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RMO 2014 Full Paper Discussion

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Regional Mathematical Olympiad-2014

Time: 3 hours

Instructions:

\(\bullet\) Calculators (in any form) and protractors are not allowed.

\(\bullet\) Rulers and compasses are allowed.

\(\bullet\) Answer all the questions.

\(\bullet\) All questions carry equal marks. Maximum marks: 102.


Questions:-

1 Let ABC be an acute-angled triangle and suppose \(\angle ABC\) is the largest angle of the triangle. Let R be it's circumcentre. Let the circumcircle of triangle ARB cut AC again in X. Prove that RX is perpendicular to BC.


2 Find all real numbers \(x\) and \(y\) such that \[x^2+2y^2+\dfrac{1}{2} \leq x(2y+1)\]


3 Prove that there does not exist any positive integer \(n<2310\), such that \(n(2310-n)\) is a multiple of \(2310\).


4 Find all positive real number triplets \((x,y,z)\) which satisfy \[2x-2y+\dfrac{1}{z}=\dfrac{1}{2014}; 2y-2z+\dfrac{1}{x}=\dfrac{1}{2014}; 2z-2x+\dfrac{1}{y}=\dfrac{1}{2014}\]


5 Let \(ABC\) be a triangle. Let \(X\) be on the segment \(\overline{BC}\), such that \(AB=AX\). Let \(AX\) meet the circumcircle \(\Gamma\) of triangle \(ABC\) again at \(D\). Show that the circumcentre of \(\Delta BDX\) lies on \(\Gamma\).


6 For any natural number \(n\), let \(S(n)\) denote the sum of digits of \(n\). Find the number of all 3-digit numbers \(n\) such that \(S(S(n))=2\).


RMO was held in various centres in India on 7th December. You can take it as a test. Please let me know your score. 17 students will be selected from my state, Rajasthan. Pls Like and Reshare and Enjoy!

Note by Satvik Golechha
2 years, 5 months ago

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(4) Add all three equations.

\(\frac {1}{x}+\frac {1}{y}+\frac {1}{z}=\frac {3}{2014}\)

Multiply the first equation by z, second by x, third by y and add all the three new equations to get \(\frac {x+y+z}{2014}=3 \implies x+y+z=3×2014\).

Thus \((x+y+z)(\frac {1}{x}+\frac {1}{y}+\frac {1}{z})=9\).

But Cauchy-Schwarz inequality tells us that the expression is greater than or equal to 9 with equality iff corresponding terms have the same ratio ie. x=y=z. Hence x=y=z=2014 is the only solution Joel Tan · 2 years, 5 months ago

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@Joel Tan Is it wrong if we open 3/2014 and compare values? Rohan Rajpal · 1 year, 5 months ago

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@Satvik Golechha Proof to the \(3\)rd question:For \[n(2310-n)\] to be a multiple of \(2310,\)both of them have to contribute factors of \(2310.\)Let us take \[2310=n*x\] thus, \(n\) and \(x\) are both factors of \(2310.\)Now,let us replace \(n\) by \[\dfrac{2310}{x}\] simplifying gives \[\dfrac{2310^{2}*(x-1)}{x^{2}}.\]But,\[\dfrac{2310^{2}}{x^{2}}=n^{2}.\]Putting this into the expression gives\[:n^{2}*(x-1)=n^{2}*x-n^{2}.\]But,\[n*x=2310.\]Substituting gives,\[2310*n-n^{2}.\]But,\(n^{2}\) can't be divisible by \(2310\) as in the prime factorisation of \(2310\) all the numbers are distinct,then how can a square exist.Hence proved. Adarsh Kumar · 2 years, 5 months ago

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@Adarsh Kumar This I did by using the fact that 2310 is square free and that NCERT procedure that if p divides n^2, then p divindes n. Thus showing 2310 is a factor of n. And hence n can never be less than it Subrata Saha · 2 years, 5 months ago

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@Subrata Saha I did it like shubham saha (that's his name :P ) . I was really doubtful about this one as I couldn't imagine an RMO question to be of such(easy) level. Salmaan Shahid · 2 years, 5 months ago

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@Subrata Saha Precisely! But do the RMO examiners give the full 17 marks for such a short method? Utkarsh Chaturvedi · 1 year, 5 months ago

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@Subrata Saha awesome method!! Adarsh Kumar · 2 years, 5 months ago

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Proof for 2: The inequality is equivalent to (x - 2y)^2 + (x - 1)^2 <= 0. But a squared number is always greater than or equal to 0, which implies that there is a real number in the equality case, which implies that we solve the equation (x - 2y)^2 + (x - 1)^2 = 0 and this is possible when x = 2y = 1, implying the ordered pair (x, y) = (1, 1/2). John Ashley Capellan · 2 years, 5 months ago

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Problem 5. Easy one. Any point \(V\) on \(\bigcirc ABC\) satisfies

\[\measuredangle BVD=\measuredangle BAD=\measuredangle BAX=2\measuredangle BXA=2\measuredangle BXD\]

so \(\bigcirc ABC\) is the locus of all points \(V\) such that \(\measuredangle BVD=2\measuredangle BXD\). The conclusion is now evident.

(All the angles are directed mod \(\pi\).) Jubayer Nirjhor · 2 years, 5 months ago

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Do the books prescribed by hbcse suffice for rmo preparation?. I am in class 9 now, and looking forward to write rmo next year. Can anyone suggest good preparation material for the exam? Sidharth Nair · 1 year, 3 months ago

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Anybody got the sixth onr Devang Patil · 1 year, 5 months ago

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Short solution to #1: Let \(D\) be the foot of the perpendicular from \(B\) to \(AC\). It is well-known that \(BR\) and \(BD\) are isogonal. Then since \(AXRB\) is cyclic and we have \(\angle RXC=\angle RBA=\angle CBD=90^\circ-\angle BCA\). Now let \(XR\) intersect \(BC\) at a point \(P\); then since \(\angle RXC+\angle PCX=90^\circ\) we easily get \(\angle XPC=90^\circ\). Done. David Altizio · 2 years, 5 months ago

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I gave RMO from UP and got 4 questions correct , will i reach stage II (INMO)? Devang Agarwal · 2 years, 5 months ago

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@Devang Agarwal Was it the same paper as this one? If yes, then you may have a chance, but many got 4 correct. If it was any other paper, you have a good chance. All the best. Satvik Golechha · 2 years, 5 months ago

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Q1: Draw RM perpendicular to AB. RM bisects /ARB since RA = RB = the circumradius of Tr. ABC. Extend XR to meet BC in N. Now /ARM=/BRM=C and /MAR=/MBR=90 - C. Hence /_ XBR= /XAR= 90-B since quad XABR is concyclic. Now /ABX = (90 - C)- (90-B)=B-C) which is a +ve quantity since B is the largest /_ in the triangle. Therefore /ARX=B - C. This makes /MRX= C+B-C = B. In other words, quad. BMRN is concyclic and thus RN or RX is perpendicular to BC since /_RMB=90 by construction. One Top · 2 years, 5 months ago

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@Satvik, If you've solved Q1, will you please post the solution here? One Top · 2 years, 5 months ago

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@One Top I did it by extending perpendicular from \(R\) to \(\overline{BC}\) and cutting \(\overline{AC}\) in \(X\), and then proved that \(ARBX\) is a cyclic quadrilateral. Satvik Golechha · 2 years, 5 months ago

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I have uploaded the Mumbai region Paper here Pranshu Gaba · 2 years, 5 months ago

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Proof for 3: We consider the equation 2310n - n^2 = 2310k which implies that n^2 is congruent to 0 (mod 2310) with the condition that n^2 < 2310^2. Consider the prime factors of 2310 = 5 . 2 . 7 . 11 . 3 in which it has no factor that is a perfect square which implies that there is no n that satisfies the divisibility and n^2 < 2310^2.... John Ashley Capellan · 2 years, 5 months ago

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What could be the expected cutoff for this set of RMO paper ? Abhigyan Shekhar · 2 years, 5 months ago

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@Abhigyan Shekhar Well, I think in MP region the cutoff would not exceed 50 marks anyway Abhigyan Shekhar · 2 years, 5 months ago

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@Abhigyan Shekhar I guess around 4 questions right outta 6. This year paper was a bit easier than last year. Satvik Golechha · 2 years, 5 months ago

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Answer to some of the questions

2) \(1, \dfrac{1}{2}\)

4)\(x\)=\(y\)=\(z\)=2014

6) \(100\) Parth Lohomi · 2 years, 5 months ago

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Why 17? Earlier, it used to be 30 right? Krishna Ar · 2 years, 5 months ago

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@Krishna Ar Exactly, it used to be 30 till last year. And I can't possible answer "Why?". Satvik Golechha · 2 years, 5 months ago

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@Satvik Golechha Well, perhaps government has understood that people in RAJASTHAN don't want to be mathematician but IITian. Just joking... Subrata Saha · 2 years, 5 months ago

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@Subrata Saha Well, actually, it's partly true. People don't even know 'mathematician' is someone; everyone's running after IIT. And it'sn't their fault; in their opinion IIT will get them jobs, and money. Satvik Golechha · 2 years, 5 months ago

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@Krishna Ar Dude, you got any idea how many from Delhi region get selected?? Ashu Dablo · 2 years, 5 months ago

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@Ashu Dablo 30(Not too sure though- You can check last yr's result +cutoff) Krishna Ar · 2 years, 5 months ago

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The question paper of Delhi was different. Aneesh Kundu · 2 years, 5 months ago

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@Aneesh Kundu This is of Rajasthan. May I please get the Delhi paper? Thanks. Satvik Golechha · 2 years, 5 months ago

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@Satvik Golechha The paper was very easy compared to previous years'. Aneesh Kundu · 2 years, 5 months ago

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@Satvik Golechha Jharkhand's paper is similar to Rajasthan 's. Subrata Saha · 2 years, 5 months ago

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@Satvik Golechha I'm posting all of them Aneesh Kundu · 2 years, 5 months ago

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@Aneesh Kundu There are many sets of question papers. Mahimn Bhatt · 2 years, 5 months ago

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The sixth one is easy. Here's how to solve it: S(S(n)) = 2, therefore the values of S(n) can be 2, 11, or 20. (you can check, these are the only possible values.

Case I: S(n) = 2, possible values = 101, 110, 200

Case II: S(n) = 11 -Subcase I: when H = 1 (H is the hundreds digit), for T = 1 to 9 and the corresponding values of O (tens and ones), we get 9 values (119, 128, 137, 146, 155, 164, 173, 182, 191) -Subcase II: when H = 2, here, the order of T and O goes from T = 0 to 9 and the corresponding O values, this produces 10 values (i.e. 209, 218, 227, 236, 245, 254, 263, 272, 281, 290) If you go for further values of H, you will see that the number of integers satisfying S(n) = 11 reduces by 1 each time (from H = 2 to H = 9) the summation of values in this case therefore is 9 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 = 61

Case III: S(n) = 20, similar to Case II, instead that the value of H is from 2 to 9 and that the no. of permissible integers increases by 1. You can go over the subcases yourself. The summation comes out to be 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

Grand total of three cases = 3 + 61 + 36 = 100. Utkarsh Chaturvedi · 1 year, 5 months ago

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I attempted 5 and all of them are 100 percent correct? what are my chances? Siddharth Kumar · 2 years, 5 months ago

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@Siddharth Kumar Was not able to solve the first question... and also had a different method for question 2, did it by discriminant and functions. Siddharth Kumar · 2 years, 5 months ago

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@Satvik Golechha This paper is of which state? Pranjal Jain · 2 years, 5 months ago

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@Pranjal Jain Here is the paper for Rajasthan region. Pranjal Jain · 2 years, 5 months ago

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What is the answer for Q.6? Ankush Tiwari · 2 years, 5 months ago

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Comment deleted Dec 08, 2014

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@Satvik Golechha S (n) is congruent to n mod 9, so n is congruent to 2 mod 9. However, the sum of digits of any such n is less than or equal to 9+9+2=20. The sum of digits of a number less than or equal to 20 cannot be greater than 1+9=10, so for all n <1000 congruent to 2 mod 9, S (S (n))=2. Thus answer is (1000-100)/9=100. Joel Tan · 2 years, 5 months ago

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@Joel Tan Well, many have have counted to get the answer.Only few got the answer correct but I don't think the approach is right. Abhigyan Shekhar · 2 years, 5 months ago

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@Joel Tan That's really cool. All of us were making cases and counting. Extra marks for extra Elegance! Satvik Golechha · 2 years, 5 months ago

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@Satvik Golechha Problem 6 is a simplified version of IMO 1975 Problem 4 Ishan Singh · 2 years, 5 months ago

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@Satvik Golechha that could be done by multinomial theorem we have to find coeff of \(x^{20},x^{11}\) and we will get 100. And Multinomial gives ordered and all distinct pairs so no need to worry for duplicacy. And also 3 pairs when S(n)=2. For 11------> 61 numbers. For 20------>36 numbers . For 2-------->3 numbers. Gautam Sharma · 2 years, 5 months ago

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@Gautam Sharma Did it by the same method Prakher Gaushal · 1 year, 7 months ago

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@Satvik Golechha I too added wrong & got 103 @Satvik Golechha !! Btw what would be cutoff?? Harsh Shrivastava · 2 years, 5 months ago

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@Harsh Shrivastava I expect cutoff to be around 4.685954 questions out of 6. Many are getting 4 correct. Satvik Golechha · 2 years, 5 months ago

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@Satvik Golechha In Q6) I showed every steps but counted wrong so what will i get for that question?@Satvik Golechha Harsh Shrivastava · 2 years, 5 months ago

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@Harsh Shrivastava Exactly the case with me! I made three cases, and made some error(s) while counting. I expect them to give around 1/3 of the total marks for the question. How many questions did you get right in RMO? Satvik Golechha · 2 years, 5 months ago

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@Satvik Golechha Almost three :( What about you?? Harsh Shrivastava · 2 years, 5 months ago

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@Harsh Shrivastava 4 completely right, one 75% correct, one 20% correct. Satvik Golechha · 2 years, 5 months ago

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@Satvik Golechha That's Good!!!! Harsh Shrivastava · 2 years, 5 months ago

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@Harsh Shrivastava Marks are OK, but since we have only 15 selections from Rajasthan, there's quite a lesser chance of selections. Satvik Golechha · 2 years, 5 months ago

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Comment deleted May 06, 2015

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@Parth Lohomi Did u give RMO?? Subrata Saha · 2 years, 5 months ago

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Comment deleted May 06, 2015

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@Parth Lohomi U are surely going to qualify, I hope u are in 8 and at this stage 4.5 in RMO, its outstanding. Subrata Saha · 2 years, 5 months ago

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@Subrata Saha Actually I am in \(10^{th}\) Parth Lohomi · 2 years, 5 months ago

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@Parth Lohomi Then too youhave a good grip over maths, as its difficult for a 10th grade student for doing all this Mahimn Bhatt · 2 years, 5 months ago

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Ans to Q6) 173 Kunal Jain · 2 years, 3 months ago

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@Kunal Jain Sorry, but the answer is \(100\). Harsh Shrivastava · 2 years, 3 months ago

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