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A light bulb is hanging in the first floor of the room. There are three switches in the ground floor room. One of these switches belongs to ...
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THE ANSWER: 39 times. 1, 2, 4, 14, 21, 22, 24, 31, 32, 34, 40, 41 (twice), 42 (twice), 43, 44 (twice), 45, 46, 47, 48, 49, 51, 52, 54, 61, 6...
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Snapshot About the Exam The Mathematics Olympiad is organized by NBHM (National Board for Higher Mathematics) and is currently run in associ...
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1. Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red or yellow. If you take any 5 balls of ... -
Start it.. Let X= Y 2X= 2Y Hence 2X- X= 2Y- Y => 2X- 2Y= X- Y => 2(X-Y)= X-Y => 2=1 Now it has been proved that 2=1.
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Find a nine digit numbers, using the numbers 1 to 9, and using each number once without repeats, such that; the first digit is a number divi...
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1 x 8 + 1 = 9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 =98765 123456 x 8 + 6 =987654 1234567 x 8 + 7 =9876543 123...
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1. Two boxes contain 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 two 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 have the same size (radius).
2. For all positive real numbersa, b, cprove that
3. A square sheet of paperABCDis so folded that Bfalls on the mid-point MofCD. Prove
that the crease will divideBCin the ratio 5 : 3.
4. Find the remainder when 21990
is divided by 1990.
5. Pis any point inside a triangle ABC. The perimeter of the triangle AB+BC+CA= 2s.
Prove that
s < AP+BP+CP <2s.
6. Nis a 50 digit number (in the decimal scale). All digits except the 26th digit (from the left)
are 1. IfNis divisible by 13, find the 26th digit.
7. A censusman on duty visited a house which the lady inmates declined to reveal their individual
ages, but said — “we do not mind giving you the sum of the ages of any two 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”. The gave the sums as follows : 30,33,41,58,66,69. The censusman
took these figures and happily went away. How did he calculate the individual ages of the ladies
from these figures.
8. If the circumcenter and centroid of a triangle coincide, prove that the triangle must be equi-
lateral.
Snapshot About the Exam
The Mathematics Olympiad is organized by NBHM (National Board for Higher Mathematics) and is currently run in association with the Homi Bhabha Centre for Science Education, Mumbai. NBHM then selects and trains the Indian team for participation in the International Mathematical Olympiad every year. While NBHM coordinates this Maths Olympiad at the national level, regional bodies (mostly voluntary) play an important role at different stages.
Details
The Indian National Mathematical Olympiad (INMO) examination is conducted in February every year for students of all classes. It is a 4-hour written test that is set centrally and is common throughout the country. It comprises 6 problems to be solved in the areas of algebra, geometry, number theory and combinatorics. 30 students qualifying the exam receive a certificate of merit and are given a month’s mathematics training. Then 6 top students selected on the basis of 5 selection tests get to represent India in the International Mathematical Olympiad.
The Mathematics Olympiad Programme leading to participation in the International Mathematical Olympiad consists of the following stages:
Stage 1
Regional Mathematical Olympiad (RMO) is held in each region normally between September and the first Sunday of December each year. There will be at least one centre in each district of the region. The RMO is a 3-hour written test containing about 6 to 7 problems. Each regional coordinatorhas the freedom to prepare his/her own question paper or to obtain the question paper from NBHM. The regions opting for the NBHM question paper hold this contest on the 1st Sunday of December. On the basis of the performance in RMO, a certain number of students from each region are selected to appear for the second stage.
Stage 2
Indian National Mathematical Olympiad (INMO) INMO is held on the first Sunday of February each year at various Centres in different regions. Only students selected on the basis of RMO from different regions are eligible to appear for INMO. INMO is a 4-hour written test. The question paper is set centrally and is common throughout the country. The top 30-35 performers in INMO receive a certificate of merit.
Stage 3
The INMO certificate awardees are invited to a month long training camp (junior batch) (International Mathematical Olympiad Training Camp (IMOTC) which is conducted in May-June each year. In addition, INMO awardees of the previous year that have satisfactorily gone through postal tuition throughout the year are invited again for a second round of training (senior batch).
Stage 4
International Mathematical Olympiad (IMO)
The team selected at the end of the camp, a "leader" and a "deputy leader," represent India at the IMO that is normally held in July in a different member country of IMO each year. The leader and deputy leader are chosen by NBHM from among mathematics teachers/researchers involved in the Mathematics Olympiad activity. IMO consists of two written tests held on two days with a gap of at least one day. Each test is of four-and-a-half-hours duration. Travel to IMO venue and return takes about two weeks. India has been participating in IMO since 1989. Students of the Indian team who receive gold, silver and bronze medals at IMO receive a cash prize of Rs. 5,000/-, Rs. 4,000/- and Rs. 3,000/- respectively, from NBHM during the following year at a formal ceremony at the end of the training camp.The Ministry of Human Resource Development (MHRD) finances international travel of the eight-member Indian delegation connected with international participation. NBHM finances the entire in-country programme and takes care of other expenditureStudents aiming for selection for participation in IMO should note that RMO is the first essential step for the programme. To appear for RMO, students should get in touch with the RMO co-ordinator of their region well in advance, for enrolment and payment of a nominal fee.
Eligibility
Only students selected on the basis of the RMO (Regional Mathematical Olympiad) from different regions are eligible to appear for INMO. All high school students up to class XII are eligible to appear for RMO
How To Prepare
The syllabus for Mathematics Olympiads (regional, national and international) is pre-degree college mathematics. The topics to be covered are: number systems, arithmetic of integers, geometry, quadratic equations and expressions, trigonometry, co-ordinate geometry, systems of linear equations, permutations and combinations, factorisation of polynomials, inequalities, elementary combinatorics, probability theory, number theory, infinite series, complex numbers and elementary graph theory. Calculus and statistics are not included. The typical areas for problems are: number theory, geometry, algebra and combinatorics. Even though the syllabus is spread over class 9-12 levels, the questions are exceptionally high level in difficulty and sophistication. The difficulty level increases from RMO to INMO to IMO.
Important Dates
The Indian National Mathematical Olympiad (INMO) examination is conducted in February every year for students of all classes. The Regional Mathematical Olympiad (RMO) is held in each region normally between September and the first Sunday of December each yea
Start it..
Let X= Y
2X= 2Y
Hence
2X- X= 2Y- Y
=> 2X- 2Y= X- Y
=> 2(X-Y)= X-Y
=> 2=1
Now it has been proved that 2=1.
Find a nine digit numbers, using the numbers 1 to 9, and using each number once without repeats, such that; the first digit is a number divisible by 1. The first two digits form a number divisible by 2; the first three digits form a number divisible by 3 and so on until we get a nine digit number divisible by 9.
You might try, for example, the number 923,156,784. But this number doesn't work — the first three digit number, 923, is not divisible by 3. Can you find a nine digit number that works?
A light bulb is hanging in the first floor of the room. There are three switches in the ground floor room. One of these switches belongs to that light bulb. The light bulb is not lit and the switches are in off state. There is only one chance to visit the room. How can it be determined which of these switch is connected to the light bulb.
Solution:
First turn ON the first switch and leave it for few minutes. Then turn OFF the first switch and ON the second switch. Now enter the first floor room. If the light bulb is lit, the second switch must be connected to it. If it is not lit, it might the first or the third switch. Now touch the light bulb, it is hot it will be the connected to the first switch. Nor if it is cold, then it should be the third one.
Solution:
First turn ON the first switch and leave it for few minutes. Then turn OFF the first switch and ON the second switch. Now enter the first floor room. If the light bulb is lit, the second switch must be connected to it. If it is not lit, it might the first or the third switch. Now touch the light bulb, it is hot it will be the connected to the first switch. Nor if it is cold, then it should be the third one.
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