• Placement Questions
• /
• Maths olympiad

maths olympiad Latest Exam Pattern - maths olympiad Sample Question with Solutions Page 17

(#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.)

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (0)   (0) Submit Your Solution

(#M40021952) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Find all primes p and q such that p^2 + 7pq + q^2 is the square of an integer.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved Read Solution (1)

Is this Puzzle helpful?   (1)   (2) Submit Your Solution

(#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.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (0)   (0) Submit Your Solution

(#M40021950) MATHS OLYMPIAD QUESTION ALGEBRA Keep an EYE

Find all real values of a for which the equation
x^4 - 2ax^2 + x + a^2 - a = 0 has all its roots real.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (2)   (2) Submit Your Solution

(#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?

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (0)   (3) Submit Your Solution

(#M40021948) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

The internal bisector of angle A in a triangle ABC with AC > AB, meets the circumcircle
􀀀 of the triangle in D. Join D to the centre O of the circle 􀀀 and suppose DO meets AC
in E, possibly when extended. Given that BE is perpendicular to AD, show that AO is
parallel to BD.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (1)   (0) Submit Your Solution

(#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.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (0)   (0) Submit Your Solution

(#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.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (0)   (0) Submit Your Solution

(#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.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (0)   (0) Submit Your Solution

(#M40021944) MATHS OLYMPIAD QUESTION GEOMETRY Keep an EYE

Let AC be a line segment in the plane and B a point between A and C. Construct isosceles triangles PAB and QBC on one side of the segment AC such that APB = BQC = 120 and an isosceles triangle RAC on the otherside of AC such that ARC = 120. Show that PQR is an equilateral triangle.

Asked In Maths Olympiad MAN (12 years ago)

Unsolved

Is this Puzzle helpful?   (1)   (1) Submit Your Solution