Question bank IMSO Maths 2015

How many three-digit positive integers $\overline{abc}$ are there such that $a\leq b\leq c$?

Twenty-four positive numbers are arranged on a circle, where each number is equal to the product of its two neighbors. If the two neighboring numbers are 3 and 4, what is the sum of all twenty-four numbers?

Let $a,b$ and $c$ be different positive integers such that $1=\frac12+\frac13+\frac17+\frac1a+\frac1b+\frac1c$. What is the smallest possible value of $a+b+c$?

There are 100 plates arranged in a circle. Susan, going around the circle, places a candy on every 15th plate. If she keeps on doing it until candies can no longer be placed on an empty plate, how many plates remain empty?

What is the largest integer less than or equal to the expression $$\frac1{\frac1{1985}+\frac1{1986}+\frac1{1987}+\cdots+\frac1{2015}}?$$

There are 81 soldiers lined up in a row with numbers 1 to 81. In each of the rounds, the soldiers call out 1, 2, 3, 1, 2, 3, 1, 2, 3, .... All soldiers who count 1 and 3 are removed from the line. The process continues until only one soldier is left. What is the number of that last soldier left on the line?

The diagram below shows a square $ABCD$ of side $6$ cm. Given that $E$ is the midpoint of $AB$ and points $F$ and $G$ are on $BC$ such that $BF=FG=GC$. What is the total area, in cm², of the shaded region?

In triangle $ABC$, $\angle ACB=45^\circ$ and $BC=24$ cm. The length of the altitude from $A$ to $BC$ is $16$ cm. Points $B^\prime$ and $C^\prime$ are on the line of $BC$ such that $BC=B^\prime C^\prime$. Suppose $AB=A^\prime B^\prime$ and $AC=A^\prime C^\prime$, as shown in the diagram below. If the area of triangle $OCC^\prime$ is $\frac13$ of the area of triangle $ABC$, what is the length, in cm, of $BB^\prime$?

In a four-digit number, the thousands digit is larger than the units digit, which is not zero, while the hundreds digit is larger than the tens digit. A new four-digit number is obtained from the original number by reversing the order of the digits. How many possible differences of the original and new number are there?

There are three fractions, all of which are in lowest terms. The the ratio of their numerators, which are all positive integers, are in the ratio $3:2:4$ while the ratio of their denominators, which are also all positive integers, are in the ratio $5:9:15$. If the sum of these three fractions is $\frac{28}{45}$, what is the sum of their denominators?

There are $12$ points around the $4\times4$ square as shown in the diagram below. How many triangles are there such that the vertices are chosen from those points and at least $1$ interior angle is equal to $45^\circ$?

In triangle $ABC$, points $D$ and $E$ are on $BC$ such that $BD:DE:EC=2:1:1$. If point $M$ is on $AC$ such that $\frac{CM}{MA}=\frac13$ and $BM$ intersects $AD$ and $AE$ at points $H$ and $G$ respectively. Find $BH:HG:GM$.

From a 16 cm by 18 cm piece of paper, a 3 cm by 3 cm square is cut off from each corner as shown in the diagram below. At most how many 3 cm by 4 cm rectangles can be cut off from the remaining part of this piece of paper?

In a $2\times3$ table, the three numbers written in the first row are all positive integers. Now, the first number in the second row is equal to the first number in the first row, the second number in the second row is equal to the product of the first two numbers in the first row and the third number in the second row is equal to the product of all three numbers in the first row. If the sum of the three numbers in the second row is $125$, find all possible values of the sum of the three numbers in the first row.

The diagram shows a $6\times6$ chessboard with $4$ squares containing black circles and $4$ squares containing white circles. Use four different colours to divide the diagram into $4$ identical regions by going along the lines so that each region will be exactly of the same size and shape; and each region contains $1$ black circle and $1$ white circle. The regions may be rotated but not reflected. The circles do not have to be in the same relative positions within the region.

The sum of the ages of three boys is equal to the sum of the ages of three girls, and the product of the boys’ ages is equal to the product of the girls’ ages, too. All six children are of different ages, and all are under 20. Suppose that their ages are all integers, and the age of the second oldest boy is 14, what is the age of the second oldest girl?

The diagram shows nine pieces of puzzles, including one that has the shape that looks like the letter P, and one that looks like the letter U. All nine pieces may be rotated or reflected.



(a) Consider the two diagrams below. The diagram on the left shows a $5\times5$ square with a hole in the shape of the letter U. The one on the right shows the double-sized letter U. Choose five different pieces, including the P-shaped one, to construct each of the diagrams. Find one solution for each diagram.



(b) Consider the other two diagrams below. This time, the diagram on the left shows a $5\times5$ square with a hole in the shape of the letter P. The one on the right shows the double-sized letter P. Choose five different pieces, including all four pieces which are not used in (a) to construct each of the diagrams. Find one solution for each diagram.

The diagram shows nine circles joined by five solid lines and two dashed lines. Fill in the circles using each of the numbers $1,2,3,4,5,6,7,8$ and $9$ exactly once, such that the number in the middle circle of a solid line is equal to the sum of the numbers in the other circles on the line, and the number in the middle circle of a dashed line is equal to the difference of the numbers in the other circles on the line. Find all solutions.

The diagram shows a bird-shaped piece consisting of two unit squares and three half-squares joined edge to edge. It may be rotated or reflected.



The next diagram shows a Greek cross enclosed by four identical bird-shaped pieces. When two pieces come into contact, they must do so along an entire edge or an entire diagonal. Find as many diagrams as you can that can be enclosed by the four bird-shaped pieces.

Anne asks her teacher his age. He replied: ‘My age now is a square number but after my birthday it will be a prime number’. If his age is greater than 20 but less than 65, how old is he now?