IMAS 2019–2020 Junior Division — First Round

What is the simplified value of $(2019-2020)^{2020}+(2018-2019)^{2019}$?

The figure below shows $AB=BC=CD$, $AC=BD$ and $\angle A=80^\circ$. What is the size of $\angle ACD$, in degrees?

In triangle $ABC$, $\angle A$ is the right angle. The line through $A$ perpendicular to $BC$ intersects $BC$ at $D$. Lines through $D$ intersect $AB$ and $AC$ perpendicularly at $E$ and $F$, respectively. What is the total number of right triangles in the figure?

One third of an acute angle and its complementary angle sums up to $60^\circ$, what is the supplementary angle of this acute angle, in degrees?

There is an integer $n$, such that $2020\times\left(\frac{10}{101}\right)^n$ is an integer. How many possible $n$’s are there?

The school restaurant sells three kinds of lunch boxes, the prices of which are 15 dollars, 13 dollars and 11 dollars, respectively. On a certain day, the total income of the restaurant is 2019 dollars. Which of the following could be the possible number of lunch boxes sold on that day?

In an exam, 25 students obtained excellent scores in Language. The number of students with excellent scores in math is 2 more than that in the Language. The students with both excellent scores in Math and Language scores are one third of all students. Every student obtained at least one excellent score. How many students are there in total?

Define the operation $a*b=\frac{a+b+2019}{a-b+2019}$. If $a*2019=7$, what is the value of $a$?

Non-negative integer $x$ satisfies $|2x+5|\le15\times(1+x+x^2+\cdots+x^{2019})^0$. What is the sum of all such $x$’s?

In a triangle $ABC$, $AB=AC$. Line segments $AD$ and $AE$ trisect $\angle BAC$, as shown in the figure. If $\angle ADB=110^\circ$, what is $\angle ABC$ in degree?

When $x=5$, $mx^5+nx^3+kx+2$ is 100. What is the value of $mx^5+nx^3+kx+2$ when $x=-5$?

In the figure below, the distance between two adjacent vertices is 1 cm. What is the area of the shaded portion of the figure, in cm$^2$?

Each square in the figure below is filled with a positive integer so that the sum of each cross (five squares) is divisible by 3.

Cross shape:



What is the minimum value of $A+B+C+D+E$?

Ten-digit number $\overline{A20192020B}$ is divisible by 9, where $A$, $B$ are digits. If the quotient has the ten-thousands digit 0, what is the value of $|A-B|$?

Let $x$ be the integers such that $\frac{21}{2+x}$ is an integer. How many such $x$’s are there?

Given that $|a|=3$, $|b|=5$, $|c|=7$, $|d|=11$, what is the least possible value of the expression $ab+ac+ad+bc+bd+cd$?

In rectangle $ABCD$, $AB=\sqrt3$ cm, $BC=3$ cm. Put a right triangle $DCE$ outside $ABCD$ and along edge $CD$ and $\angle CDE=30^\circ$. Flip along $AC$ such that $B$ lands at $K$. Join $KA$, $KC$, $KE$, $KD$, as in the figure. What is the area of the shaded region, in cm$^2$?

Given that $x=\sqrt{19-8\sqrt3}$, which of the following equalities holds true?

Mickey has a large gridded paper. Each small square has side of length 1 cm. He draws a circle with a center at some vertex of a square with radius 6 cm. How many small whole squares are inside the circle?

Suppose $a,b,c,d,e,f,g,h,i,j$ is an arrangement of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 with different letters representing different numbers. At most how many of the ten fractions $\frac ab,\frac bc,\frac cd,\frac de,\frac ef,\frac fg,\frac gh,\frac hi,\frac ij,\frac ja$ can be reduced to integer values?