Mostly are out of syllabus, like Cauchy-Schwarz Inequality, AM-GM Inequality, but Number Theory(Modular Arithmetic, Diophantine Equation) and Euclidean Plane Geometry do cover on syllabus of STPM further maths...

9th Dec 2013: Updated questions and answers...

OMK 2013 Sulong

Section A

1. A large cube is divided into 99 cubes with integer lengths, 98 of them with side 1. Find the volume of the large cube.

2. Let M be the set of all nine-digit positive integers that contain each digit from 1 to 9 once. Find the highest common factor of all elements of M.

3. What is the remainder when 5^5555 is divided by 10000?

4. Given a trapezium with perpendicular diagonals and height 12. The length of one of its diagonals is 15. Find the area of the trapezium.

5. The digits of 2013 can be rearranged to form an arithmetic progression. Determine the number of four-digit positive integers with this property.
(Note: An arithmetic progression might have common difference 0.)

6. Determine the smallest prime factor of 8051.


Section B

1. Given a triangle ABC. The midpoints of AB, BC, CA are C1, A1, B1 respectively.
Construct another triangle DEF with side lengths equal to the lengths of AA1, BB1, CC1.
(a) Prove that the ratio of the area of triangle DEF to the area of triangle ABC is a constant, regardless of the choice of triangle ABC.
(b) Find the area of triangle DEF if AB = 13, BC = 14, CA = 15.


2. Prove that there exist integers a1, a2, a3, …, a2013, b, all greater than 1, such that
(a1!)(a2!)(a3!)…(a2013!) = b!


3. A sequence x1, x2, x3, … is defined as follows: x1 = 1, x2 = 143, and
xn+1 = 5(x1 + x2 + … + xn)/n for all n ≥ 2.
Prove that all terms of the sequence are integers.


Hints & Answers:


This post has been edited by ystiang: Dec 9 2013, 05:53 PM

P(X < 500) = 0.01

Show basically means prove. Don't forget [shown].

Edited for: wrong info... sorry T_T

This post has been edited by ystiang: Nov 29 2013, 07:56 AM

Try to participate OMK if you intend to study form 6.
OMK considered as very hard, IMO I can't imagine my brain blank 4 hours...

For Malaysia, Top OMK performers are selected to attend the training camps, and the final IMO representatives are selected based on the students' performance in the camps and race. (I think only 6 in the final stage, and 3~4 of them are "bumiputera".

This post has been edited by ystiang: Aug 6 2013, 10:39 PM

You ask the same question to me several months ago... But I think my answer is different now.

Quiet enjoying... especially maths =)

My target: 3.5 or above.

Quit working this sem, and rapidly done all the PBS, focusing on study now.

The only 'tak syok' thing in this sem so far is my monthly test math paper, originally can get perfect score, but so much careless T_T

Since P(Z < z) = 0.01

Since the probability is less than 0.5, z must be negative.

P(Z < -z) = 1 - 0.01 = 0.99

From the normal distribution table, P(Z < 2.326) = 0.99

Thus, P(Z < -2.326) = 0.01

(500-510)/SD = -2.326
-10/SD = -2.326

SD = -10/-2.326 = 4.3g [shown]


This post has been edited by ystiang: Nov 29 2013, 07:55 AM

Bravo. Me too haven't had any formal training but surely you're more success than me.

For Section A, I think I only have one question correct, yup that prime factor.
Actually, 8051 = 8100 - 49 = (90^2) - (7^2) = (90-7)(90+7) = 83*97

Q3 should be easy though but I wrote 625 and the answer is 8125 TvT
Q1, Q4, Q5... just randomly put some answers...

Section B, I hate those prove, prove and prove...

I hope I can. I'm upper six this year, should be last year for me to participate =(

P(X >= 1) = 1 - P(X < 1) = 1 - P(X = 0) = 0.8

Yup, binomial distribution.

|-2/(x+1)| < 1

-1 < -2/(x+1) < 1

0 < -2/(x+1) + 1 or -2/(x+1) - 1 < 0

(x-1)/(x+1) > 0 or -(x+3)/(x+1) < 0

By number line or sign table, you'll get

{x| x<-3 or x>1}



This post has been edited by ystiang: Nov 14 2013, 06:35 PM

Your working don't have any error. Essentially you can verify your answer with calculator.

Trigonometric identities and Complex numbers in exponential form do cover in STPM Mathematics T syllabus.

Others are also in the STPM Further Mathematics syllabus which only a few canditates take for each year.

What they cut in the new syllabus for Maths T are law of sines, law of cosines, deductive geometry... Which are moved to Further Maths.



This post has been edited by ystiang: Dec 13 2013, 03:34 AM

Oh ya, and Heron's formula too~

But the Plane Geometry in further maths is more likely logical reasoning, a lot of 'prove'.
And there are a few new theorems, Apollonius’, Ptolemy’s, Menelaus’ and Ceva’s theorem.

I'm terrible with this chapter... :/

Slowly in progress, so lazy to study in the holiday.
I'm done with kinematics in both one and two dimensions, now studying kinetics and dynamics.

This post has been edited by ystiang: Dec 15 2013, 10:23 PM

A standard way to solve this type of problem in STPM, taking the logarithm at both sides:



Hence, the least value of n is 5.

This post has been edited by ystiang: Dec 25 2013, 05:39 PM

Is it the answer:


Brilliant & Interesting
May I share it out later on my FB?


This post has been edited by ystiang: Dec 31 2013, 06:09 PM

Conditional Probability
The probability that the event A occurs given that event B has occured is
P(A|B) = P(A ∩ B) / P(B)
(denominator is always the condition, notice the word given, if)


Independent events
Two events are said to be independent if the outcome of one cannot influence the outcome of the other in any way.
P(A ∩ B) = P(A) × P(B)
P(A|B) = P(A) or P(B|A) = P(B)

Example: STPM 2013 Mathematics (M) P2
It is known that of the customers buying laptops, 75% buy an antivirus software, 40% buy an additional memory card and 30% buy both.
a) Find the probability that a randomly selected customer does not buy any of those items.
b) Find the probability that a randomly selected customer buys an antivirus software given that an additional memory card is bought.
c) Determine whether the event "an additional memory card is bought" is independent of the event "an antivirus software is bought.

Information given:


Solution:


This post has been edited by ystiang: Jan 16 2014, 05:22 PM

Will that human become disabled or even die with that shooting force? XDD

Btw, been not touching mechanics for a months...

Recently I'm studying discrete mathematics. (Very interesting =D)

Plenty of new theorem and corollary though.

This post has been edited by ystiang: Jan 24 2014, 08:37 PM

Notice that's ln(Q_0 - Q),

Since d/dx [ln (ax+b)] = a/(ax+b)
∫ a/(ax+b) dx = ln |ax+b| + c

In general,
∫ [f'(x) / f(x)] dx = ln |f(x)| + c

Hints/Ex.:
d/dx of (2 - x) = -1

This post has been edited by ystiang: Jan 26 2014, 12:26 AM

Event A is independent of both B and C,
P(A ∩ B) = P(A)P(B) = 0.0002
P(A ∩ C) = P(A)P© = 0.0008

Events B & C are mutually exclusive,
P(B ∩ C) = 0 or
P(B ∪ C) = P(B) + P© = 0.05

P(A or B or C) = P(A ∪ B ∪ C)
=P(A) + P(B) + P© - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) = 0.02 + 0.01 + 0.04 - 0.0002 - 0.0008 - 0 = 0.069

You may use two-way table or Venn diagram to illustrate your problem.

28th Jan 2014: Updated and corrected my wrong solutions...
Sorry for giving the wrong information!

f(x) = (x+1)/(x-1) , x ≠ 1

(f^2)(x) = [(x+1)/(x-1) + 1] / [(x+1)/(x-1) - 1] = x
(f^2)(2) = 2

Since (f^2)(x) = x, (f^-1)(x) = f(x)

You may verify it:


(f^2)(2) = 3(f^-1)(k)
2 = 3[(k+1)/(k-1)]
By cross multiplying, you'll get k = -5.


Q1
A monotone increasing function is always increasing, ie. f(x) is always increasing as x increase. ie. f(a) < f(b) for all a < b. The derivative of monotone increasing function is always positive.


Q2
Notice that a circle equation.

BTW, Critical_Fallacy, can you teach me how to space out in mathURL LaTeX input?

Thanks for Critical_Fallacy for stating my mistakes~

This post has been edited by ystiang: Jan 28 2014, 08:03 PM

Oops, I've drawn the wrong graph.



Just plot the graph with wolfram widgets,
http://www5a.wolframalpha.com/Calculate/MS...df=RangeControl

This post has been edited by ystiang: Jan 27 2014, 08:44 PM