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Frequently Asked Maths Olympiad Latest Exam Pattern - Maths Olympiad Sample Question with Solutions Page 17

(#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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(#M40021978) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Consider a 20-sided convex polygon K, with vertices A1,A2, . . . ,A20 in that order. Find the number of ways in which three sides of K can be chosen so that every pair among them has at least two sides of K between them. (Forexample (A1A2,A4A5,A11A12) is an admissible triple while (A1A2,A4A5,A19A20)is not.)

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(#M40021977) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

A natural number n is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting k from n and the larger one is obtained by adding l to n. Prove that n+kl is a perfect square.

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(#M40021976) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let (a1, a2, a3, . . . , a2011) be a permutation (that is a rearrangement) of the numbers
1, 2, 3, . . . , 2011. Show that there exist two numbers j, k such that 1  j < k  2011 and aj + j = ak + k.

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(#M40021975) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle. Let D, E, F be points respectively on the segments BC, CA, AB such that AD, BE, CF concur at the point K. Suppose BD/DC = BF/FA and ADB = AFC. Prove that ABE = CAD

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(#M40021970) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find all real numbers a for which the equation
x^2 + (a + 2)x + 1 = 3|x| has exactly three distinct real solutions in x.

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(#M40021968) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find the number of ordered triples (x, y, z) of nonnegative integers satisfying the
conditions:
(i) x = y + z;
(ii) x + y + z = 100.

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