**MATHMAA**

KVS Sample Papers has KVS *math olympiad* sample papers with *answers* which help the students who appear for KVS Junior Mathematical Olympiad .

Here are the questions and solutions .

Sample paper 1 :

**Time: 3hrs** ** M.M:100**

__Each question carry 10 marks.__

1Q) If a, b , c are measures which form a triangle for all n=2,3,4...etc, Prove that \(\sqrt[n]{a}, \sqrt[n]{b}, \sqrt[n]{c}\)

also form a triangle .

2Q) A square sheet of paper PQRS is so folded that the point Q fall on the mid point M of RS. Prove that the crease will divide BC in the ratio 5:3 .

3Q) Prove that in any triangle ABC, if one angle is \(120^{\circ}\), the triangle formed by the feet of the angle bisectors is a right angled.

4Q) a) Find all the integers which are equal to 11 times the sum of their digits.

**KVS Sample Papers**

b) Prove that \(3a^{4}-4a^{3}b+b^{4} \geq 0 \), for all real numbers a and b.

5Q) In triangle ABC, the area is \(\frac{1}{2}bc sq.units \). AD is a median to BC. Prove that \(\angle ABC = \frac{1}{2}\angle ADC\).

6Q) Solve the inequality, |x-1|+|x+1| < 4.

7Q) Prove that, if the coefficients of the quadratic equation \(ax^{2}+bx+c =0\) are odd integers, then the roots of the equation cannot be rational numbers .

8Q) Given real numbers x, y, and z such that x+y+z=3, \(x^{2}+y^{2}+z^{2}=5 , x^{3}+y^{3}+z^{3}=7\) . Find the value of \(x^{4}+y^{4}+z^{4}\).

**math olympiad sample papers**

9Q) If x and y are positive numbers such that x+y=1, find the maximum value of \(x^{4}y+xy^{4}\).

10Q) Given three non collinear points A, H, G. Construct a triangle With A as vertex, H as Orthocentre and G as centriod .

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