IMAS 2020–2021 Junior Round 2

If $P=\frac{2021}{2020}-\frac{2020}{2021}$ and $Q=\frac1{2020}\times\frac1{2021}$, then what is the value of $\frac PQ$?

How many non-negative integers $x$ satisfy $|3x+2|\le20$?

In the figure, $ABC$ is a right triangle such that $\angle ACB=90^\circ$, $AC=6$, $CB=8$ and $CDEF$ is a square. If $AF$ meets $DE$ at point $K$, then what is the value of $\frac{DK}{EK}$?

Emily bought some number of fruit candies and milk candies for a total of 100 dollars, where each fruit candy costs 3 dollars while each milk candy costs 8 dollars. It is known that the number of fruit candies bought is less than the number of milk candies bought. How many candies did Emily bought in total?

Given that $x^2=x+1$, which of the following algebraic equations is equal to $x^8$?

Define $(a,b,c,d)=ab-cd$, where $a,b,c,d$ are positive integers. Given that $\frac{(a,3,2,1)}{(3,2,1,a)}=5$, determine the value of $a$.

For any real number $x$, we define $[x]$ to be the greatest integer that does not exceed $x$ and $\{x\}$ to be the fractional part of $x$ such that $\{x\}=x-[x]$. If $2[x]+3\{x\}=\frac{69}{5}$, then what is the value of $x$?

If $xyz>0$ so that $x+y=\frac{xy}{2}$, $y+z=\frac{yz}{3}$ and $z+x=\frac{zx}{4}$, then what is the value of $z$?

Let $a,b,c$ be positive integers such that $a>b>c$. It is known that $b$ is a single digit number, $c$ is a single digit prime number and $a^2+b^2+c^2=2021$. What is the value of $a+b+c$?

In the diagram, fill each of the “○” with numbers so that the sum of the four vertices of all squares that can be found are equal. If the middle number is filled as $\frac nm$, where $m$ and $n$ are both positive numbers that are relatively prime. What is the value of $m+n$?

In the figure below, $ABCD$ and $ADEF$ are both parallelograms and the areas of triangles $ABF$ and $ACF$ are 20 cm² and 50 cm², respectively. What is the area, in cm², of parallelogram $ADEF$?

Let $a,b,c,d$ be positive integers such that $a>b>c>d$ which satisfy the following system of equations: $$\begin{cases}a+b+c+d=2020\\a^2-b^2+c^2-d^2=2020.\end{cases}$$ How many ordered quadruples $(a,b,c,d)$ are there that satisfy the conditions?

Let $\overline{abcd}$ and $\overline{cdab}$ be two 4-digit numbers such that $a,b,c,d$ are four different digits from 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. If $M$ is the greatest common factor of both $\overline{abcd}$ and $\overline{cdab}$, then what is the largest possible value of $M$?

In the figure below, $ABCD$ is a convex quadrilateral such that $M$ is the midpoint of $CD$ and $\angle AMB=135^\circ$. Prove that $AD+BC+\frac{\sqrt2}{2}CD\ge AB$. (A detailed solution is required.)

There exists a positive integer $N$ such that $N+1$ is a multiple of 24. Prove that the sum of all the positive factors of $N$ is also a multiple of 24. (A detailed solution is required.)