7/29/2019 Inmo Question Paper 2013 Previous year Question Papers
of Indian National Mathematical Olympiad (INMO) with s
1/2
28 - 2013: 4 3 2013: (protractors)
(rulers) (compasses) 1. 12 R :
l1 2 P , 1 O1 l2 1 Q , 2 O2 l1 l2 K KP = KQ, PQR
2. m, n p 5
m(4m2+ m + 12) = 3(pn - 1)
3. a, b, c, d a b c d x4 ax3 bx2 cx d = 0
4. n {1, 2, 3, ..., n} S(good) S tn, {1, 2, 3, ..., n} (good)
tnn
5. ABC O,H G
OD, BC HE, CA DBC ECA AB F ODC,HEAGFB C
6. a,b,c,x,y,z
a + b + c =x +y +zabc =xyz a x
7/29/2019 Inmo Question Paper 2013 Previous year Question Papers
of Indian National Mathematical Olympiad (INMO) with s
2/2
28th Indian National Mathematical Olympiad-2013
Time: 4 hours February 03, 2013Instructions:
Calculators (in any form) and protractors are not allowed.
Rulers and compasses are allowed.
Answer all the questions. All questions carry equal marks.
Answer to each question should start on a new page.
Clearlyindicate the question number.
1. Let 1 and 2 be two circles touching each other externally at
R.
Let l1 be a line which is tangent to 2 at P and passing
through
the centre O1 of 1 . Similarly, let l2 be a line which is
tangent to1 at Q and passing through the centre O2 of 2 . Suppose
l1 and
l2 are not parallel and intersect at K. If KP = KQ, prove that
the
trianglePQR is equilateral.
2. Find all positive integers m, n and primesp 5 such that
m(4m2+ m + 12) = 3(pn - 1) .
3. Let a, b, c, d be positive integers such that a b c d.
Prove
that the equation x4
ax3
bx2
cx d = 0 has no integersolution.
4. Let n be a positive integer. Call a nonempty subset S of
{1, 2, 3, . . . , n} good if the arithmetic mean of the elements
of
S is also an integer. Further let tn denote the number of
good
subsets of {1, 2, 3, . . . , n}. Prove that tn and n are both
odd or
both even.
5. In an acute triangle ABC, O is the circumcentre, H
theorthocentre and G the centroid. Let OD be perpendicular to
BC
andHE be perpendicular to CA, withD on BCandE on CA. Let
Fbe the mid-point of AB. Suppose the areas of triangles ODC,
HEA and GFB are equal. Find all the possible values of C .
6. Let a,b,c,x,y,z be positive real numbers such that
a + b + c =x +y +z and abc =xyz. Further, suppose that
a x
