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KVPY and AMTI Second Level Test: Inter Level

This note has been used to help create the KVPY Exam Preparation wiki

Last weekend was incredibly busy, as I, along with many others wrote two competitive exams: the AMTI Inter NMTC, Second Level, and the KVPY

First, the AMTI Second Level

This is the paper:

  1. a) If \(x\) and \(y\) are positive reals such that \(x^{2014}+y^{2014}=1\), prove that \[(\displaystyle\sum^{1007}_{k=1}\frac{1+x^{2k}}{1+x^{4k}})(\displaystyle\sum^{1007}_{k=1}\frac{1+y^{2k}}{1+y^{4k}})<\frac{1}{(1-x)(1-y)}\]

    b) The angles of a triangle are in arithmetical progression. The altitudes of this triangle are also in arithmetical progression. Show that the triangle is equilateral.

  2. \(ABC\) is an acute angle triangle in which the three sides are unequal. \(L,M,N\) are respectively the midpoints of the sides \(BC,CA,AB\). The perpendicular bisectors of the sides \(AB,AC\) intersect \(AL\) at \(D,E\) respectively. \(BD, CE\) cut at \(F\) inside the triangle. Show that \(A,M,N,F\) are concyclic.

  3. \(S\) is a set of all positive integers, such that for any two distinct members of \(S\), \(a\) and \(b\), \(\frac{a+b}{GCD(a,b)}\) is also a member of \(S\). Find and describe all such finite sets.

  4. a) Find all real number triples \((x,y,z)\) which satisfy \[3(x^2+y^2+z^2)=1, x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\]

    Ab) Show that all the numbers \(1\) to \(16\) can be written in a line, but not in a circle, so that the sum of any two adjacent numbers is a perfect square.

  5. A certain number is the product of three different prime factors, the sum of whose squares is \(2331\). There are \(7560\) number (including unity) which are less than the number and prime to it. The sum of all its divisors (including unity and the number) is \(10560\). Find the number.

  6. A rectangular parallelepiped is given, such that its intersection with a plane is a regular hexagon. Prove that the rectangular parallelepiped is a cube

  7. A triangle \(ABC\) is given. The midpoints of the sides \(AC\) and \(AB\) are \(B_1\) and \(C_1\) respectively. The in centre of the \(\triangle ABC\) is \(I\). The lines \(B_1I\) and \(C_1I\) meet the sides \(AB,AC\) at \(C_2,B_2\) respectively. If the areas of \(\triangle ABC\) and \(\triangle AB_2C_2\) are equal, find the measure of the angle \(BAC\)

  8. If \((a+b)\) and \((a-b)\) are relatively prime integers for some natural numbers \(a,b\), find the greatest common divisor of \(2a+(1+2a)(a^2-b^2)\) and \(2a(a^2+2a-b^2)(a^2-b^2)\)

I would like to know how other Brilliant people who qualified did in their levels

The other exam was the KVPY

I found the exam rather tough. How did you do?

After checking the answer key, I am getting around 45. So, unfortunately, I will not be qualifying.

But how did you do? Post a comment about your KVPY experience or AMTI experience.

Note by Nanayaranaraknas Vahdam
2 years, 10 months ago

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For the third question, is the answer {a,a(a-1)} where a runs from 3 to infinity Ronq Vader · 2 years, 10 months ago

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Anyone know what is the use of NMTC round 2? Ronq Vader · 2 years, 10 months ago

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@Ronq Vader No use. Only Rank is given+Certificate of Merit/Excellence Krishna Ar · 2 years, 9 months ago

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For the 4th apply muirhead Y Kousthubh · 1 month ago

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4(B) For the numbers to be arranged in a circle all the numbers must have at least to other distinct numbers from the set so that their sum is a perfect square. But here 8 has only 1 partner which is 1 and 16 has only one partner which is 9. So these to need to be at the terminal points of the line and so this set can't be arranged into a circle. Kushagra Sahni · 1 year, 6 months ago

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For how much was the test conducted? Naren Ezhil · 1 year, 11 months ago

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@Naren Ezhil 10 crores Kushagra Sahni · 1 year, 6 months ago

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Very useful Arun Kumar · 1 year, 11 months ago

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The KVPY shortlist and the AMTI second level results have come out. I have qualified in both. I had also seen @Aditya Raut on both lists Nanayaranaraknas Vahdam · 2 years, 9 months ago

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@Nanayaranaraknas Vahdam Yeah.I saw, this is pretty cool!. How many did you solve in NMTC? Your rank is awesome. Krishna Ar · 2 years, 9 months ago

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@Krishna Ar How did you do RMO? Nanayaranaraknas Vahdam · 2 years, 9 months ago

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@Krishna Ar I think I attempted all of them, and got 6 right Nanayaranaraknas Vahdam · 2 years, 9 months ago

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@Nanayaranaraknas Vahdam Woa!!!!!! 6/8 right!!! still 15th AIR :/ . RMO , I didnt give. I gave GMO rather. Look into my note! How was your RMO paper? I hear it was kind of mixed (easy on the whole , yet some questions were tough) Krishna Ar · 2 years, 9 months ago

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@Krishna Ar It was quite easy Nanayaranaraknas Vahdam · 2 years, 9 months ago

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@Nanayaranaraknas Vahdam How many? Krishna Ar · 2 years, 9 months ago

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@Krishna Ar Probably 5 and a half on 6 Nanayaranaraknas Vahdam · 2 years, 9 months ago

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@Nanayaranaraknas Vahdam I bow to thee. Krishna Ar · 2 years, 9 months ago

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@Nanayaranaraknas Vahdam Never mind, I found it in google search Aditya Raut · 2 years, 9 months ago

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@Aditya Raut What region are from, for RMO? Nanayaranaraknas Vahdam · 2 years, 9 months ago

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@Nanayaranaraknas Vahdam Whoa ! I didn't know this!!! Thank you very much for letting me know. What's the website? Aditya Raut · 2 years, 9 months ago

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What do you think the cutoffs will be? I'm probably getting ~40. Haven't really checked. Went horribly. They made us sit in benches meant for 4th grade students. EDIT:- Did you take the 11th one or the 12th one? I took the 11th one. Siddhartha Srivastava · 2 years, 10 months ago

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@Siddhartha Srivastava Wait, are you a 11th grader? :O Krishna Ar · 2 years, 10 months ago

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@Krishna Ar Yes. Why the :O? Siddhartha Srivastava · 2 years, 10 months ago

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@Siddhartha Srivastava I thought you weren't one and was thinking that you worked in some Asdds too :P Krishna Ar · 2 years, 10 months ago

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@Krishna Ar "Don't believe everything you read on the Internet." - Abraham Lincoln. Siddhartha Srivastava · 2 years, 10 months ago

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@Siddhartha Srivastava -_- (I hope you know what this is for) Krishna Ar · 2 years, 10 months ago

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@Krishna Ar What do you mean? Siddhartha Srivastava · 2 years, 10 months ago

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@Siddhartha Srivastava I took the 11th one as well.

Last year's cutoff was 53 or so. I think they are reducing the cutoff this year. Maybe it will be 50. Nanayaranaraknas Vahdam · 2 years, 10 months ago

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@Nanayaranaraknas Vahdam I am hoping they reduce the cutoff to 40. xD Everyone I know said their paper went horrible. Can't find anyone who actually said their paper went well. Siddhartha Srivastava · 2 years, 10 months ago

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@Siddhartha Srivastava The highest anyone I know got is 56, which is above the cutoff Nanayaranaraknas Vahdam · 2 years, 10 months ago

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AMTI was tough. Krishna Ar · 2 years, 10 months ago

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@Krishna Ar You don't say! The KVPY Math was waaay tougher than previous years. But maybe to counter that, the Biology section was strictly of 10th level. Oh and BTW, ask me no questions and I shall tell you no lies! B.S.Bharath Sai Guhan · 2 years, 10 months ago

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@B.S.Bharath Sai Guhan Agreed. The 2 marks questions were fine though the Geometry in 1 markers put me off. BTW, why no questions-no lies ? :P And...how much are you expecting? (Was this why, no question-no lies? _ Krishna Ar · 2 years, 10 months ago

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@Krishna Ar Well I gave the exam in 11th and I did horribly bad scoring just 52. Actually there are a couple of mistakes in the answer key and I think I could have scored 65 if I hadn't been so nervous. Soham Dibyachintan · 2 years, 10 months ago

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@Krishna Ar What was the Junior paper like? Nanayaranaraknas Vahdam · 2 years, 10 months ago

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@Nanayaranaraknas Vahdam Not too hard, Not too trivial. Lots of lengthy problems with one being improperly worded. It definitely wasnt a good experience. Maybe you can look at Siddhart G's set which consists of few problems from the paper. How many did you solve? Krishna Ar · 2 years, 10 months ago

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@Krishna Ar @krishna ..you in 11th ? Gaurav Sharma · 2 years, 10 months ago

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@Gaurav Sharma No, 10th. Krishna Ar · 2 years, 10 months ago

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@Krishna Ar then how did u appear fr KVPY ? Isn't it supposed to be fr 11th nd 12 th standard students ? Gaurav Sharma · 2 years, 10 months ago

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@Gaurav Sharma I just saw the paper :P Krishna Ar · 2 years, 10 months ago

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