06 May Mathletics 2013 (Category III) – Assam Academy of Mathematics
(Class IX and X)
Time: 3 Hours
[Answer the following questions in English or in mother tongue. Two students of the group will discuss the solutions of the problems and write answer in a khata for the group. No third person can help the group to solve the problems]
Answer all ten questions
1. Find the smallest number when divided by 3 gives a remainder 1, when divided by 4 gives remainder 2, when divided by 5 gives a remainder 3 and when divided by 6 gives remainder 4.
2. If a>b show that
3. Two tangents are drawn to a circle at X and Y from a point C. A line AB is drawn tangent to the circle at a point P randomly chosen on the circle’s circumference between X and Y.
If YC=XC=10 units find the perimeter of the triangle ABC.
4. ABC is an isosceles triangle. XY is drawn parallel to the base cutting the sides in X and Y. Show that B, C, X, Y lie in a circle.
5. In a triangle ABC, AB=AC. A transversal intersects AB and AC internally at K and L respectively. It intersects BC produced at M. If KL=2LM, find
6. If show that
8. Prove that is a composite number for all values of n>1.
9. Prove that for every positive integer n,
is divisible by 10.
10. Find the remainder when is divided by 7.