IMSO 2021 Mathematics - International Short Answer

In year 2021, Ada's little brother is 10 years old. If Ada's age in 2021 is the sum of all the digits of the year she was born in, how old is she in 2021?

Points $A$, $B$, and $C$ are on a straight line such that $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$, and lines $EC$ and $BF$ intersect at $G$. What is the measure, in degrees, of angle $EGB$?

The twelve-digit number $\overline{155a710b4c16}$ is divisible by $396$, where $a,b,c$ are three distinct digits. What is the value of $a+b+c$?

Each cell of the $3\times4$ grid below contains a positive integer less than $9$ such that the sums of the four numbers in each row are all equal, while the sums of the three numbers in each column are all equal as well. What number must be placed in the gray cell?

A computer store sells four types of laptop ($A,B,C,D$) and has sold a total of $2021$ units. It is known that $20$ type $A$ laptops cost as much as $7$ type $B$ laptops, $4$ type $A$ laptops cost as much as $1$ type $C$ laptop, and $4$ type $A$ laptops cost as much as $3$ type $D$ laptops. If the same amount was collected from each type, how many type $A$ laptops were sold?

In the expression $(e\times7)\oplus f=19$, the symbol $\oplus$ can represent any one of the operations $+,-,\times,\div$, and $e,f$ are both one-digit non-zero positive integers. What is the sum of all possible values of $e+f$?

The first $999$ positive integers are listed on a board. In each turn, Peter selects any two numbers from the list and replaces them with their positive difference. He keeps doing this until only one number is left. What is the largest possible value of this number?

In the diagram below, how many different squares can be formed which contain the unit square with the star inside it?

Cammy listed all possible distinct five-digit positive even integers that can be formed using each of the digits $1,3,4,5,9$ exactly once. What is the sum of all the integers on Cammy's list?

Consider the product of all the positive divisors of $120$: $1\times2\times3\times4\times\cdots\times120$. It is known that this product is divisible by $2^k$ for some integer $k$. What is the largest possible value of $k$?

In the diagram below, Andy used a metal rod to form a regular hexagon whose side $AB$ is $16$ cm long. He then divides the hexagon into two equal parts, using one part to construct a square and the other to construct an isosceles triangle with a base of $18$ cm. What is the sum of the areas, in square centimetres, of the newly created square and triangle?

A regular triangle, a regular quadrilateral, a regular pentagon, and a regular hexagon each have interior angles measured in integer degrees, as shown below. In total, how many kinds of regular polygons have interior angles measured in integer degrees?

The numbers $1$ to $6$ are printed on the six faces of each of seven cubes, and the numbers on any two opposite faces sum to $7$. Arrange the seven cubes as shown below so that the numbers on any two touching faces sum to $8$. What number does the star on the upper face of the topmost cube represent?

Seven points divide a circle into seven equal arcs. How many obtuse triangles can be formed by selecting three of these seven points as vertices?

The positive integers from $1$ to $45$ inclusive are divided into $5$ groups of $9$ numbers each. What is the largest possible average of the medians of these $5$ groups?

Note: The median of a finite set of numbers is the middle number when the numbers are listed from smallest to greatest or vice versa.

Find the smallest positive integer that is divisible by $18$ and has exactly $18$ different positive divisors.

I have a digital clock like the one shown below. During the $12$-hour period from 03:00 AM to 2:59 PM, for how many minutes does the digit $3$ appear at least once?

There are $32$ coins, all with different weights. At least how many weighings on a standard two-pan balance are needed to determine both the heaviest and the second-heaviest coins?

In the diagram below, $ABCD$ is a square and $M$ is the midpoint of $AB$. Line $AC$ intersects $DM$ at $E$. If the area of quadrilateral $BCEM$ is $400$ square centimetres, what is the area, in square centimetres, of square $ABCD$?

In a target-shooting gallery, nine prizes worth $1$ to $9$ dollars are hung on three strings as shown below. The objective is to hit the targets, shown as shaded circles, above the prizes. If a target is hit, the player wins everything below it, and all targets and prizes below it become unavailable for the next player. Alice plays first, followed by Brian and then Colin. Each hits two targets. Alice wins $18$ dollars in total, Brian $13$ dollars, and Colin $14$ dollars. Which targets were hit by Colin?