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maths olympiad Latest Exam Pattern - maths olympiad Sample Question with Solutions Page 12

(#M40022013) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.

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(#M40022012) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABC be an acute-angled triangle, and let D, E, F be points on BC, CA, AB respectively such that AD is the median, BE is the internal angle bisector and CF is the altitude. Suppose FDE = C, DEF = A and EFD = B. Prove that ABC is equilateral.

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(#M40022011) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Let f : Z ! Z be a function satisfying f(0) = 0, f(1) = 0 and
(i) f(xy) + f(x)f(y) = f(x) + f(y);
(ii)f(x y) − f(0)


f(x)f(y) = 0,
for all x, y 2 Z, simultaneously.
(a) Find the set of all possible values of the function f.
(b) If f(10) 6= 0 and f(2) = 0, find the set of all integers n such that
f(n) 6= 0

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(#M40022010) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Define a sequence hf0(x), f1(x), f2(x), . . .i of functions by
f0(x) = 1, f1(x) = x,
􀀀
fn(x)

2
− 1 = fn+1(x)fn−1(x), for n  1.
Prove that each fn(x) is a polynomial with integer coefficients

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(#M40022009) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Let p1 < p2 < p3 < p4 and q1 < q2 < q3 < q4 be two sets of prime numbers such that p4 + p1 = 8 and q4 + q1 = 8. Suppose p1 > 5 and q1 > 5. prove that 30 divides p1 + q1.

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(#M40022007) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Call a natural number n faithful, if there exist natural numbers a < b < c such that a divides
b, b divides c and n = a + b + c.
(i) Show that all but a finite number of natural numbers are faithful.
(ii) Find the sum of all natural numbers which are not faithful.

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(#M40022006) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find all functions f : R + R such that
f(x + y)f(x - y) = f(x) + f(y)

2 - 4x^2f(y), (1)
for all x, y - R, where R denotes the set of all real numbers.

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(#M40022005) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let ABCD be a quadrilateral inscribed in a circle. Let E, F, G, H be the midpoints of the arcs AB, BC, CD, DA of the circle. Suppose AC· BD = EG · FH. Prove that AC, BD, EG, FH are concurrent.

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(#M40022004) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show
that one can select four among these five such that they are the vertices of a trapezium.

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(#M40022003) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that
BD = CE = AF and BDF = CED = AFE. Prove that ABC is equilateral

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