# Tear the question paper, then throw it away

After the APMOPS, here are some questions that I translated:

1. A magician made a six-digit number $$A$$, and the digit sum of $$A$$ be $$B$$. The magician called out a spectactor to evaluate $$A$$-$$B$$. The spectactor said out 5 numbers, namely $$0,2,4,6$$ and $$8$$.The magician successfully revealed the last number. What number is it? Explain your reason.

*Bonus question: Find the minimum and maximum value for $$A$$.

1. The eight vertexes of an octagon are attached with circles, each required to fill up the numbers from 1 to 8. Can the sum of four consecutive attaching circles be:

a)larger than 16?

b) larger than 17?

If possible, find a way of doing so; if not, explain your reason.

1. Wong cycled from station A to station B. Buses form station A and B each give out a bus at the same interval time (e.g. when station A gives out a bus every 30 minutes, station B does the same), but at different times. Every 6 minutes Wong meets up with a bus coming from the opposite, and every 9 minutes he is overtaken by a bus traveling at the same direction with him. It is known that all the buses from stations A & B take 50 minutes to travel to the other side ( it means that buses from station A travel 50 minutes to station B, and vice versa ). How long does it take for Wong to travel from station A to B?

4.In the figure below, 3 different heights of the triangle move from the bases to the vertexes of the triangle. P is a point in the triangle such that another 3 lines move from point P to the triangle, causing each of the lines to be parallel with AD, BE and CF respectively. If AD=2010 cm, BE= 2013 cm, CF = 2016 cm and PR = 1005 cm, PS= 671 cm, find the length of PT. (Oh, and I forgot, each line extending to the base is straight, or 90 degrees)

Feel free to discuss! Enjoy!

Note by Bryan Lee Shi Yang
3 years, 9 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Q1: The last number is $$7$$. $$A$$ can be expressed as $$(10^5\times a+10^4\times b+10^3\times c+10^2\times d+10\times e+f)$$ where $$a,b,c,d,e$$ and $$f$$ are the digits of $$A$$.

$$B$$ can be expressed as $$(a+b+c+d+e+f)$$.

Now, $$A-B=99999a+9999b+999c+99d+9e$$.

$$A-B=9(11111a+1111b+111c+11d+e)$$ implying that $$A-B$$ is a multiple of $$9$$.

Since, $$A-B$$ is a multiple of $$9$$, the digit sum of $$A-B$$ must also be divisible by $$9$$ (Divisibility Test of $$9$$).

Now, we are provided with $$5$$ digits of $$A-B$$ and we have to find the sixth one. The sum of these $$5$$ digits comes out to be $$20$$ and so for $$A-B$$ to be divisible by $$9$$, the sixth digit should be $$7$$ so that it sums out to be a multiple of $$9$$.

NOTE: There is a trick here. If the spectator would have called out $$2,4,6,7$$ and $$8$$ and then we were to find out the sixth digit, we could have $$2$$ different answers namely $$0$$ and $$9$$. Can you figure it out why?

- 3 years, 9 months ago

Because 2+4+6+7+8 IS a multiple of 9

- 3 years, 9 months ago

Yes, it has to be a multiple of $$9$$. This is true for all numbers and can be generalized easily for $$n$$-digit numbers.

- 3 years, 9 months ago