SMO 2017 Open Round 2 List

Cut off for Rd 2 ~13

SMO 2017 Open Questions

You can find past year SMO papers (2005-2013, Round 1 and 2 with full solutions) here.

SMO 2017 (Open Section) Answers

1. 3

2. 2

3. 4032

4. 11

5. 7

6. 7

7. 2017

8. 10

9. 10

10. 32

11. 43 (assuming solution exists)

12. 7

13. 45

14. 3

15. 1008

16. 2017

17. 4036

18. 35

19. 3

20. 45

21. 6

22. 121

23. 3025

24. 12

25. 28

## 45 comments:

q12 is 14?

q14 I got 2018

q8 i got 12

q22 I got 115

q19 i got 1

q22 i got 106

q17 I got 2018

I think q19 should be 1 too.

why is q19 3 and not 8

q13 should be 60.

Rearranging the given equation, we get a^2 = b(b-c)

If you construct a circle with centre at point A and with radius of c, you can directly apply the above equation to show that BC is tangent to the circle at B (by tangent-secant theorem). From there, x = 60 degrees.

why isn't q8 14?

@PSJH q13 should be 45.

You can use cosine rule and sine rule to show (you should get sinC = 1/sqrt(2)) and C=135 deg is rejected

Q22 I got 121, filled the board with one color, then fit 6 new colors into each row, 6*20+1, the best I can do.

What do you think of this year's paper? Is it easier or more difficult that previous years?

qn 12 I got 17

and an 17 I got 8072? isn't it correct?

Fixing 1st term does not give unique way

23. I got 3027

Apparently you have to mod 20 the numerator if you want to divide by 2.

@PSJH Q13 should be 45 not 60 just draw an accurate diagram

Why do you think will be the cut-off for bronze?

question 19 i got 8

units digits of r = 2017^2017 = 7^2017 = 7^2016 * 7 = 1*7 = 7

ar = 7*8/2 = (2)8

You would need the last 2 digits of 2017^2017 to confirm the units digit. Thus, by finding the last 2 digits, which are 77, you get 77*78/2 = 3003 and 3 is the units digit.

For question 19, you just need 2017^2017 mod 20, which comes out to 17 and is easier to generate. Furthermore, the pattern of a_n repeats after a cycle of 20 numbers.

yea what is the cut off for bronze and hm?

will 9 marks get me anything haha

Bronze for 9 marks maybe. You'll probably need 14-15 marks to get silver and get into second round

Is 10 marks good enough to get something?

By the way, I'm curious: how did you manage to supply all the correct answers within 1 hour after the paper ended anyway?

We can bring home our paper and we probably had our answers recorded and the answers are provided by some math whiz so they should all be correct

q22 i got 134 by adding7*7+7*7+6*6

http://sms.math.nus.edu.sg/Competitions/SMO2017/SMO%20Open%20R2.pdf

The results are out! Anyone knows the cut off point?

I got 10 didn't get in

I got 12, didn't get in. My friend got 13 and got in.

I got 3026...

I got 14 and got in.

I also got 12 and did not get in

9 honorable mention

10-12 bronze

13 silver

How did you know all these results?

Why q4 is 11?

I have know... Sorry

Q11, there is no need to assume solution exists.

The question said that all of a, b, c and d are positive.

You can therefore show all the quantities required during manipulations,

not only exist, but are positive.

Q4) Note that you are integrating the floor function of x. Since floor(x) = 0 for 0 < x < 1, integrating from 0 to 1 gives 0. Similarly, integrating from 1 to 2 gives 1. Integrating floor(x) from 0 to integer a gives 0 + 1 + ... + (a - 1). Solving, a = 11

11) When Lim Jeck says "assuming solution exists", he means that it is one of those "flawed questions" where the scenario is impossible. For more examples of such questions, see SMO(J) 2014 Q26 or SMO(J) 2015 Q35.

The issue here is that if a, b, c, d are all positive and a + b + c + d = 8, then a / (b + c + d) + ... + d / (a + b + c) cannot be equal to 3/10 because the expression is naturally greater than 4/3.

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