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Frequently Asked Maths Olympiad Latest Exam Pattern - Maths Olympiad Sample Question with Solutions Page 20

(#M40021938) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Given any nine integers show that it is possible to choose, from among them, four
integers a, b, c, d such that a + b − c − d is divisible by 20. Further show that such a
selection is not possible if we start with eight integers instead of nine.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40021937) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

If a, b, c are positive real numbers such that abc = 1, prove that
ab+c bc+a ca+b ≤ 1.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40021936) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Show that the equation
x^2 + y^2 + z^2 = (x - y)(y - z)(z - x)
has infinitely many solutions in integers x, y, z.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40021935) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle in which no angle is 90◦. For any point P in the plane
of the triangle, let A1,B1,C1 denote the reflections of P in the sides BC,CA,AB
respectively. Prove the following statements:
(a) If P is the incentre or an excentre of ABC, then P is the circumcentre of A1B1C1;
(b) If P is the circumcentre of ABC, then P is the orthocentre of A1B1C1;
(c) If P is the orthocentre of ABC, then P is either the incentre or an excentre of
A1B1C1.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40021933) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Let a, b, c be three real numbers such that 1 ≥ a ≥ b ≥ c ≥ 0. Prove that if  is a
root of the cubic equation x3 + ax2 + bx + c = 0 (real or complex),then || ≤ 1.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40021932) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

In a convex quadrilateral PQRS, PQ = RS, (√3+1)QR = SP and RSP−SPQ =
30◦. Prove that
PQR − QRS = 90◦.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40021931) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

If a, b, c, x are real numbers such that abc 6= 0 and
xb + (1 - x)ca=xc + (1 - x)ab=xa + (1 - x)bc
then prove that either a + b + c = 0 or a = b = c.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40021930) MATHS OLYMPIAD QUESTION INTEGER Keep an EYE

Solve for integers x, y, z:
x + y = 1 - z, x^3 + y^3 = 1 - z^2.

Asked In Maths Olympiad MAN (12 years ago)

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(#M40019955) MATHS OLYMPIAD QUESTION PRIME NUMBERS Keep an EYE

find whether the given no. is prime or not

84597857

Asked In Maths Olympiad (12 years ago)

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(#M40019250) MATHS OLYMPIAD QUESTION applitude test Keep an EYE

Q. The cost of house is 123000. The tax on it will be increasing 7% percent every year. After 50 months one man buy the house how much he should pay for the house including the repair work for it Rs 12345

Asked In Maths Olympiad Harsh Poddar (12 years ago)

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