Problem Garden

Mathematics is not a spectator sport. In this portal, we have gathered (and are adding) problems, discussions and challenges that you may try your hands on.

I.S.I 2019 Subjective Problem -4

Understand the problem Let  be a twice differentiable function such thatShow that there exist  such that  for all . I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2019. Subjective Problem no. 4 calculus  8.5 out of 10Problems In CALCULUS OF...

Sum Of 1’S C.M.I UG-2019 Entrance

Understand the problem Find the sum 1+111+11111+1111111+.....1....111(2k+1) ones   C.M.I UG-2019 entrance examAlgebra 3.5 out of 10 challenges and trills of pre college mathematics    Start with hintsDo you really need a hint? Try it first!can you some how...

C.M.I-2019 Geometry problem

Understand the problemlet O be a point inside a parallelogram ABCD such that \(\angle AOB+\angle COD =180\) prove that \(\angle OBC =\angle ODC\) C.M.I (Chennai mathematical institute UG-2019 entrance    Geometry 5 out of 10challenges and thrills of pre college...

Triangle in complex plane – ISI 2019 Obj P8

This problem from ISI Entrance 2019 is an interesting application of complex numbers in geometry. Try your hands on this!

Four Points on a Circle, ISI Entrance 2017, Subjective Problem no 2

Understand the problem Consider a circle of radius 6 as given in the diagram below. Let \(B,C,D\) and \(E\) be points on the circle such that \(BD\) and \(CE\), when extended, intersect at \(A\). If \(AD\) and \(AE\) have length 5 and 4 respectively, and \(DBC\) is a...

The Product of Digits, ISI Entrance 2017, Subjective Solution to problem – 5.

Understand the problem  Let \(g : \mathbb{N} \to \mathbb{N} \) with \( g(n) \) being the product of digits of \(n\).        (a) Prove that \( g(n)\le n\) for all \( n \in \mathbb{N} \) .        (b) Find all \(n \in \mathbb{N} \) , for which \( n^2-12n+36=g(n) \)....

Three Primes, ISI Entrance 2017, Subjective solution to Problem 6.

Understand the problemLet \(p_1,p_2,p_3\)  be primes with \(p_2\neq p_3\), such that \(4+p_1p_2\) and \(4+p_1p_3\) are perfect squares. Find all possible values of \(p_1,p_2,p_3\).   Start with hintsDo you really need a hint? Try it first!Let \(4+p_1p_2=m^2\) and...

System of n equations, ISI Entrance 2008, Solution to Subjective Problem No. 9.

Understand the problem For \(n\ge3 \), determine all real solutions of the system of \(n\) equations :                                                \(x_1+x_2+\cdots+x_{n-1}=\frac{1}{x_n}\)                                                                ...

C.M.I. 2019 Entrance – Answer Key, Sequential Hints

CMI (Chennai Mathematical Institute) Entrance 2019, Sequential hints, answer key, solutions.

A Trigonometric Substitution, ISI Entrance 2019, Subjective Solution to Problem – 6 .

Understand the problem For all natural numbers\(n\), let          \(A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}\)           (\( n\) many radicals) (a) Show that for \(n\ge 2,  A_n=2\sin \frac{π}{2^{n+1}}\). (b) Hence, or otherwise, evaluate the limit              ...

Math Olympiad

Problems and discussions from various math olympiads including RMO, INMO (India), USAMO, AMC (United States), BMC and more.

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Informative Articles

Selected articles on books, learning methods, and scholarship opportunities.

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ISI and CMI Entrance Solutions and Problems

Problems and Solutions from Test of Mathematics at 10+2 Level, previous year ISI and CMI Entrances.

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College Mathematics

Problems and discussions from TIFR, M.Math, Subject GRE, IIT JAM and more.

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