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 circumcircle of triangle $ A_1 B_1 C_1 $ passes through A. Prove that $ B_1, C_1, I, I_1 $ are concyclic, where $ I_1 $ is the incentre of triangle […]
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 $(3^{2n} = (9^n)^2 )$ […]
Try this beautiful problem from Algebra on Quadratic equation from AMC-10A, 2002. You may use sequential hints to solve the problem.
Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Arbitrary Arrangement. You may use sequential hints.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Remainders and Functions.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Rational Numbers.
This problem is a simple application of the sequential definition of continuity from ISI MStat 2015 PSB Problem 1.
This is a problem from ISI MStat Examination,2019. This tests one's familiarity with the simple and multiple linear regression model and estimation of model parameters and is based on the Invariant Regression Coefficient.
This is a problem from the ISI MStat Entrance Examination, 2019. This primarily tests one's familiarity with size, power of a test and whether he/she is able to condition an event properly.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.
Try this beautiful problem from the Pre-RMO, 2017 based on ratio and proportion. You may use sequential hints to solve the problem.
Try this beautiful problem from Number Theory based on largest possible value from AMC-10A, 2004. You may use sequential hints to solve the problem.