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 […]
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 […]
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.
Cyclic Quadrilaterals are often important objects in a Geometry problem. Recognizing them can lead to a path to the solution. Pre RMO 2017 Problem 13 Solution is a part of our Pre-RMO problem solving series. Also visit: Math Olympiad Program of Cheenta
Tools in Geometry is very useful for pre regional mathematical olympiad, regional mathematical olympiad as well as I.S.I. & C.M.I entrance.
Cheenta is introducing open seminar for students interested in Advanced Mathematics and preparing for Pre-RMO and ISI Entrance Students. Know more..
It is interesting to represent Algebraic Identities using Geometric objects. In other words the Relationship between algebra and geometry. One easy example is presented in this video. Think! Relationship between Algebra and Geometry Also Visit: Thousand Flowers Program Try to draw the following algebraic identities using geometric objects \(a^2 - b^2 = (a+b)(a-b) \) \( (a+b)^3 = a^3 + […]
Try this beautiful problem from Geometry: Ratios of the areas of Triangle and Quadrilateral from AMC-10A. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on real numbers. Use sequential hints if required.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Digits and Integers.
This is a beautiful problem from ISI MStat 2016 PSB (sample) Problem 1 based on area bounded by the curves. We provide a detailed solution with the prerequisites mentioned explicitly.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Complex numbers and Sets.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Consecutive Positive Integers.
Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on a derivative of Function. You may use sequential hints to solve the problem.
Try this beautiful Logarithm Problem From Singapore Mathematics Olympiad, SMO, 2011 (Problem 7). You may use sequential hints to solve the problem.
Try this beautiful problem from the Pre-RMO, 2017 based on Sides of Quadrilateral. You may use sequential hints to solve the problem.
This is a beautiful problem from ISI MStat 2016 PSB (sample) based on Vectorspace . It uses several concepts to solve it . We provide detailed solution with prerequisites mentioned explicitly .