SEAMO 2020 — Paper D — Grade 7–8

Which number is largest?

If $|x-10|+x-10=0$, find the range of x.

How many triangles are in the figure?

Form all distinct three-digit numbers using digits one, two, three and four without repetition. What is their average?

Find the sum of coefficients of $(x^2+2x+3)^3(3x^2+2x+1)$.

How many of $1/99,2/99,3/99,\ldots,98/99$ are in lowest terms?

Find the area of square ABCD from the perpendicular staircase lengths shown.

A teacher asks for the LCM of $4875,4563,N$, with N a positive integer. A student reads the first number as $4275$ but obtains the correct LCM. Find the least N.

Evaluate $\sqrt{31\times30\times29\times28+1}$.

How many terms of $20,22,24,\ldots,2020$ are divisible by three but not five?

An arithmetic progression has common difference one and $a_1+\cdots+a_{98}=137$. Find $a_2+a_4+\cdots+a_{98}$.

Positive reals a,b satisfy $1/a+1/b=5/(a-b)$. Evaluate $(a/b+b/a)^2$.

Evaluate $\sqrt{1+2\sqrt{1+2\sqrt{1+2\sqrt{1+\cdots}}}}$.

How many integers between zero and $10^4$ have digit sum five?

Two circles centered at M,N are tangent to line l at A,B. Given $AB=25,MN=24$ and $[ABNM]=300$, find the product of their radii.

Arrange all five-digit numbers made from digits one through five without repetition in ascending order. Find the hundredth number.

Count positive integers $n\le2020$ satisfying $\lfloor n/2\rfloor+\lfloor n/3\rfloor+\lfloor n/6\rfloor=n$.

A circle is inscribed in a triangle of sides three, four and five. Find its radius.

For $1\le x\le10$, let $f(x)=|x-2|-|x-3|+|2x-12|$. Find the sum of its minimum and maximum values.

How many positive integers $n\lt 100$ make $(n^2+4n+6)/(n+3)$ a fraction in lowest terms?