Math Olympiad – Singapore Maths Tuition

US First Place in International Math Olympiad! China Drops to Second (Related to Banning of Math Olympiad?)

Congratulations to USA for their First Position in the IMO, a position traditionally held by China! China has been holding the 1st position in the IMO for 21 years.

News: Indian-Origin Students Help US Win Math Olympiad After 21 Years

Indian-Origin Students Help US Win Math Olympiad After 21 Years

Washington: Two Indian-origin students have helped the US win the prestigious International Mathematical
Olympiad after more than two decades.

Shyam Narayanan, 17, and Yang Liu Patil, 18, were part of the six-member US team that won the renowned award after a gap of 21 years. India was ranked 37th in the competition.

Some provinces in China, e.g. Beijing, Chengdu have curiously banned Math Olympiad, and one may wonder does this have an effect in China’s drop in ranking? Having a state wide ban on Math Olympiad would have the result of lowering the number of students taking Math Olympiad, shrinking the talent pool, as well as giving Math Olympiad a stigma and a bad reputation. Students in China are known to be very talented in Math Olympiad, but with such a severe ban, they may forgo Math Olympiad altogether.

Source: http://english.cri.cn/7146/2012/01/20/2702s677257.htm

In 2005, the Ministry of Education issued a regulation forbidding state-run primary and junior middle schools from offering Olympic math courses. It later cancelled the policy of including Olympic math on school entrance examinations. Likewise in 2010, the ministry cancelled a regulation that the winners of Mathematics Olympiads could be recommended for admission to junior middle schools to remove some of the heavy study burden from students.

The Chengdu government has achieved a huge success since it issued harsh regulations banning Olympic math training in 2009. Local authorities prohibit state-run schoolteachers from working part-time to teach Olympic math and have removed school headmasters who give weight to Olympic math performance in student admissions.

For most primary students in Chengdu, this came as a huge relief.

“I feel like a caged bird been set free.”

“I have more time to do physical exercises and have fun. And I can cultivate my own hobbies.”

The students’ parents were also relieved.

This banning of Math Olympiad is indeed very harmful. Instead, Math Olympiad should be made optional so that students who are interested in it can still participate in it, while students who are not interested can learn something else. Banning learning, Math Olympiad, or tuition simply does not make sense! Countries who are interested in promoting STEM (Science, Technology, Engineering, Math) should be actively promoting Math Olympiad instead of banning it. Hopefully China may reverse its ban for Math Olympiad, as China’s huge talent pool and surplus of brilliant students makes it naturally easy to get 1st in Math Olympiad, provided students are given an incentive to pursue Math Olympiad.

Singapore Math Olympiad Results

Source: https://www.imo-official.org/team_r.aspx?code=SGP&year=2015

Singapore did relatively well also, with one gold, 4 silver, and 1 bronze, with an overall 10th position. Congratulations Singapore!

To learn more about Math Olympiad, read my earlier blog post on Recommended Maths Olympiad Books for Self Learning.

Count Down: The Race for Beautiful Solutions at the International Mathematical Olympiad

Each summer, hundreds of seemingly average teens from around the world gather for the International Mathematical Olympiad, a chance to race the clock and one another in the quest for elegant mathematical solutions. In Count Down, the National Book Award finalist Steve Olson sets out to crack the secret of what makes these students such nimble problem solvers. He follows the six U.S. contestants from their free-time games of Ultimate Frisbee to the high-pressure rounds of the competition. In each he finds a potent mix of inspiration, insight, competitiveness, talent, creativity, experience, and, perhaps most important, an enduring sense of wonder. As he observes the Olympians, Olson delves into common questions about math culture and education, exploring why many American students dread geometry, why so few girls pursue competitive math, and whether each of us might have a bit of genius waiting to be nurtured.

Permutation Math Olympiad Question (Challenging)

March’s Problem of the Month was a tough one on permutations. Only six people solved it! (Site: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm)

The question goes as follows:

In each step one can choose two indices $1\leq k,l\leq 100$ and transform the 100 tuple $(a_1, \cdots, a_k, \cdots, a_l, \cdots, a_{100})$ into the 100 tuple $(a_1, \cdots, \frac{a_k}{2}, \cdots, a_l+\frac{a_k}{2}, \cdots, a_{100})$ if $a_k$ is an even number. We say that a permutation $(a_1, \cdots, a_{100})$ of $(1, 2, \cdots, 100)$ is good if starting from $(1,2,\cdots, 100)$ one can obtain it after finite number of steps. Find the total number of distinct good permutations of $(1, 2, \cdots, 100)$ .

The official solution is beautiful and uses induction.

Personally, I used a more brute force technique to get the same answer using equivalence class theory which I learnt in my first year of undergraduate math! It is not so bad in this question, since n is only 100, but for higher values of n the approach in the official solution would be better.

If you are looking for recommended Math Olympiad books, check out this page. In particular, if you are looking for more Math Olympiad challenges, do check out this book Mathematical Olympiad Challenges. In fact, any book by Titu Andreescu is highly recommended as he is the legendary IMO (International Math Olympiad) coach that led the USA team to a perfect score!

NUS High Selection Test (DSA)

Official Website: http://www.nushigh.edu.sg/admission-n-outreach/admissions/eligibility-n-admissions-process

The official website, unfortunately, doesn’t tell much about how the NUS High Selection Test / DSA is like, in particular the format of the exam.

However, from online sources from students who took the test, we can have a glimpse of what the NUS High Selection Test (DSA) is like.

Disclaimer: I have not taken the NUS High Selection Test (DSA) before, and I am only listing down suggested format of the tests based on the online sources. I have taken the GEP Selection Test (both round 1 and round 2) though, at Primary 3.

Source 1: http://wwwdontmesswith6a.blogspot.sg/2011/06/nus-high-selection-test.html

This is a highly reliable blog post by the sister of Lim Jeck, a highly skilled Math Olympiad Participant who has achieved perfect score at IMO. From the blog post, we can tell that:

The Math Paper is 1 and a half hours.
Math Paper is “ok” (easier than NMOS) Do take note that the blogger is very good at math, so “easy” is subjective.
Math Paper has 7 pages, inclusive of cover page and last page.
23 Non-MCQ questions, where you have to shade the integer answer. (Do bring a pencil!)
“The first few Math questions are easy, like P6 Math questions. One of the easiest Math questions is, the average of 3 numbers is given, you add another 2 numbers and you get another given average, you have to find the sum of the 2 numbers added. There are varying marks for different questions. I think the harder questions carry 4 marks.”
(Again, easy is subjective, what is easy for a Olympiad Gold Medalist may not be easy at all)
“Total marks for Maths and Science are 55 and 30 respectively.For Maths, max of answer shades is 4, so max answer may be 9999. Maths questions carry 1 mark, 2 marks, 3 marks and 4 marks. Think Q23 (last qn) is a 4-mark question.”
(We can assume that due to the format of this test, all answers are integers!)

Read more at http://wwwdontmesswith6a.blogspot.sg/2011/06/nus-high-selection-test.html to get an idea of the original post and how the Science NUS High DSA (supposedly more difficult than the Math NUS High DSA) is like.

To deal with difficult NUS High DSA problems (last few questions of the Selection Test), most likely the student has to be trained in Math Olympiad. A book like The Art of Problem Solving Volume 1: The Basics AND Basics Solution Manual (2 Volume Set would be ideal in beginning the journey in Math Olympiad. Note that Math Olympiad is nothing like normal school math, and even a fresh university graduate in a math-related major say Engineering/Accounting would have great problems solving a Primary 6 Math Olympiad question, if he doesn’t have the necessary Math Olympiad background!

If you are also interested in preparing for GEP (Primary 3 or Secondary 1 intake), do check out my most popular page on Recommended Books for GEP.

Other blogs with info on the NUS High DSA Selection Test:

Update (2016): Check out this Pattern Recognition (Visual Discrimination) book that is a guided tutorial for training for GEP / DSA Tests!

Solution to HP A4 Printer Paper Mysterious Question

A while ago, I posted the HP A4 Paper Mysterious Question which goes like this:

Problem of the Week

Suppose $f$ is a function from positive integers to positive integers satisfying $f(1)=1$ , $f(2n)=f(n)$ , and $f(2n+1)=f(2n)+1$ , for all positive integers $n$ .

Find the maximum of $f(n)$ when $n$ is greater than or equal to 1 and less than or equal to 1994.

So far no one seems to have solved the question on the internet yet!

I have given it a try, and will post the solution below!

If you are interested in Math Olympiad, it is a good idea to invest in a good book to learn more tips and tricks about Math Olympiad. One excellent Math Olympiad author is Titu Andreescu, trainer of the USA IMO team. His book 104 Number Theory Problems: From the Training of the USA IMO Team is highly recommended for training specifically on Number Theory Olympiad questions, one of the most arcane and mysterious fields of mathematics. He does write on other Math Olympiad subjects too, like Combinatorics, so do check it out by clicking the link above, and looking at the Amazon suggested books.

Now, to the solution of the Mysterious HP A4 Paper Question:

We will solve the problem in a few steps.

Step 1

First, we will prove that $\boxed{f(2^n-1)=n}$ . We will do this by induction. When $n=1$ , $f(2^1-1)=f(1)=1$ . Suppose $f(2^k-1)=k$ . Then,

$\begin{aligned} f(2^{k+1}-1)&=f(2(2^k)-1)\\ &=f(2(2^k-1)+1)\\ &=f(2(2^k-1))+1\\ &=f(2^k-1)+1\\ &=k+1 \end{aligned}$

Thus, we have proved that $f(2^n-1)=n$ for all integers n.

Step 2

Next, we will prove a little lemma. Let $g(x)=2x+1$ . We will prove, again by induction, that $\boxed{g^n (1)=2^{n+1}-1}$ . Note that $g^n(x)$ means the composition of the function g with itself n times.

Firstly, for the base case, $g^1(1)=2+1=3=2^2-1$ is true. Suppose $g^k (1)=2^{k+1}-1$ is true. Then, $g^{k+1}(1)=2(2^{k+1}-1)+1=2^{k+2}-1$ . Thus, the statement is true.

Step 3

Next, we will prove that if $y<2^n-1$ , then $f(y)<n$ . We will write $y=2^{\alpha_1}x_1$ , where $x_1$ is odd. We have that $x_1<2^{n-\alpha_1}$ .

$\begin{aligned} f(y)&=f(2^{\alpha_1} x_1)\\ &=f(x_1) \end{aligned}$

Since $x_1$ is odd, we have $x_1=2k_1+1$ , where $k_1<2^{n-\alpha_1-1}$ .

Continuing, we have

$\begin{aligned} f(x_1)&=f(2k_1+1)\\ &=f(2k_1)+1\\ &=f(k_1)+1 \end{aligned}$

We will write $k_1=2^{\alpha_2}x_2$ , where $x_2$ is odd. We have $x_2<2^{n-\alpha_1-\alpha_2-1}$ .

$\begin{aligned} f(k_1)+1&=f(2^{\alpha_2}x_2)+1\\ &=f(x_2)+1 \end{aligned}$

where $x_2=2k_2+1$ , and $k_2<2^{n-\alpha_1-\alpha_2-2}$ .

$\begin{aligned} f(x_2)+1&=f(2k_2)+1+1\\ &=f(k_2)+2\\ &=\cdots\\ &=f(k_j)+j \end{aligned}$

where $k_j=1$ , $1=k_j<2^{n-\alpha_1-\alpha_2-\cdots-\alpha_j-j}$ .

Case 1: All the $\alpha_i$ are 0, then $y=2(\cdots 2(k_j)+1=g^j(1)=2^{j+1}-1$ . Then, $j+1<n$ , i.e. $j<n-1$ .

Thus, $f(y)=f(k_j)+j<1+n-1=n$ .

Case 2: Not all the $\alpha_1$ are 0, then, $1=k_j<2^{n-\alpha_1-\alpha_2-\cdots-\alpha_j-j}\leq 2^{n-j-1}$ . We have $2^0=1<2^{n-j-1}$ , thus, $0<n-j-1$ , which means that $j<n-1$ . Thus, $f(y)=f(k_j)+j<1+n-1=n$ .

Step 4 (Conclusion)

Using Step 1, we have $f(1023)=f(2^{10}-1)=10$ , $f(2047)=f(2^{11}-1)=11$ . Using Step 3, we guarantee that if $y<2047$ , then $f(y)<11$ . Thus, the maximum value of f(n) is 10.

Ans: 10

Is there a set of 2015 consecutive positive integers containing exactly 15 prime numbers?

Check out this intriguing Math Olympiad Number Theory question: Is there a set of 2015 consecutive positive integers containing exactly 15 prime numbers?

For instance, the number of primes in the set {1,2,3,…,2014,2015} is 305. This can be computed by entering $\pi (2015)$ in WolframAlpha.

The solution to this problem can be obtained at: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm (February 2015 Problem of the Month)

To learn more about techniques for Math Olympiad style questions (including Number Theory and more), check out this book Mathematical Olympiad Treasures by noted author Titu Andreescu. Truly a treasure trove of useful tips and techniques.

Math Olympiad Question: Sum of powers of two

Is there a number that is a sum of two powers of 2, and ends with 3 zeros?

The answer is yes!

For example, $2^3+2^{253}$ =14474011154664524427946373126085988481658748083205070504932198000989141205000

How about n zeroes?

Check out last month’s problem of the month at: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm

Question: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1409q.pdf

Solution: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1410a.pdf

Featured book:

104 Number Theory Problems: From the Training of the USA IMO Team

This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and in mathematical research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas in writing to explain how they conceive problems, what conjectures they make, and what conclusions they reach. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory.

HCI Confession Page Math Joke

Source: https://www.facebook.com/HwaChongConfessions

“Today I asked the girl I like on FB to help me do math prob
9x-7i>3(3x-7u)
9x-7i>9x-21u
-7i>-3(7u)
-i>-3u
=
i<3u

but she go put the ans as 3u>i ruining my whole plan T.T”
-HCJC Student (M)

Math Girls

Math isn’t hard. Love is.
Currently in its eighteenth printing in Japan, this best-selling novel is available in English at last. Combining mathematical rigor with light romance, Math Girls is a unique introduction to advanced mathematics, delivered through the eyes of three students as they learn to deal with problems seldom found in textbooks. Math Girls has something for everyone, from advanced high school students to math majors and educators.

Praise for Math Girls!

“…the type of book that might inspire teens to realize how much interesting mathematics there is in the world—not just the material that is forced upon them for some standardized test.” “Recommended”
—CHOICE: Current Reviews for Academic Libraries

“Imagine the improbable: high-school students getting together on their own — not in a Math Club or Math Circle, not in preparation for any Math Olympiad or “regular” test, not on the advice of any of their teachers, not as part of any organized program — to talk about pure math, math more interesting than the math found in their textbooks. The three students in this book do that for the sheer love of it. That to me is the beauty and fascination of this novel for young people, mostly young people interested in math.”
—Marion Cohen, Arcadia University, MAA Reviews

“Sometimes the math goes over your head—or at least my head. But that hardly matters. The focus here is the joy of learning, which the book conveys with aplomb.”
—Daniel Pink, NYT and WSJ best-selling author of Drive and A Whole New Mind

“if you have a…teenager who’s really into math, this is a really interesting choice”
—Carol Zall, Public Radio International, The World

“Math Girls provides a fun and engaging way to learn and review mathematical concepts…the characters’ joy as they explore and discover new and old ideas is infectious.” —review, “Experiments in Manga” blog

Reviews from amazon.co.jp

“As a physics major, math has always been a painful tool to use and nothing more. But Math Girls changed the way I look at mathematics. Now I actually find it interesting!”
— “Au”

“Math Girls is a fun read, but I was surprised to find that it’s also a serious math book chock full of careful explanations. I hope that people who think they don’t like math will read it. Even when the formulas go over your head, just following the story gives you a great feel for how fun math can be.”
— “Nyanta”

“I got hooked on this book during summer vacation, and had a great time reading it by the pool. It was so good that I read it twice, the second time while working out the problems on the hotel stationary.”
— “Kei0210”

“Advanced math, explained in a playful way. But it’s not just a textbook, with dry solutions to problems. It’s a bittersweet story, with mathematics telling part of the tale. A brilliant comparison between the uncertainties of youth and the absolute proofs of symbols and numbers.”
— Shiori Oguchi

Are Math competitions good?

Source: http://www.dnaindia.com/academy/report-do-math-competitions-inspire-students-to-gain-proficiency-in-the-subject-1966588

Check out the above website to see the pros and cons of Math competitions, and whether they inspire students to be better at Math.

The most important is to enjoy doing Math, as Math is fun!

Math Olympiad Contest Problems for Elementary and Middle Schools, Vol. 1