Equations and Inequalities

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Diffrentiation and integration

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On the Sunday morning, I'd like make new post about Integral and diffrentiation

Calculus – differentiation, integration etc. – is easier than you think. Here's a simple example: the bucket at right integrates the flow from the tap over time. The flow is the time derivative of the water in the bucket. The basic ideas are not more difficult than that. Calculus analyses things that change, and physics is much concerned with changes. For physics, you'll need at least some of the simplest and most important concepts from calculus. Fortunately, one can do a lot of introductory physics with just a few of the basic techniques.
So stick with us: differentiation really is just subtracting and dividing, and integration really is just multiplying and adding. This short introduction is no substitute, however, for a good high school calculus course: we shall take some short cuts of which mathematicians may disapprove.
Differentiation: How rapidly does something change? T…
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IMO 2014 problems and Solutions

On, Thursday, 12 May 2016. 1 : 48 pm. i'd like to make an solution of math which i copy from the book.

Problem 1. Let a0 < a1 < a2 . . . be an infinite sequence of positive integers. Prove that there exists a unique integer n ≥ 1 such that an < a0 + a1 + a2 + · · · + an n ≤ an+1.
 Problem 2. Let n ≥ 2 be an integer. Consider an n × n chessboard consisting of n 2 unit squares. A configuration of n rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer k such that, for each peaceful configuration of n rooks, there is a k × k square which does not contain a rook on any of its k 2 unit squares.
 Problem 3. Convex quadrilateral ABCD has ∠ABC = ∠CDA = 90◦ . Point H is the foot of the perpendicular from A to BD. Points S and T lie on sides AB and AD, respectively, such that H lies inside triangle SCT and ∠CHS − ∠CSB = 90◦ , ∠T HC − ∠DT C = 90◦ . Prove that line BD is tangent to the circumcircle of triangle T SH…
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