IMAS 2020–2021 Junior Division — First Round

Given that $|x-2019|+(y+2020)^2=0$, what is the numerical value of $(x+y)^y$?

A square, with side length 6 cm, is divided into 6 congruent small rectangles as shown in the figure. What is the length of the diagonal, in cm, of each small rectangle?

It is known that the sum of a positive integer and 4 is a multiple of 3. If the sum of the same positive integer and 3 is a multiple of 7, what is the smallest possible value of this positive integer?

In the figure, the area of triangle $ABC$ is 2020 cm$^2$. If points $D$, $E$ and $F$ are on sides $BC$, $BA$ and $AC$, respectively, such that $CDEF$ is a parallelogram and its area is $\frac{50}{101}$ of triangle $ABC$. What is the area, in cm$^2$, of the shaded region?

There are three boxes that are painted red, yellow and blue respectively. We are to put 2 identical apples and 2 identical pears into these three boxes such that no two identical fruit can be placed into the same box. If empty boxes are allowed, how many different placement methods can we have?

In the figure below, it is given that $AB\parallel DE$, $AB=AC$, $CD=DE$, $\angle B=80^\circ$ and $\angle E=70^\circ$. What is the measure, in degrees, of $\angle ACD$?

Let $x$ and $y$ be real numbers such that $|x|=5$, $|y|=7$ and $x\lt y$. What is the value of the algebraic expression $x^2+xy-y^2$?

Define the operation $a*b=\frac{a-2020}{b+2020}$. If $(a*2020)+(2020*a)=1$, what is the value of $a$?

In the figure below, $ABC$ is an equilateral triangle with side length 1 cm, where a straight line intersects $AB$, $BC$ and the extension of $AC$ at points $D$, $E$ and $F$, respectively. If $BD=CE=CF$, what is the length, in cm, of $BD$?

If the cost to buy 2 pens, 3 ballpens and 5 notebooks in a stationery store is 44 dollars and the cost to buy 3 pens, 7 ballpens and 10 notebooks is 81 dollars. How much does it cost to buy 1 pen, 3 ballpens and 4 notebooks?

Given two positive integers such that $\frac47$ of one number is equivalent to $\frac25$ of the other. What is the minimum possible sum of these two numbers?

Andy used some number of regular pentagonal cardboards of the same size to form a solid which is shown on the diagram below. How many edges does this solid have?

There are 43 pearls with a hole in the center that are to be stringed together using a thread such that the thread will form the letters "I", "M", "A", and "S", as shown in the figure below. If each pearl can only be passed through once by the thread, how many from the four letters cannot be done?

John wants to know the weight of a bottle of fruit juice using a two-pan balance having the following items, namely a Teddy Bear doll, a Snoopy doll, a Mickey Mouse doll and 5 kg weight as follows:
- If John places the Teddy Bear doll and the bottle of juice on the left plate, and places both the Snoopy and Mickey Mouse dolls on the right plate, it will be balanced.
- If John places the Snoopy doll, Mickey Mouse doll and the bottle of juice on the left plate, and places the Teddy Bear doll and the 5 kg weight on the right plate, it will be balanced.
What is the weight, in kg, of the bottle of juice?

Given that $a$, $b$ and $c$ are positive integers, which among the following values may the expression $(a+b+c)(a+b-c)(a-b+c)(-a+b+c)$ take?

Using the numbers $1,2,\ldots,2020$, let the sum of all the numbers that are divisible by 3 be $X$ and the sum of all the numbers that are divisible by 2 but not by 3 be $Y$. What is the value of $X-Y$?

What is the product of all real numbers $x$ that satisfies the expression $(|x-2|-|x-6|)(|x-6|-|x-12|)(|x-12|-|x-2|)=0$?

How many pairs of positive integers $(a,b)$, where $a\lt b$, are there which satisfies the expression $\frac1a+\frac1b=0.3$?

In the figure on the right, $ABCDE$ is a pentagon where $\angle BAC=\angle CAD=\angle DAE=40^\circ$, $AC\perp BD$ and $AD\perp CE$. Extend sides $BC$ and $ED$ to meet at point $F$. What is the measure, in degrees, of $\angle BFE$?

Given a four-digit integer $\overline{abcd}$ and a positive integer $k$ which satisfies $\overline{abcd}=k(a+b+c+d)^2$, what is the minimum value of $k$?