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ABC is a right angled triangle with BC as hypotenuse. The medians drawn to BC and AC are perpendicular to each other. If AB has length 1cm, find the area of triangle ABC.
(a) Find the smallest positive integer such that it has exactly 100 different positive integer divisors, including 1 and the number itself. (b) A rectangle can be divided into n equal squares. The same rectangle can also be divided into $$(n+76)$$ equal squares. Find n.
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Prove that $$1^n+2^n+3^n+\cdots+15^n$$ is divisible by 480 for all odd $$n \ge 5$$.
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Is it possible to have 19 lines in a plane such that (1) no three lines have a common point and (2) they have exactly 95 points of intersection? Validate.
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In a trapezium ABCD with AB parallel to CD, the diagonals intersect at P. The area of $$\triangle ABP$$ is $$72\text{cm}^2$$ and area of $$\triangle CDP$$ is $$50\text{cm}^2$$. Find the area of the trapezium.
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Let $$a<b<c$$ be three positive integers. Prove that among any 2c consecutive positive integers there exist three different numbers x, y, z such that $$abc$$ divides $$xyz$$.
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(a) Let m, n be positive integers. If $$m^3+n^3$$ is the square of an integer, then prove that $$(m+n)$$ is not a product of two different prime numbers. (b) a, b, c are real numbers such that $$ab+bc+ca=-1$$. Prove $$a^2+5b^2+8c^2 \ge 4$$.
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ABCD is a quadrilateral in a circle whose diagonals intersect at right angles. Through O, the centre of the circle,$$GOG'$$ and $$HOH'$$ lines are drawn parallel to AC and BD respectively, meeting AB, CD at G, H and DC, AB produced at $$G'$$, $$H'$$. Prove GH, $$G'H'$$ are parallel to BC and AD respectively.
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