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Frequently Asked Maths Olympiad Latest Exam Pattern - Maths Olympiad Sample Question with Solutions Page 19

(#M40021953) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Find the number of positive integers x which satisfy the condition
h x
99 i = h x
101 i.
(Here [z] denotes, for any real z, the largest integer not exceeding z; e.g. [7/4] = 1.)

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(#M40021951) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let BE and CF be the altitudes of an acute triangle ABC, with E on AC and F on AB. Let O be the point of intersection of BE and CF. Take any line KL through O with K on AB and L on AC. Suppose M and N are located on BE and CF respectively, such that KM is perpendicular to BE and LN is perpendicular to CF. Prove that FM is parallel to EN.

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(#M40021949) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

(i) Consider two positive integers a and b which are such that aabb is divisible by 2000.
What is the least possible value of the product ab?
(ii) Consider two positive integers a and b which are such that abba is divisible by 2000.
What is the least possible value of the product ab?

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(#M40021947) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Suppose hx1, x2, . . . , xn, . . .i is a sequence of positive real numbers such that x1 x2
x3 x^n , and for all n
x1^1+x4^2+x9^3+..... +xn^2
n 1.
Show that for all k the following inequality is satisfied:
x1^1+x2^2+x3^3+ ...... +xk^k
n 3.

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(#M40021946) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Solve the equation y^3 = x^3 + 8x^2 - 6x + 8, for positive integers x and y.

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(#M40021945) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Produce AP and CQ to meet at K. Observe that AKCR is a rhombus and BQKP is a parallelogram. Put AP = x,CQ = y. Then PK = BQ = y, KQ = PB = x and AR = RC = CK = KA = x + y. Using cosine rule in triangle PKQ, we get PQ^2 = x^2 + y^2 - 2xy cos120 = x^2 + y^2 + xy. Similarly cosine rule in triangle QCR gives QR^2 = y^2 +(x+y)^2 - 2xy cos60 = x^2 +y^2+xy and cosine rule in triangle PAR gives RP^2 = x^2 + (x + y)^2 - 2xy cos 60 = x^2 + y^2 + xy. It follows that PQ = QR = RP.

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(#M40021942) MATHS OLYMPIAD QUESTION FUNCTION Keep an EYE

›HW:8'VXHM:701N>fBiHI4~70jm:²lp+VY

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(#M40021941) MATHS OLYMPIAD QUESTION FUNCTION Keep an EYE

0/713:P]25[ 246^/
'79468/
:67';=UYHM

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(#M40021940) MATHS OLYMPIAD QUESTION FUNCTION Keep an EYE

Let R denote the set of all real numbers. Find all functions f : R → R satisfying the
condition
f(x + y) = f(x)f(y)f(xy)
for all x, y in R.

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(#M40021939) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle and D be the mid-point of side BC. Suppose DAB = BCA
and DAC = 15◦. Show that ADC is obtuse. Further, if O is the circumcentre of
ADC, prove that triangle AOD is equilateral.

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