SOALAN REGIONAL OLYMPHIAD MATEMATHICS (RMO 1990)

RMO means Regional Matemathic Olymphiad, where the participants may come from some counteries in that region. For example, for the region of Asia, the selected participants may come from Malaysia, Singapore, Hong Kong, India and so on. . They are given 3 hours to solve alll the questions given. They are all 7 questions.  By average, they must allocate 45 minutes for each question.

Faham ke tak Bahasa Inggeris di atas???   Anda akan diberikan 7 soalan RMO 1990. Kebiasaannya peserta diberi masa 3 jam untuk menjawabnya atau secara puratanya 45 minit untuk setiap soalan.

However, you and all readers of this blog are given no limit of time to solve the problem. Whats the importance is that you can solve it.

We start with Q1. Do yourselves at home.  If you can solve it, you are the hero of matemathic. Then shift to Q2, Q3 and so on. Just follow the hieracchy of the question. 

Q1. A censusman on duty visited a house which the lady inmates declined to reveal their individual ages, but said, "we don't mind giving you the sum of the ages of any 2 ladies you may choose". Thereupon the censusman said, "in that case, please give me the sum of the ages of every possible pair of you. They gave the sum as follow, 30, 33, 41, 58, 66, 69. The censusman took these figures & hapilly went away. How did he calculate the individual age of these ladies from these figures???  

[Seorang pembanci yang sedang bertugas pergi ke sebuah rumah di mana para gadis di  dalamnya enggan mendedahkan umur mereka masing2, sebaliknya bersetuju memberi jumlah umur bagi mana2 pasangan 2 gadis yang dipilih leh pembanci. Maka kata pembanci, sila berikan jumlah umur bagi mana2 pasangan yang ada. Dia diberi data berikut, 30,33, 41, 58, 66, 69. Dia mengambil data ini & berlalu pergi. Bgaimanakah dia dapat mengetahui umur setiap gadis itu?]

From cikgu zain...the clue i got, there are 4 girls at all. Says that a, b, c & d. Make the simeltanous equation from them. For  example, a+b = 30 and so on. The solve it.  Then you can get the respective value for each lady. Thanks a lot to ahmad Wafi Razali for sharing his idea. Tq    

Now, we go to Q2.  It is quite difficult.

Q2 : A square sheet of paper ABCD is so folded that B falls on the midpoint, M of CD. Prove that the cease will divide BC in the ratio 5 : 3

[Satu kertas empat segi sama ABCD dilipat sebegitu rupa supaya titik B terjatuh ke atas titik tengah CD, iaitu M. Buktikan bahawa garisan lipatan itu akan membahagi BC  dengan nisbah 5 : 3]

How about the next question??? Look at Q3. Very tough.

Q3. For all positive real number a,b,c,  prove that

[Untuk semua nombor nyata positif a,b,c, buktikan bahawa]

      a /  (b+c)   + b / (c + a)  + c / ( a+ b) > = 1.5

Well, i will give you a simple clue for it. For the minumun value, assume that a=b=c=1. For the maximum value, assume that only a =3, but b = c = 1. Put the value of a, b & c in both cases. Then you can prove it.   

 
 Now go to Q4. Look easy but very tricky. . 

Q4.Find the remainder when 2 with the power of 1990 is divided by 1990

[Cari nilai baki bila 2 kuasa 1990 di bahagi dengan 1990]

The answer is 1.99. Those who can answer it, let show me your anwer. You can call me at 0139148154 / write your answer in my fb. Gud luck. Thank you. 

The following question is very difficult. You need a very good imagination.

Q5. 2 boxes contains between them 65 balls of several different sizes. Each ball is white, black, red or yellow.. If you take any 5 balls of the same colour, at least 2 of them will always be of the same size (radius). Prove that there are at least 3 balls which lie in the same box have the same colour and the same radius???

[2 kotak mengandungi sejumlah 65 guli yang berbeza saiznya. Setiap guli berwarna samada putih, hitam. merah atau kuning. Jika anda  mengambil 5 guli yang sama warnanya, sekurang2nya 2 daripada mereka akan mempunyai saiz yang sama. Tunjukkan bahawa terdapat sekurang2nya 3 guli di dalam kotak yang sama di mana mereka mempunyai warna dan saiz yang sama]

Now go to Q6. It is similiar to the past one, Q3. 

Q6.P is any point inside a triangle ABC. The perimeter of the triangle is AB+BC+CA = 2s. Prove that
[P adalah sebarang titik dalam segitiga ABC. Perimeter bagi segitiga tersebut ialah AB+BC+CA =2s. Buktikan bahawa]

                            s < AP + BP + CP < 2s

Q7. N is a 50 digit number (in a decimal scale). All digits except the 26th digit (from the left) are 1. If N is divisible by 13, find the 26th digit.

[N adalah sekumpulan 50 digit nombor (dalam bentok titik perpuluhan). Semua digit kecuali yang ke 26 (bermula dari kiri) adalah 1. Jika N boleh dibahagi dengan 3, cari digit yang ke 26 tersebut?].
 

Disiapkan pada 23.12.2011    3.11 pm
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