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maths olympiad Latest Exam Pattern - maths olympiad Sample Question with Solutions Page 13

(#M40022001) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be an acute-angled triangle with altitude AK. Let H be its ortho-centre and O
be its circum-centre. Suppose KOH is an acute-angled triangle and P its circum-centre.
Let Q be the reflection of P in the line HO. Show that Q lies on the line joining the
mid-points of AB and AC.

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(#M40022000) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

How many 6-tuples (a1, a2, a3, a4, a5, a6) are there such that each of a1, a2, a3, a4, a5, a6 is from the set {1, 2, 3, 4} and the six expressions a2 j - ajaj +1 + a2 j+1
for j = 1, 2, 3, 4, 5, 6(where a7 is to be taken as a1) are all equal to one another?

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(#M40021999) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Find all non-zero real numbers x, y, z which satisfy the system of equations:
(x^2 + xy + y^2)(y^2 + yz + z^2)(z^2 + zx + x^2) = xyz,
(x^4 + x^2y^2 + y^4)(y^4 + y^2z^2 + z^4)(z^4 + z^2x^2 + x^4) = x^3y^3z^3.

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(#M40021998) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Find all natural numbers n > 1 such that n2 does not divide (n - 2)!.

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(#M40021997) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle with circum circle. Let M be a point in the interior of triangle ABC which is also on the bisector of A. Let AM, BM, CM meet in A1, B1, C1 respectively. Suppose P is the point of intersection of A1C1 with AB; and Q is the point of intersection of A1B1 with AC. Prove that PQ is parallel to BC.

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(#M40021995) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle in which AB = AC and CAB = 90◦. Suppose M and
N are points on the hypotenuse BC such that BM2 + CN2 = MN2. Prove that
MAN = 45◦

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(#M40021993) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

m'F[Uw^),§-=Pq7g-h-=PA,F[,‰,ÆÎ

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(#M40021992) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find the sum of all 3-digit natural numbers which contain at least one odd digit
and at least one even digit.

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(#M40021991) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Show that 32008 + 42009 can be wriiten as product of two positive integers each of
which is larger than 2009182.

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(#M40021990) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Show that there is no integer a such that a^2 - 3a - 19 is divisible by 289.

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