8/16/2019 Problems From Various Olympaid
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MATHEMATICS OLYMPIAD PROGRAMME IN INDIA
Organized by the
NATIONAL BOARD OF HIGHER MATHEMATICSDEPARTMENT OF ATOMIC
ENERGY
GOVERNMENT OF INDIAAnushakti Bavan, Mumbai
http : //www.geocities.com/olympiad pondicherry
Question Paper 4
1)(2005 India Regional Mathematics Olympiad ) If x, y
are integers, and 17 dividesboth the expressions x2−
2xy + y2−5x + 7y and x2− 3xy + 2y2 + x− y, then prove
that17 divides xy − 12x + 15y.
2)(2005 India Regional Mathematics Olympiad ) In a triange
ABC , let D be the
midpoint of B C . If ADB
= π4
and ACD = π6
, determine BAD.
3)(2005 India Regional Mathematics Olympiad ) Determine all
triples (a,b,c) of positive integers such that a ≤ b ≤ c
and a
+ b + c + ab + bc + ca =
abc + 1.
4)(1997 Czech and Slovak Republics National Mathematics
Olympiad)Let ABC bea triangle with sides
a,b,c and corresponding angles α,β,γ . Prove
that the equalityα = 3β implies the equality (a2
− b2)(a − b) = bc2, and determine wether the conversealso
holds.
5)(2005 India National Mathematics Olympiad ) Let α
and β be positive integerssuch that
43
197 < α
β < 17
77. Find the minimum possible value of β .
6)(2005 India National Mathematics Olympiad ) Let p,
q,r be positive real numbers,not all equal, such that some two
of the equations px2 + 2qx + r = 0 qx2
+ 2rx + p = 0rx2 + 2 px + q = 0 have a
common root, say α. Prove that (a) α is real and
negative; and
(b) the third equation has non-real roots.
7)(1996 Russia National Mathematics Olympiad ) Show that if the
integers a1,...,amare nonzero and for each k = 0,
1,...,m (n < m−1) a1 + a22
k + a33k + ... + amm
k = 0,then the sequence a1,...,am contains at
least n + 1 pairs of consecutive terms havingopposite
signs.
8)(1996 Romania National Mathematics Olympiad ) Find all prime
numbers p, q forwhich the
congruence α3 pq ≡ α (mod 3 pq )
holds for all integers α.
9)(2003 India National Mathematics Olympiad ) Let AB
C be a triangle with sidesa,b,c. Consider a triangle
A1B1C 1 with sides equal to a +
b2
, b + c2
, c + a2
. Show that[A1B1C 1] ≥
9
4[ABC ] where [XY Z ] denotes the area of the triangle
X Y Z .
10)(2003 India National Mathematics Olympiad ) Suppose
that p is a prime greaterthan 3. Find all pairs of
integers (a, b) satisfying the
equation a2+3ab+2 p(a+b)+ p2 = 0.
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