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# 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.

## Looks can be deceiving

Understand the problemFind all non-zero real numbers  which satisfy the system of equations:Indian National Mathematical Olympiad 2010AlgebraMediumAn Excursion in MathematicsStart with hintsDo you really need a hint? Try it first!When a polynomial equation looks...

## IMO, 2019 Problem 1 – Cauchyish Functional Equation

This problem is a patient and intricate and simple application of Functional Equation with beautiful equations to be played aroun with.

## A sequence of natural numbers and a recurrence relation

Understand the problemDefine a sequence  by ,  andfor  For every  and  prove that  divides. Suppose  divides  for some natural numbers  and . Prove that  divides Indian National Mathematical Olympiad 2010 Number Theory Medium Problem Solving Strategies by Arthur Engel...

## Linear recurrences

Linear difference equationsA linear difference equation is a recurrence relation of the form $latex y_{t+n}=a_1y_{t+n-1}+a_2y_{t+n-2}+\cdots +a_ny_t+b$. If $latex b=0$, then it is called homogeneous. In this article, we shall also assume $latex t=0$ for...

## 2013 AMC 10B – Problem 5 Maximizing the Difference:

This is based on simple ineqaulities on real numbers.

## An inductive inequality

Understand the problemGiven  and  for all , show that  Singapore Mathematical Olympiad 2010 Inequalities Easy Inequalities by BJ Venkatachala Start with hintsDo you really need a hint? Try it first!Use induction. Given the inequality for $latex n=k$, the inequality...

## A search for perfect squares

Understand the problemDetermine all pairs  of positive integers for which  is a perfect square.Indian National Mathematical Olympiad 1992 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!First consider $latex n=0$....

## INMO 1996 Problem 1

Understand the problema) Given any positive integer , show that there exist distint positive integers  and  such that  divides  for ; b) If for some positive integers  and ,  divides  for all positive integers , prove that .Indian National Mathematical Olympiad...

## Trigonometric substitution

Understand the problemLet  with . Prove thatDetermine when equality holds.Singapore Team Selection Test 2004InequalitiesMediumInequalities by BJ VenkatachalaStart with hintsDo you really need a hint? Try it first!Show that there exists a triangle $latex \Delta ABC$...

## The non-existence of a polynomial

Understand the problemIf  is a polynomial with integer coefficients and , , , three distinct integers, then show that it is impossible to have , , .Indian National Mathematical Olympiad 1986AlgebraEasyAn Excursion in MathematicsStart with hintsDo you really need a...

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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