State Level Mathematics Competition-2013-Category-IV : Assam Academy of Mathematics

01 September 2013

(Class XI and XII)

 

Marks: 10 X 10 = 100

Time: 1.30 pm to 4.30 pm

Answer the following ten questions

1. Prove that 4(x_1^4+ x_2^4+dots + x_{14}^4)=7(x_1^3+ x_2^3+dots + x_{14}^3) has no solution in positive integers.

(Hint: Suppose on the contrary sum_{k=1}^{14}(x_k^4-frac{7}{4}x_k^3)=0. Also use sum (x_k-1)^4.)

 

2. Find all non negative integers a, b, c, d, n that satisfy

a^2+b^2+c^2+d^2=7.4^n.

(Hint: Put n=0 and use 2^2+1^2+1^2+1^2=7.)

 

3. Let ABCD be convex quadrilateral. Suppose that the lines AB and CD intersect at E and the lines AD and BC intersect at F. Prove that the following statements are equivalent

(i) a circle is inscribed in ABCD

(ii) BE+BF=DE+DF

(iii) AE-AF=CE-CF

(Hint: Inscribe a circle in the quadrilateral ABCD to touch the quadrilateral at K, L, M and N.)

 

Image Source : Shutterstock

Image Source : Shutterstock

4. A and B are two points situated on the same side of a line XY. Find the position of a point M on the line such that the sum AM+MB is minimal.

(Hint: Suppose B´ is the reflection of B across the line XY. M is the point of intersection of AB´ and XY.)

 

5. Let a, b, c be non-zero real numbers such that a+b+cneq 0 and frac{1}{a}+frac{1}{b}+frac{1}{c}=frac{1}{a+b+c}.

Prove that frac{1}{a^n}+frac{1}{b^n}+frac{1}{c^n}=frac{1}{a^n+b^n+c^n}.

 

OR

Prove that is p is a prime the sqrt{p} is an irrational number.

 

6. Let a, b, c be positive numbers such that abc=1. Prove that

frac{1}{a^3(b+c)}+frac{1}{b^3(a+c)}+frac{1}{c^3(a+b)geqfrac{3}{2}}.

(Hint: Use inequality frac{a^2}{x}+frac{b^2}{y}geqfrac{(a+b)^2}{x+y} in frac{frac{1}{a^2}}{ab+ac}+frac{frac{1}{b^2}}{ab+bc}+frac{frac{1}{c^2}}{ac+bc}.)

 

7. Find all functions f:Qrightarrow Q such that

f(x+y)+f(x-y)=2f(x)+2f(y), for all rationals x, y.

 

OR

The sum of two integers is 52 and their L.C.M. is 168. Find the numbers.

 

8. 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.

 

9. If a_1leq a_3leqdots leq a_n be n real numbers such that sum_{j=1}^na_j=0.

Show that na_1a_2+sum_{j=1}^na_j^2leq 0.

 

OR

Prove that in a triangle ABC,angle A=2angle B if and only if a^2=b(b+c).

 

10. The number of class of 27 pupils each goes swimming on some of the days from Monday to Friday in a certain week. If each pupil goes atleast twice, show that there must be two pupils who go swimming on exactly the same days.

 

OR

Let ABC be an acute angled triangle; AD be the bisector of angle BAC with D on BC; BE be the altitude from B on AC. Show that angle CED>45^{circ}.

 

[ad#ad-2]

No Comments

Leave a Reply