SMO 2021 Junior Round 1

Question 1

Let $a$ and $b$ be real numbers satisfying $a\lt 0\lt b$. Which of the following is not true?

$a^2b>0$
$ab^2\lt 0$
$\frac ab>0$
$b-a>0$
$|a-b|>0$

The following diagram shows a system of balances hanging from the ceiling with three types of weights. The balances tip down to the heavier side. If we use $\square\lt \triangle$ to represent $\square$ is lighter than $\triangle$, which of the following is true?

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$\square\lt \circ\lt \triangle$
$\square\lt \triangle\lt \circ$
$\triangle\lt \square\lt \circ$
$\triangle\lt \circ\lt \square$
$\circ\lt \square\lt \triangle$

Question 3

Let $x=2^{20}\cdot3^5$, $y=2^5\cdot5^{10}$ and $z=7^{10}$. Which of the following is true?

$x>y>z$
$x>z>y$
$y>z>x$
$y>x>z$
$z>x>y$

Question 4

In the diagram, six circles are tangent to each other. If the radius of the largest circle is $1$ and the radii of the four medium sized circles are equal, what is the radius of the smallest circle?

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$\sqrt2-1$
$3-2\sqrt2$
$2-\sqrt2$
$6-4\sqrt2$
None of the above

Question 5

Which of the following is closest to the value of $$\frac1{\sqrt2+1}+\frac1{\sqrt3+\sqrt2}+\frac1{\sqrt4+\sqrt3}+\cdots+\frac1{\sqrt{2021}+\sqrt{2020}}\ ?$$

Question 6

Let $x$ be a positive integer. Suppose that the lowest common multiple of $x$ and $14$ is $42$ and the lowest common multiple of $x$ and $33$ is $66$. What is the value of $x$?

Question 7

What are the last four digits of the sum $$1+22+333+4444+\cdots+\underbrace{999999999}_{\text{nine 9s}}\ ?$$ Give your answer as a 4-digit number.

Question 8

How many distinct triples of positive integers $(a,b,c)$ satisfy $1\le a\le b\le c$ and $$\frac1a+\frac1b+\frac1c=1\ ?$$

Question 9

Given five consecutive positive integers, if the product of the largest and the smallest integer is $2021$, what is the sum of the five integers?

Question 10

The numbers from $1$ to $2021$ are concatenated from left to right and the result is read as an integer $$12345678910111213\cdots201920202021.$$ What is the remainder when this number is divided by $6$?

Question 11

In the figure below, each distinct letter represents a unique distinct digit such that the arithmetic holds. If S represents $6$ and E represents $8$, what number does SIX represent?
$$\begin{array}{rrrrrr}&S&E&V&E&N\\&S&E&V&E&N\\+&&&S&I&X\\\hline T&W&E&N&T&Y\end{array}$$

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Question 12

What is the value of $$\sqrt{(219)(220)(221)(222)+1}\ ?$$

Question 13

Let $A,B,\ldots,I$ be unknowns satisfying
$$\begin{aligned}A+B+C&=1,\\B+C+D&=2,\\C+D+E&=3,\\D+E+F&=4,\\E+F+G&=5,\\F+G+H&=6,\\G+H+I&=7.\end{aligned}$$
What is the value of $A+E+I$?

Question 14

If $x$ is a 3-digit number, we define $M(x)$ and $m(x)$ respectively as the largest and smallest positive number that can be formed by rearranging the three digits of $x$. For example, if $x=123$, then $M(123)=321$ and $m(123)=123$. If $y=898$, then $M(898)=988$ and $m(898)=889$.

Given that $z$ is a 3-digit number that satisfies $z=M(z)-m(z)$, what is the value of $z$?

Question 15

How many integers $k$ are there such that the quadratic equation $$kx^2+20x+20-k=0$$ has only integer solutions?

Question 16

In the following diagram, $ABCD$ is a quadrilateral inscribed in a circle with centre $O$. If $|AB|=|BC|=6$, $|AD|=14$ and $CD$ is a diameter, what is the length of $|CD|$?

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Question 17

The diagram below shows a piece of cardboard in the shape of an equilateral triangle with side length $36$ cm. Six perpendicular cuts of length $2\sqrt3$ cm are made to remove the corners in order to fold the cardboard into a tray whose base is an equilateral triangle and height is $2\sqrt3$ cm. What is the volume of the tray in cm$^3$?

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