Number Patterns

This week's Columbus State University's Math contests are about number patterns.

Problem of the Week:
0, -2, -2, 0, 4, 10, ___, ___

Algebra in Action:
1.5, 1.0, 0.83, 0.75, 0.7, ___, ___

Middle School Madness:
1, 4, 9, 61, 52, 63, 94, ___, ___

Elementary Brain Teaser:
2, 5, 11, 17, 23, 31, 41, 47, ____, ____

Algebra in Action's pattern is more challenging than the rest as only about 1/4 of participants got the answer correct. You should be able to get the other 3 answers quickly. Just in case you can't solve the challenging one, check the solution here by highlighting the text -> 1.5 (3/2), 1.0 (4/4), 0.83 (5/6), 0.75 (6/8), 0.7 (7/10), 0.67 (8/12), 0.64 (9/14)

Question from AMC10A 2010

Someone posted a question (#18) from AMC10A on the cbox. It's easier and less messy to explain it here rather than on the cbox:

Qn: Bernardo randomly picks distinct numbers from the set (1,2,3,4,5,6,7,8,9) and arranges them in descending order to form a 3-digit number. Silvia randomly picks distinct numbers from the set (1,2,3,4,5,6,7,8) and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?

Solution:
Consider two scenarios: (a) Bernardo picks a '9' among the 3 numbers that she picks from her set of 9 numbers (b) Bernardo does not pick any '9'

Bernardo picks ‘9’:
When picking 3 numbers from the set (1,2,3,4,5,6,7,8,9), probability that Bernardo would pick a ‘9’ is 3/9 or 1/3. When this happens, Bernardo’s number (which has to be 9**, since the 3-digit number is to be arranged in descending order) will always be greater than Silvia’s number (obvious, since Silvia's greatest possible 3-digit number can only be 876!).

Bernardo does not pick ‘9’:
Number of different 3-digit numbers: 8C3 = 56
Probability that both Bernardo and Silvia pick the same 3-digit number is 1/56
Probability that both Bernardo and Silvia do not pick the same 3-digit number is 55/56
If both Bernardo and Silvia pick different number, probability that Bernardo’s number is bigger than Silvia’s number is ½ (similarly, probability that Bernardo’s number is smaller than Silvia’s number is also ½)

Therefore the probability that Bernardo’s number is larger than Silvia’s number is:
(Bernardo picks ‘9’) 1/3 + (Bernardo does not pick ‘9’) 2/3 x ½ x (55/56)
= 37/56

APMO 2010 (Revealed)

My sister and I took the APMO today. We were told not to reveal anything about it, not the name, not the number of questions, how long is it, whether it is a test or not, and how many questions I did. The APMO is held at RI from 0800 to 1200, which is about geometry, number theory, combinatorics and algebra. We ate alot of food at RI. There are five questions in the test and we can only discuss the APMO 7 days later.

APMO - Asia Pacific Maths Olympiad

today - 9 March 2010

AIME I 2010

This morning I took the AIME I at NUS High Theatrette. I did 13 questions out of 15, but I may have made careless mistakes because I did not have time to check my answers.

I think this year's AIME is about as tough as AMC 12B. AMC 12B is considered tough because there is very little time (75 minutes for 25 questions). For AIME, we were given 3 hrs to do 15 questions. The questions were of course harder than AMC 12, but I did more percentage of questions in AIME than in AMC 12 since I was given more time to solve the problems. I would rather find the product of 2 10-digit numbers than find 100 products of 2 1-digit numbers.

Update on 21 March - answers for AIME I are found here. I think I have made 3 careless mistakes and got only 10 correct answers. I got the hardest question (#15) correct but the easiest question (#1) wrong :-(

**** 2010

My sister and I took the **** today. We were told not to reveal anything about it, not the name, not the number of questions, how long is it, whether it is a test or not, and how many questions * did. The **** is held at ** from **** to ****, which is about ********, ****** ******, ************* and *******. We ate alot of food at **. There are **** questions in the test/not test and we can only discuss the **** * day(s) later.

My sister will decrypt the above message 1 week later.

SIMO training

We did some combi questions today led by our new IMO gold medalist trainer from Bulgaria. His name is Vladimir Barzov, and he has won 3 gold medals in the IMO 2000 to 2002. He has an accent and his handwriting is sometimes hard to read. But at least he taught us, unlike some other trainers who just left us alone to do the pr0blems and then asked one of us to present it.

I heard that this year there will be two rounds of selection tests to select the IMO team, one in April and one in May. In previous years, there was only one selection test in May.

Year 3 Level Briefing to Parents

NUS High principal, Dr Hang, informed parents that there would be two new Major subjects introduced - Computing Studies and Economics. The current Year 3 students would be able to opt for these Majors when they are in Year 5 & 6. I think this is a good initiative by NUS High. At present the Major modules and Major with honours (advanced modules designed at university undergraduate level) are: Maths, Physics, Chemistry, Biology; Major modules only are: Geography, History, Music and Fine Arts. The new initiative provides more options for students who have passion in Computing Studies or Economics to further pursue their dreams.

Dr Hang also shared with us that for the first batch of 88 NUS High graduates ('08), all made it into the university with many having multiple offers from both local and overseas universities. Details on the university offers are here. Among those who have already enrolled in the university (i.e. 88 minus those who are currently serving National Service), about half chose local universities (the majority is predictably with NUS) while the rest opted for an overseas education.

As NUS High is affiliated with NUS, I think it would be good that eventually it can be a through-train from Year 1 to a bachelor's degree at NUS. This would likely make NUS High an even more attractive choice for top PSLE students, as currently IP schools only offer a through-train system from Sec 1 to JC2, and students have to take 'A' levels/IB before applying for the universities. There could also be a more seamless module acceleration not just within NUS High, but leading all the way to a degree course. It would make it easier for high-ability students to get double degrees or even Masters/Doctor's degrees in NUS, within a shorter timeframe.
(posted by LJ's mum)