NMTC Stage 2 KAPREKAR Sub Junior Level 2015

(a) The diagram above contains 13 boxes. The numbers in the second and the twelfth boxes are respectively 175 and 70. Fill up the boxes with natural numbers such that
(i) the sum of all numbers in all 13 boxes is 2015,
(ii) the sum of the numbers in any three consecutive boxes is always the same.
      The solution must contain the steps how you arrive at the numbers.
(b) If x, y, z are real and unequal numbers, prove that $$2015x^{2}+2015y^{2}+6z^{2} > 2(2012xy+3yz+3zx)$$.

If a, b, c are reals such that $$a+b=4$$ and $$2c^2-ab = 4\sqrt{3}c-10$$, find the numerical values of a, b and c.

When $$a=2^{2014}$$ and $$b=2^{2015}$$, prove that $$\left\{\frac{\frac{(a+b)^{2}+(a-b)^{2}}{b-a}-(a+b)}{\frac{1}{b-a}- \frac{1}{a+b}} \right\}\div \left\{\frac{(a+b)^{3}+(b-a)^{3}}{(a+b)^{2}-(a-b)^{2}} \right\}$$ is divisible by 3.

Prove that the feet of the perpendiculars drawn from the vertices of a parallelogram onto its diagonals are the vertices of a parallelogram.

ABC is an acute angled triangle. P, Q are points on AB and AC respectively such that area of $$\triangle APC$$ equals area of $$\triangle AQB$$. A line is drawn through B parallel to AC and meets the line through Q parallel to AB at S. QS cuts BC at R. Prove that $$RS=AP$$.

(a) A man is walking from a town A to another town B at a speed of $$4\text{km/hr}$$. He started one hour before a bus starts. The bus travelling at $$12\text{km/hr}$$. The man on the way got into the bus and travels 2 hours and reached town B. What is the distance between town A and town B? (b) A point P is taken within a rhombus ABCD such that $$PA=PC$$. Show that B, P, D are collinear.

If $$(x+y+z)^3 = (y+z-x)^3+(z+x-y)^3+(x+y-z)^3+kxyz$$, find the numerical value of k. Deduce the following result. If $$a=2015, b=2014, c= \frac{1}{2014}$$ Prove that $$(a+b+c)^{3} -(a+b-c)^{3}-(b+c-a)^{3} -(c+a-b)^{3}-23abc = 2015$$