Try this beautiful problem from TOMATO Objective no. 27 based on Closure of a set of even numbers. Problem: Closure of a set of even numbers S is the set whose elements are zero and all even integers, positive and negative. Consider the 5 operations- [1] addition; [2] subtraction; [3] multiplication; [4] division; and […]
Try this beautiful problem from TOMATO Objective no. 13 based on Calendar Problem. This problem is useful for BSc Maths and Stats Entrance Exams. Problem: June 10, 1979, was a SUNDAY. Then May 10, 1972, was a (A) Wednesday; (B) Friday; (C) Sunday; (D) Tuesday; Solution: In a (non-leap) year there are 365 days. $365 […]
Try this beautiful problem from TOMATO Subjective Problem no. 173 based on the Sum of Polynomials. Problem : Sum of polynomials Let [latex] {{P_1},{P_2},...{P_n}}[/latex] be polynomials in [latex] {x}[/latex], each having all integer coefficients, such that [latex] {{P_1}={{P_1}^{2}+{P_2}^{2}+...+{P_n}^{2}}}[/latex]. Assume that [latex] {P_1}[/latex] is not the zero polynomial. Show that [latex] {{P_1}=1}[/latex] and [latex] {{P_2}={P_3}=...={P_n}=0}[/latex] Solution : As [latex] […]
This problem is from the Test of Mathematics, TOMATO Subjective Problem no. 172 based on the Round Robin tournament. Problem : Suppose there are [latex] {k}[/latex] teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the [latex] {i^{th}}[/latex] team loses [latex] […]
This is a Test of Mathematics Solution Subjective 87 (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 visit: I.S.I. & C.M.I. Entrance Course of Cheenta Problem: Let \(P(z) = az^2+ bz+c\), […]
This is a beautiful problem based on Some Direct Inequalities from Test of Mathematics Subjective Problem no. 80. Problem: Some Direct Inequalities If \(a,b,c\) are positive numbers, then show that \(\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b}\geq a+b+c\) Solution: This problem can be solved using a direct application of the Titu's Lemma but we will instead prove the lemma first using […]
This is a beautiful problem based on Solving Equations from Test of Mathematics Subjective Problem no. 20. Problem : Solving equations If \(\ a,b,c,d\) satisfy the equations $$a+7b+3c+5d=0,$$ $$8a+7b+6c+2d=-16,$$ $$2a+6b+4c+8d=16,$$ $$5a+3b+7c+d=-16,$$ then \(\ (a+d)(b+c)\) equals \(\ (A)16 \quad (B)-16\quad (C)0 \quad\) (D)none of the foregoing numbers Solution: $$a+7b+3c+5d=0\dots(1),$$ $$8a+7b+6c+2d=-16\dots(2),$$ $$2a+6b+4c+8d=16\dots(3),$$ $$5a+3b+7c+d=-16\dots(4),$$ \(\ (1)-(3)\), and \(\ […]
Cauchy Schwarz Problem: Let be a polynomial with non-negative coefficients.Prove that if for ,then the same inequality holds for each . Discussion: Cauchy Schwarz's Inequality: Suppose for real numbers (\ a_{i},b_{i}), where (\ i\in{1,2,\dots,n}) we can say that $${\sum_{i=1}^{n}a_{i}^2}{\sum_{i=1}^{n}b_{i}^2}=\sum_{i=1}^{n}{a_{i}b_{i}}^2$$. Titu's Lemma: Let (\ a_{i},b_{i}\in{\mathbb{R}}) and let (\ a_{i},b_{i}>0) for (\ i\in{1,2,\dots,n}) $$\sum_{i=1}^{n}\frac{a_{i}^2}{b_{i}}\ge\frac{{\sum_{i=1}^{n}a_{i}}^2}{\sum_{i=1}^{n}b_{i}}$$ Proof of Cauchy Schwarz's Inequality: We […]
PROBLEM: Given and are two quadratic polynomials with rational coefficients. Suppose and have a common irrational solution. Prove that for all where is a rational number. SOLUTION: Suppose the common irrational root of (\ f(x)) and (\ g(x)) be (\sqrt{a}+b). Then by properties of irrational roots we can say that the other root of both of […]
Problem: Let ( \ k) be a fixed odd positive integer.Find the minimum value of ( \ x^2+y^2),where ( \ x,y) are non-negative integers and ( \ x+y=k). Solution: According to Cauchy Schwarz's inequality, we can write, ( \ (x^2+y^2)\times(1^2+1^2) \ge)(\ (x\times1+y\times1)^2) =>( \ 2(x^2+y^2)\ge)(\ (x+y)^2) =>( \ x^2+y^2\ge) (\frac{k^2}{2}) Therefore,the minimum value of ( \ x^2+y^2) is […]
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Right Rectangular Prism.
Try this beautiful problem from the Pre-RMO, 2019 based on Greatest Integer. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1996 based on Parallelogram Problem.
Try this beautiful problem from PRMO, 2019, problem-12, based on Integer Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Sectors in Circle from AMC-10A, 2012. You may use sequential hints to solve the problem
Try this beautiful problem from Algebra: Sum of whole numbers from AMC-10A, 2012. You may use sequential hints to solve the problem
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Pyramid with Square base.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Repeatedly Flipping a Fair Coin.
Try this beautiful problem from the Pre-RMO, 2019 based on Sum of digits. You may use sequential hints to solve the problem.
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Trigonometry Simplification. You may use sequential hints.
Are you preparing for ISI MStat Entrance Exams? Here is the list of useful books for ISI MStat Entrance Exam based on the syllabus.
Answer Key (This is a work in progress, please proceed with caution. We are reviewing some of these answers). Problem 1 Three parallel lines $L_{1}, L_{2}, L_{3}$ are drawn in the plane such that the perpendicular distance between $L_{1}$ and $L_{2}$ is 3 and the perpendicular distance between $L_{2}$ and $L_{3}$ is also $3 .$ […]
Explore this beautiful book on problems useful for Math Olympiad, ISI CMI Entrance. It is written by three Russian authors. Title: Selected Problems and Theorems in Elementary Mathematics – Shklyarsky, Chentsov, Yaglom
Try these AMC 10 Geometry Questions and check your knowledge! Two right circular cones with vertices facing down as shown in the figure below contains the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius […]
Author: Kazi Abu Rousan C is hard but fast But you need to be on guard to last. Python is easy but slow But you can use it to glow. But if you have julia Beautiful rhythms will flow. ---Me Julia is a high-level, high-performance, dynamic programming language. Most of you guys have heard or […]
Dear student, In the past few years several Cheenta students reached the top 300 universities in the world. These universities include Oxford, UCLA, NUS, MIT and University of Edinburgh. We have gradually shaped a success pathway for students that works in the long run. This pathway can be useful for you as well.There two components of this success path: Component 1: Performance […]
Author: Kazi Abu Rousan There are some problems in number theory which are very important not only because they came in exams but also they hide much richer intuition inside them. Today, we will be seeing one of such problems. Sources: B.Stat. (Hons.) and B.Math. (Hons.) I.S.I Admission Test 2012 problem-2. B.Stat. (Hons.) and B.Math. […]