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maths olympiad Latest Exam Pattern - maths olympiad Sample Question with Solutions Page 15

(#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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(#M40021969) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Suppose P is an interior point of a triangle ABC such that the ratios
d(A,BC)
d(P,BC)
,
d(B,CA)
d(P,CA)
,
d(C,AB)
d(P,AB)

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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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(#M40021967) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Let a, b, c be three positive real numbers such that a + b + c = 1. Prove that among the three numbers a - ab, b - bc, c - ca there is one which is at most 1/4 and there is one which is at least 2/9

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(#M40021966) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

If n is an integer greater than 7, prove that
n^7-hn^7 is divisible by 7. Here n^7 denotes the number of ways of choosing 7 objects from among n objects; also, for any real number x, [x] denotes the greatest integer not exceeding x.


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(#M40021965) MATHS OLYMPIAD QUESTION ALGEBRA 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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