All About Math Olympiad Training & Questions

[Dharma]

mummyloves:

Yeah, keen to know which level is this qn for? :? Not so easy eh...

Hi mummyloves,

Qn is pitched at P6 level. Taken from one of the early SMOPS papers, I think.

Need to have understanding of distance, time and speed relationship. Then use ratios to solve.

[Courage]

Need :?: for the following questions:


(1) If we write whole numbers starting from 1 to form a 1000-digit number N as follows: N=12345678910111213.... what is the unit digit of N?

(2) How many whole numbers from 1 to 2009 contain the digit \"2\"?

(3) Study the following pattern:

1=1: 1(1+1)/2=1
1+2=3; 2(2+1)/2=3
1+2+3=6; 3(3+1)/2=6
1+2+3+4=10; 4(4+1)/2=10
1+2+3+4+5=15; 5(5+1)/2=15

Find the value of 22+24+26+....+200.

TIA.

[Pen88n]

Let me try to get the answers:


1) Between 1 - 99, total # of digits 189 digits
Betweem 100 - 199, total # of digits 300
Between 200 - 299, total # of digits 300
Between 300 - 399, total # of digits 300

So you can see the 1000th digit falls within number "370" and it ends at digit 3.

2) From 1- 100, there are 19 numbers with digit 2 (10 from the "20 - 29" range and 1 each from the "1 - 90 exclude 20+" range)

For 200 - 299, there are 100 numbers with digit 2

From 1 - 1000, there are 100 + 9(19) = 271 numbers with digit 2.

From 1001 - 1999, there are 271 numbers with digit 2

From 2000 - 2009, there are 10 numbers with digit 2

From 1 - 2009, total 271+271+10 = 552 numbers with digit 2

3) 22+24+26+…+200 = 200(200+2)/2 = 20200

Please advise if answers are correct. Thanks

[Courage]

Pen88n, Thanks for your reply.


My working for Q3 is

2(11+12+13+…+100)
2(1+2+3+4+5+…+100)

Using the formula,
1+2+3+4+…+100 is 100(100+1)/2 =5050

thus, 2(5050-55)=9990 since 1+2+3…+10=55

[kiasiparent]

Pen88n:

Let me try to get the answers:


1) Between 1 - 99, total # of digits 189 digits
Betweem 100 - 199, total # of digits 300
Between 200 - 299, total # of digits 300
Between 300 - 399, total # of digits 300

So you can see the 1000th digit falls within number \"370\" and it ends at digit 3.

2) From 1- 100, there are 19 numbers with digit 2 (10 from the \"20 - 29\" range and 1 each from the \"1 - 90 exclude 20+\" range)

For 200 - 299, there are 100 numbers with digit 2

From 1 - 1000, there are 100 + 9(19) = 271 numbers with digit 2.

From 1001 - 1999, there are 271 numbers with digit 2

From 2000 - 2009, there are 10 numbers with digit 2

From 1 - 2009, total 271+271+10 = 552 numbers with digit 2

3) 22+24+26+....+200 = 200(200+2)/2 = 20200

Please advise if answers are correct. Thanks

First 2 are correct
3) 45 pairs x 222 = 9990

[Pen88n]

Thanks, Courage & Kiasiparent.


Finally see \"light\" for Q3.

It's great workout for my brain

[Tang]

Pen88n:

Let me try to get the answers:


1) Between 1 - 99, total # of digits 189 digits
Betweem 100 - 199, total # of digits 300
Between 200 - 299, total # of digits 300
Between 300 - 399, total # of digits 300

So you can see the 1000th digit falls within number \"370\" and it ends at digit 3.

2) From 1- 100, there are 19 numbers with digit 2 (10 from the \"20 - 29\" range and 1 each from the \"1 - 90 exclude 20+\" range)

For 200 - 299, there are 100 numbers with digit 2

From 1 - 1000, there are 100 + 9(19) = 271 numbers with digit 2.

From 1001 - 1999, there are 271 numbers with digit 2

From 2000 - 2009, there are 10 numbers with digit 2

From 1 - 2009, total 271+271+10 = 552 numbers with digit 2

3) 22+24+26+....+200 = 200(200+2)/2 = 20200

Please advise if answers are correct. Thanks

1)
1 to 9 --> 9 digits
10 to 99 --> 2 x 90 = 180 digits
100 to 299 --> 3 x 200 = 600 digits

1000 - 9 - 180 - 600 = 211 digits

211 / 3 = 70 r 1 --> So number from 300 to 369 and 3(70) is incomplete
Hence the digit is 3.

3)
(2 + 4 + .... 200) - (2 + 4 + ....20)
= (2+200)/2 x 100 - (2+20)/2 x 10
= 10100 - 110
= 9990

[mathzgenius]

Courage:

Need :?: for the following questions:


(1) If we write whole numbers starting from 1 to form a 1000-digit number N as follows: N=12345678910111213.... what is the unit digit of N?

(2) How many whole numbers from 1 to 2009 contain the digit \"2\"?

(3) Study the following pattern:

1=1: 1(1+1)/2=1
1+2=3; 2(2+1)/2=3
1+2+3=6; 3(3+1)/2=6
1+2+3+4=10; 4(4+1)/2=10
1+2+3+4+5=15; 5(5+1)/2=15

Find the value of 22+24+26+....+200.

TIA.

My answers for the questions are:

2)355

3)55

Please advise if the answers are correct. I can teach you how to work it out mentally.

* How do you solve Q1??

Please help. I don't quite get your workings. TIA

[Merlion]

> 3)55


How can 22+24+ … +200 equal to 55? No way.
We can simply use Gauss’s method, the answer is 9990.

[Tang]

Merlion:

> 3)55


How can 22+24+ .... +200 equal to 55? No way.
We can simply use Gauss's method, the answer is 9990.

(2 + 4 + ...... + 200) - (2 + 4 + ..... + 20)
= (2 + 200)/2 x 100 - (2 + 20)/2 x 10
= 10100 - 110
= 9990