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Frequently Asked Maths Olympiad Latest Exam Pattern - Maths Olympiad Sample Question with Solutions Page 18

(#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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(#M40021964) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find all integers a, b, c, d satisfying the following relations:
(i) 1  a  b  c  d;
(ii) ab + cd = a + b + c + d + 3.

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(#M40021962) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

The circumference of a circle is divided into eight arcs by a convex quadrilateral ABCD, with
four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of
the arcs lying inside the quadrilateral are denoted by p, q, r, s in counter-clockwise direction
starting from some arc. Suppose p + r = q + s. Prove that ABCD is a cyclic quadrilateral.

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(#M40021961) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Suppose the integers 1, 2, 3, . . . , 10 are split into two disjoint collections a1, a2, a3, a4, a5 and
b1, b2, b3, b4, b5 such that
a1 < a2 < a3 < a4 < a5,
b1 > b2 > b3 > b4 > b5.
(i) Show that the larger number in any pair {aj , bj}, 1  j  5, is at least 6.
(ii) Show that |a1-b1|+|a2-b2|+|a3-b3|+|a4-b4|+|a5-b5| = 25 for every such partition.

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(#M40021960) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Let a, b, c be positive integers such that a divides b^2, b divides c^2 and c divides a^2. Prove that abc divides (a + b + c)^7.

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(#M40021959) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Solve the following equation for real x:
(x^2 + x - 2)^3 + (2x^2 - x - 1)^3 = 27(x^2 - 1)^3

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(#M40021956) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

If x, y, z are the sides of a triangle, then prove that
|x^2(y - z) + y^2(z - x) + z^2(x - y)| < xyz.

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(#M40021954) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Consider an n × n array of numbers:
0BBBB@
a11 a12 a13 · · · a1n
a21 a22 a23 · · · a2n
...
...
an1 an2 an3 · · · ann
1CCCCA
Suppose each row consists of the n numbers 1, 2, 3, . . . , n in some order and aij = aji for
i = 1, 2, . . . , n and j = 1, 2, . . . , n. If n is odd, prove that the numbers a11, a22, a33, . . . , ann
are 1, 2, 3, . . . , n in some order.

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