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+\ldots + x_{14}^4)=7(x_1^3+ x_2^3+\ldots + 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 satisfya^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+c\neq 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\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\geq\frac{3}{2}.

    (Hint: Use inequality \frac{a^2}{x}+\frac{b^2}{y}\geq\frac{(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:Q \rightarrow Q such thatf(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:N\rightarrow 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_1\leq a_3\leq\ldots \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^2\leq 0.


    Prove that in a \triangle ABC , \angle A=2 \angle 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}.