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


(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}.



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


(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.



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.



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.



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}.



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