Let G be the centroid of \(\Delta ABC\) and d be the line outside \(\Delta ABC\).Draw \(AM,BN,CP,GQ\perp d\).Prove that \(AM+BN+CP=3GQ\)

Given a + b + c = 3.Simply the fraction : \(\dfrac{a^3+b^3+c^3-3abc}{\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3}\)

Prove that if \(\dfrac{1}{x}-\dfrac{1}{y}-\dfrac{1}{z}=1\) and x = y + z,then \(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}=1\)

A coin that has a diameter of 8 cm is tossed onto a 5 x 5 grid of squares each having side length 10 cm.Given that the coin is always inside the grid,prove that the probability that no part of it touches or crosses a grid line is \(\dfrac{25}{441}\)

Find the last digit of A :

\(A=1^1+2^2+3^3+4^4+5^5+...+1234^{1234}\)

Given N silver bars (N > 37),all of different weights.The 24 lightest ones and the 13 heaviest ones weigh 45% and 26% of the total weight respectively.Find N

A.51          

B.52

C.53

D.54

E.55

The positive integers a,b and c satisfy \(\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{c^2}\)

The sum of all possible values of \(a\le100\) is :

A.315          B.615                C.680           D.555          E.620

The base of a triangular piece of paper ABC is 12 cm long.The paper is folded down over the base,with the crease DE parallel to the base of the paper.The area of the triangle that projects below the base is 16% that of the area of the triangle ABC.The length of DE,in cm,is :

A. 9.6        B. 8.4            C.7.2            D. 4.8              E.6.96

Gene and Anne have 1 coin and continue to flip it until one of them wins the game.Gene wins if the sequence "Head-Head" appears before the sequence "Tail-Head". Anne wins if the sequence "Tail-Head" appears before the sequence "Head-Head".What is Anne's probability of winning this game ?

A.1/4        B.1/3            C.1/2           D.2/3             E.3/4

Several non-overlapping isosceles triangles have vertex O in common.Every triangle shares an edge with each immediate neighbour.The smallest of the angles at O has size m⁰ (\(m\in Z^+\)) and the other triangles have angles at O of size 2m⁰,3m⁰,4m⁰ and so on.The diagram shows an arrangement of 5 such triangles.Prove that 3 is the smallest value of m for such a set of triangles exists.

Yurko saw a tractor slowly pulling a long pipe down the road. Yurko walked along beside the pipe in the same direction as the tractor,and counted 140 paces to get from one end to the other.Then he turned around and walked back to the other end,taking only 20 paces.The tractor and Yurko kept to a uniform speed and Yurko's paces were all 1 m long.Prove that the pipe is 35 m long

A regular 13-sided polygon is inscribed in a circle with center O.Triangles can be formed by choosing 3 vertices of this polygon to be the vertices of the triangle.Prove that there are 91 triangles formed in this way with the point O inside the triangle.

The numbers 1,2,3,4,...,10 are to be written around a circle in some order.Then each number will be added to its immediate neighbours to obtain 10 new numbers.Prove that 16 is the largest possible value of the smallest of these new numbers.