right diagram. How many triangles are there in the diagram?

A sculptor wanted to carve a statue from a piece of marble. On the first week, the sculptor carved out 35% of the original piece of marble. On the second week, he carved out 20% of what was left, and on the third week, he carved out 25% of the remaining block to complete the statue. The weight of the completed statue is 48.75 kg. What was the weight, in kg, of the original piece of marble?

Find the greatest 10-digit positive integer that is divisible by 36 and in which each of the digits 0 to 9 appears exactly once.

Jack wants to climb up a 26-step staircase. He is only allowed to take 2 or 4 steps for each move. In how many ways can he climb up to the top of the staircase with exactly 8 moves?

What number should be subtracted from the numerator of the fraction $\frac{537}{463}$ and added to the denominator so that the resulting fraction is equal to $\frac19$?

In the diagram below, $H$, $I$, $J$, ..., $Q$ and $R$ are 11 points on the circle such that $HI=IJ=JK=KL=LM$ and $NO=OP=PQ=QR$. If $HM$ is the diameter of the circle, find the total measure, in degrees, of $\angle HRI$, $\angle IQJ$, $\angle JPK$, $\angle KOL$ and $\angle LNM$.

A palindrome is a number which remains the same when its digits are written in the reverse order. For example, 131 is a palindrome. A car’s odometer reads 16961 km. How much further should the car travel in kilometers before the odometer reads the next palindrome?

If $\overline{abcdef}$ represents a 6-digit number, where $a$, $b$, $c$, $d$, $e$ and $f$ denote distinct digits, what is the largest possible value of $\overline{abc}+\overline{bcd}+\overline{cde}+\overline{def}$?

Alvin and Charles both walk from Town $Q$ to Town $M$ at a speed of 4.8 km/h and 5.4 km/h respectively. Bob cycles from Town $M$ to Town $Q$ at a speed of 10.8 km/h. If all three of them start at the same time, Alvin will meet Bob 5 minutes after Charles meets Bob. How many minutes will Bob take to travel from Town $M$ to Town $Q$?

In the diagram below, square $AEJH$ and square $FCGL$ overlap to form the shaded square $KJIL$. The area of $KJIL$ is $\frac14$ the area of $FCGL$ and $\frac49$ the area of $AEJH$. What fraction of the square $ABCD$ is shaded?

In the diagram below, $D$ is a point on the line $AC$ such that $BD$ is perpendicular to $AC$. Point $E$ is on the line $BC$ such that $AB$ is parallel to $DE$. The area of triangle $CDE$ is 9 $\mathrm{cm}^2$. Given that $BD=8$ cm and $DC=6$ cm, find the length, in cm, of $AD$.

$A$, $B$ and $C$ are positive integers. The sum of 160 and the square of $A$ is equal to the sum of 5 and the square of $B$. The sum of 320 and the square of $A$ is equal to the sum of 5 and the square of $C$. Find the positive integer $A$.

Eight positive integers are arranged in a row as shown below.
$$2017,\ a,\ b,\ c,\ d,\ e,\ f,\ g$$
Starting from $b$, each number is the average of the previous two numbers. If the difference between $f$ and $g$ is 1, what is the smallest possible value of $a$?