## 02 Oct State Level Mathematics Competition-2013-Category-III : Assam Academy of Mathematics

**01 September 2013**

**(Class IX and X)**

** **

Marks: 10 X 10 = 100

Time: 1.30 pm to 4.30 pm

*Answer the following ten questions*

1. Show that there does not exist a function $$f:Nrightarrow N$$ which satisfy

(a) $$f(2)=3,$$

(b) $$f(mn)=f(m)f(n)$$ for all m,n in N;

(c) $$f(m)<f(n)$$ whenever $$m<n.$$

(Hint: Suppose the contrary. Suppose $$f:Nrightarrow N$$ be such that the function satisfy (b) and (c) such that $$f(2)=3.$$ It leads to absurd calculation 256<243.)

2. Show that $$15x^2-7y^2=9$$ has no integral solutions.

(Hint: RHS is odd, x and y will be such that one of x, y is even and the other is odd.)

3. Determine the integer n for which $$n^2+19n+92$$ is a square.

(Hint: Let $$n^2+19n+92=x^2.$$ Solve for x.)

4. In a triangle ABC, AB=AC. A circle is drawn touching the circumcircle of $$triangle ABC$$ initially and also, touching the sides of AB and AC at P and Q respectively. Prove that the midpoint of PQ is the incentre of triangle ABC.

5. Prove that in an arbitrary triangle, the sum of the lengths of the altitude is less than the triangle’s perimeter.

6. Let $$x_1,x_2,dots ,x_m,y_1,y_2,dots ,y_n$$ be positive integers such that the sums $$x_1+x_2+dots +x_m, y_1+y_2+dots +y_n$$ are equal and less than $$mn.$$ Prove that in the equality

$$x_1+x_2+dots +x_m=y_1+y_2+dots +y_n$$

one can cancel some terms and obtain another equality.

7. Determine the remainder when $$3^{2^n}-1$$ is divided by $$2^{n+3}.$$

8. Prove that the diagonals of a (convex) quadrilateral are perpendicular, if and only if, the sum of the squares of one pair of opposite sides equals that of the other.

9. ABC is a triangle. $$angle A=30^{circ},angle B=60^{circ}$$ and AB=10cm. Find the length of the shorter trisector of $$angle C.$$

(Hint: Trigonometry can be used to calculate the length of the sides.)

10. A rhombus has half the area of the square with the same side length. Show that the ratio of the longer diagonal to that of the shorter one is $$2+sqrt{3}.$$