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Frequently Asked maths olympiad Latest Exam Pattern - maths olympiad Sample Question with Solutions Page 16

(#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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(#M40021988) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

A convex polygon is such that the distance between any two vertices of does not exceed 1.
(i) Prove that the distance between any two points on the boundary of does not exceed 1.
(ii) If X and Y are two distinct points inside, prove that there exists a point Z on the boundary of such that XZ + Y Z  1.

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(#M40021987) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose
BC = AB + AI. Find BAC.

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(#M40021985) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle in which A = 60. Let BE and CF be the bisectors of the angles B and C with E on AC and F on AB. Let M be the reflection of A in the line EF. Prove that M lies on BC.

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(#M40021984) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.

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(#M40021983) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find the number of 4-digit numbers(in base 10) having non-zero digits and which are divisible by 4 but not by 8.

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