olympiads.hbcse.tifr.res.in Indian National Mathematical Olympiad Question Paper INMO 2017 : Homi Bhabha Centre For Science Education
Name of the Centre : Homi Bhabha Centre For Science Education
Name Of The Exam : Indian National Mathematical Olympiad-2017
Name Of The Subject : Mathematical
Document type : Previous Question Papers
Year : 2017
Website : http://olympiads.hbcse.tifr.res.in/how-to-prepare/past-papers/
Download Model/Sample Question Paper :
INMO 2017 : https://www.pdfquestion.in/uploads/13123-inmo2017.pdf
INMO 2016 : https://www.pdfquestion.in/uploads/13123-INMO-2016.pdf
INMO 2015 : https://www.pdfquestion.in/uploads/13123-inmo2015.pdf
INMO 2014 : https://www.pdfquestion.in/uploads/13123-inmo-2014.pdf
INMO 2013 : https://www.pdfquestion.in/uploads/13123-inmo2013.pdf
Indian National Mathematical Olympiad Question Paper :
Time: 4 hours
INMO – 2017 :
Instructions :
** Calculators (in any form) and protractors are not allowed.
** Rulers and compasses are allowed.
Related : Homi Bhabha Centre For Science Education Regional Mathematical Olympiad-2016 : www.pdfquestion.in/13118.html
** All questions carry equal marks. Maximum marks: 102.
** Answer all the questions.
** Answer to each question should start on a new page. Clearly indicate the question number.
1. In the given figure, ABCD is a square sheet of paper. It is folded along EF such that A goes to a point A’ different from B and C, onthe side BC and D goes to D’. The line A’D’ cuts CD in G. Show that the inradius of the triangle GCA’ is the sum of the inradii of the triangles GD’F and A’BE.
2. Suppose n 0 is an integer and all the roots of x3 + x + 4 – (2 × 2016n) = 0 are integers. Find all possible values of .
3. Find the number of triples (x, a, b) where x is a real number and a, b belong to the set {1, 2, 3, 4, 5, 6, 7, 8, 9} such that x2 – a{x} + b = 0, where {x} denotes the fractional part of the real number x. (For example {1.1} = 0.1 = {-0.9}.)
4. Let ABCDE be a convex pentagon in which \A = \B = \C = \D = 120 and side lengths are five consecutive integers in some order. Find all possible values of AB + BC + CD. 5. Let ABC be a triangle with \A = 90? and AB < AC. Let AD be the altitude from A on to BC. Let P,Q and I denote respectively the incentres of triangles ABD, ACD and ABC. Prove that AI is perpendicular to PQ and AI = PQ.
INMO 2016 :
1. Let ABC be triangle in which AB = AC. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio AB=BC.
2. For positive real numbers a; b; c, which of the following statements necessarily implies a = b = c: (I) a(b3 + c3) = b(c3 + a3) = c(a3 + b3), (II) a(a3 + b3) = b(b3 + c3) = c(c3 + a3) ? Justify your answer.
3. Let N denote the set of all natural numbers. Dene a function T : N ! N by T(2k) = k and T(2k + 1) = 2k + 2. We write T2(n) = T(T(n)) and in general Tk(n) = Tk??1(T(n)) for any k > 1.
(i) Show that for each n 2 N, there exists k such that Tk(n) = 1.
(ii) For k 2 N, let ck denote the number of elements in the set fn : Tk(n) = 1g. Prove that ck+2 = ck+1 + ck, for k 1.
4. Suppose 2016 points of the circumference of a circle are coloured red and the remaining points are coloured blue. Given any natural number n 3, prove that there is a regular n-sided polygon all of whose vertices are blue.
5. Let ABC be a right-angled triangle with \B = 90. Let D be a point on AC such that the inradii of the triangles ABD and CBD are equal. If this common value is r0 and if r is the inradius of triangle ABC, prove that
INMO-2015 :
1. Let ABC be a right-angled triangle with \B = 90. Let BD be the altitude from B on to AC. Let P, Q and I be the incentres of triangles ABD, CBD and ABC respectively. Show that the circumcentre of of the triangle PIQ lies on the hypotenuse AC.
Solution: We begin with the following lemma :
Lemma: Let XY Z be a triangle with \XY Z = 90 + . Construct an isosceles triangle XEZ, externally on the side XZ, with base angle . Then E is the circumcentre of 4XY Z.
2. For any natural number n > 1, write the infinite decimal expansion of 1=n (for example, we write 1=2 = 0:4 9 as its infinite decimal expansion, not 0:5). Determine the length of the non-periodic part of the (infinite) decimal expansion of 1=n.
Solution: For any prime p, let p(n) be the maximum power of p dividing n; ie pp(n) divides n but not higher power. Let r be the length of the non-periodic part of the infinite decimal expansion of 1=n.