This is a problem from Indian National Mathematics Olympiad, INMO, 2013 based on Orthocenter on perpendicular bisector. Try out this problem. Problem: Orthocenter on perpendicular bisector In an acute angled triangle ABC with AB < AC the circle $latex \Gamma $ touches AB at B and passes through C intersecting AC again at D. Prove […]
Find the number of 4-tuples (a,b,c,d) of natural numbers with $latex a \le b \le c $ and $latex a! + b! + c! = 3^d $ Discussion: Number of 4-tuples The basic idea is: factorial function is faster than the exponential function in the long run. Note that all three of a, b, c […]
Find the number of 8 digit numbers sum of whose digits is 4. Discussion: Suppose the number is $latex a_1 a_2 a_3 ... a_8 $.The possible values of $latex a_1 $ are 1, 2, 3, 4. We consider these four cases. If $latex a_1 = 4 $ then all other digits are 0 (since sum […]
In this post, there are questions from Regional Math Olympiad 2013. Try out the problems. Find the number of 8 digit numbers sum of whose digits are 4.Discussion Find the number of 4-tuples (a,b,c,d) of natural numbers with $latex a \le b \le c $ and $latex a! + b! + c! = 3^d […]
"How do I involve my child in challenging mathematics? He gets good marks in school tests but I think he is smarter than school curriculum." "My daughter is in 4th grade. What competitions in mathematics and science can she participate in? How do I help her to perform well in those competitions?" "I have a […]
In math olympiads around the world, number theory problems have many recurring themes. One such theme is the 'LTE' or lifting the exponent. Selective Logarithm We define a function callled Selective Logarithm (SL) which works on all integers (positive, negative or zero). Suppose N is any integer and p be a prime then $latex SL_p […]
Let's discuss a problem based on Arithmetic of Remainders and understand the concept behind it. Consider the two number: 37 and 52 What is the remainder when we divide 37 by 7? 2 of course. And 52 produces remainder 3 when divided by 7. Suppose we want to know the remainder when the product of […]
Let's understand one-one function and differentiability with the help of a problem. Try it yourself before reading the solution. Let f be real valued, differentiable on (a, b) and $ f'(x) \ne 0 $ for all $ x \in (a, b) $. Then f is 1-1. True Discussion: Suppose f is not 1-1. Then there […]
Lets understand direct product of two subgroups with the help of a problem. This problem is useful for College Mathematics.