IMSO 2015 — Math Division — Essay

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?