This post contains the problems from Indian National Mathematics Olympiad, INMO 2008 Question Paper. Do try to find their solutions. Let $ABC$ be a triangle, $I$ its in-centre; $ A_1, B_1, C_1 $ be the reflections of $I$ in BC, CA, AB respectively. Suppose the circum-circle of triangle $ A_1 B_1 C_1 $ passes through […]
Test of Mathematics Solution Subjective 35 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also see: Cheenta I.S.I. & C.M.I. Entrance Course Problem (a) Prove that, for any odd integer n, $ […]
12. In $ (\triangle ABC, AB=AC=28)$ and BC=20. Points D,E, and F are on sides $ (\overline{AB}, \overline{BC})$, and $ (\overline{AC})$, respectively, such that $ (\overline{DE})$ and $ (\overline{EF})$ are parallel to $ (\overline{AC})$ and $ (\overline{AB})$, respectively. What is the perimeter of parallelogram ADEF?$ (\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }60\qquad\textbf{(E) }72\qquad )$ Solution: Perimeter = […]
4. Let N be an integer greater than 1 and let $ (T_n)$ be the number of non empty subsets S of ({1,2,.....,n}) with the property that the average of the elements of S is an integer. Prove that $(T_n - n)$ is always even. Sketch of the Proof: $ (T_n )$ = number of […]
3 Let $ (a,b,c,d \in \mathbb{N})$ such that $ (a \ge b \ge c \ge d)$. Show that the equation $ (x^4 - ax^3 - bx^2 - cx -d = 0)$ has no integer solution. Sketch of the Solution: Claim 1: There cannot be a negative integer solution. Suppose other wise. If possible $x= […]
1. Let $(\Gamma_1)$ and $(\Gamma_2)$ be two circles touching each other externally at R. Let $(O_1)$ and $(O_2)$ be the centres of $(\Gamma_1)$ and $(\Gamma_2)$, respectively. Let $(\ell_1)$ be a line which is tangent to $(\Gamma_2)$ at P and passing through $(O_1)$, and let $(\ell_2)$ be the line tangent to $(\Gamma_1)$ at Q and passing […]
This post contains problems from Indian National Mathematics Olympiad, INMO 2013. Try them and share your solution in the comments. Problem 1 Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles touching each other externally at $R$. Let $l_{1}$ be a line which is tangent to $\Gamma_{2}$ at $P$ and passing through the center $O_{1}$ of $\Gamma_{1}$. […]
6. Find all positive integers n such that $latex (3^{2n} + 3 n^2 + 7 )$ is a perfect square. Solution: We use the fact that between square of two consecutive numbers there exist no perfect square. That is between $(k^2 )$ and $((k+1)^2 )$ there is no square. Note that $(3^{2n} = (9^n)^2 )$ […]
5. Let ABC be a triangle. Let D, E be points on the segment BC such that BD = DE = EC. Let F be the mid point of AC. Let BF intersect AD in P and AE in Q respectively. Determine the ratio of triangle APQ to that of the quadrilateral PDEQ. Solution: Applying Menelaus' […]