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maths olympiad Latest Exam Pattern - maths olympiad Sample Question with Solutions Page 14

(#M40021989) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

In a book with page numbers from 1 to 100, some pages are torn off. The sum of
the numbers on the remaining pages is 4949. How many pages are torn off?

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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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(#M40021982) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let P1(x) = ax^2 + bx + c, P2(x) = bx^2 + cx +a, P3(x) = cx^2 + ax + b be three quadratic polynomials where a, b, c are non-zero real numbers. Suppose there exists a real number
 such that P1() = P2() = P3(). Prove that a = b = c.

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(#M40021981) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O. Suppose the area of traingle OAF is the geometric mean of those of OAB and OEF; and the area of triangle OBC is the geometric mean of those of OAB and OCD. Prove that the area of triangle OED is the geometric mean of those of OCD and OEF.

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(#M40021980) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find all pairs (x, y) of real numbers such that
16x^2 + y + 16x + y^2 = 1.

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(#M40021979) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle and let BB1, CC1 be respectively the bisectors of angle B, C with B1 on AC and C1 on AB. Let E, F be the feet of perpendiculars drawn from A onto BB1, CC1 respectively. Suppose D is the point at which the incircle of ABC touches AB. Prove that AD = EF

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