IMAS 2022–2023 — Junior Division — Round 2

What is the total number of positive divisors of the expression $12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$?

Let $a,b,c$ be real numbers such that $a+b+c=0$ and $abc=1$. What is the value of $a^3+b^3+c^3$?

A book contains 213 problems from 12 years of competitions. At first there were 25 problems per year, then it became 16 problems per year, and finally 20 per year. What was the number of years when there were 25 problems?

A circular pizza is sliced by using two perpendicular cuts, of lengths $16$ cm and $x+6$ cm, as shown in the diagram below. Alice ate the two white pieces and Bob ate the two shaded pieces. Who ate more pizza? What is the difference between their areas, in $\mathrm{cm}^2$?

In a certain class, more than 40%, but less than 45% of the students are girls. If one of the boys decided to leave the class, the percentage of girls would still be under 45%, but if a new girl joined the class, the percentage of girls would be higher than 45%. What is the smallest possible number of students in the class?

What is the coefficient of $x^2$ in the product $(1-x)(1+2x)(1-2^2x)(1+2^3x)(1-2^4x)(1+2^5x)(1-2^6x)(1+2^7x)$?

What is the minimum possible surface area, in $\mathrm{cm}^2$, of a rectangular box having a volume strictly greater than $1000\ \mathrm{cm}^3$ if each side length, in cm, is an integer?

Let $A$ and $B$ be disjoint subsets of $\{1,2,\ldots,100\}$ of equal size, where $n$ belongs to $A$ and $2n+2$ belongs to $B$. What is the maximum number of elements in $A\cup B$?

Initially, there is a rook on each of the $36$ squares of a $6\times6$ chessboard. In each move, one may take away any rook which attacks an odd number of other rooks still on the chessboard. What is the maximum number of rooks that can be removed? (Note: The rook can move any number of spaces horizontally or vertically.)

Determine the largest integer $n$ such that $4^{27}+4^{100}+4^n$ is the square of an integer.

A small movie theater has 30 seats; their numbers are indicated on the tickets. The first 21 people who come to a show take 21 seats without paying attention to the seat numbers. Edmond is the 21st person to arrive. But the remaining 9 ticket holders are perfectionists. If any of them finds his or her assigned seat occupied, the person sitting there is evicted; that person then looks for his or her proper seat and evicts the occupier; and so forth. This migration ends with the spectator whose assigned seat is unoccupied. Find the probability, in percentage, that Edmond will not have to change his seat. Enter only the numerical percentage, without the percent sign.

Three students write on the blackboard three 2-digit squares next to each other. At the end they observe that the 6-digit number obtained is also a square. Find the sum of all possible 6-digit numbers.

As shown in the figure below, triangle $ABC$ is isosceles with $AB=AC$. It is given that $D$ is the foot of the altitude from vertex $A$ and $E$ lies on side $AB$ such that $\angle ACE=\angle ECB=18^\circ$. If $AD=24$ cm, find the length, in cm, of $CE$.

A Mathematics test consists of 4 problems, where each problem is being graded independently with integer points between 0 and 10. How many ways can the total points for this test be exactly 31? Show a detailed solution.

Andy has to pass through an underground maze as in the diagram below. Each square means a room. Andy can pass through a dashed segment to enter the adjacent room. If there is no segment between adjacent rooms, there is an open door between them. Andy will move from 1 to 6 rooms by rolling a die. The enchantment on the maze makes Andy drop one, two or three rows if he lands on a room with a door. For example, if Andy rolls a 2 from room S, he would drop to the second row; if Andy rolls a 2 from room X, he would drop to the third row.

Andy is required to move from left to right on odd rows and from right to left on even rows. At the end of a row, the enchantment requires Andy to move down to the next row. For example, when Andy rolls a 5 from room X, he would enter the second row at the fourth move and drop to the third row at the fifth move.



As Andy arrives in room F, he stops. What is the minimum number of dice rolls required to let Andy move from room S to room F? Explain your reasoning in detail and draw the route map.

Route-map template: