Problem Let ABC be an acute-angled triangle, let D, F be the mid-points of BC, AB respectively. Let the perpendicular from F to AC and the perpendicular at B to BC meet in N. Prove that ND is equal to circum-radius of ABC. Theorems and tools The discussion uses the following Theorems: Midpoint Theorem: The line […]
TIFR 2017 Math Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects. The image is a front cover […]
This is a problem from ISI B.Stat-B.Math Entrance Exam 2018, Subjective Problem 7. It is based on Bases, Exponents and Role reversals. I.S.I. Entrance 2018 Problem 7 Let $(a, b, c)$ are natural numbers such that $(a^{2}+b^{2}=c^{2})$ and $(c-b=1)$. Prove that(i) a is odd.(ii) b is divisible by 4(iii) $( a^{b}+b^{a} )$ is divisible by […]
Problem In a quadrilateral $ABCD$. It is given that $AB=AD=13$, $BC=CD=20$, $BD=24$. If $r$ is the radius of the circle inscribable in the quadrilateral, then what is the integer close to $r$? Hint 1: First, notice that the quadrilateral is a kite. Diagonals of a kite bisect each other (Prove this!) If $X$ is the point […]
A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the third. The number of pages in the second volume is 50 more than in the first volume, and the number pages in the third volume is one and a half times that in […]
Pre RMO 2018 Find the problems, discussions and relevant theoretical expositions related to Pre-RMO 2018. Problems of Pre RMO 1. A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the third. The number of pages in the second volume is […]
Problem Suppose (a, b) are positive real numbers such that (a \sqrt{a}+b \sqrt{b}=183 . a \sqrt{b}+b \sqrt{a}=182). Find (\frac{9}{5}(a+b)). Hint 1 This problem will use the following elementary algebraic identity: $(x+y)^3=x^3+y^3+3 x^2 y+3 x y^2$ Can you identify what is x and what is y? Hint 2 background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none"] Set $x=\sqrt{a}, y=\sqrt{b}$. Then the […]
In this post we have discussed AMC 10A 2018 problem number 13.