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Problems worth
thinking about.

A growing archive of problems from every level we teach — each with hints and a full solution. Try before you reveal.

From the founder’s blog · Number theory

Find the remainder when is divided by 7.
Hint 1
Hint 2
Solution

N is divisible by 7, so the remainder is 0. Original post ↗ ∎

From the founder’s blog · Series / JEE Advanced

Evaluate
Hint 1

Sum over all triples first, then remove the triples where two or more indices are equal.

Hint 2
Solution

By inclusion–exclusion, S = 81/208. Original post ↗ ∎

Number theory · Class 9+

Find the last two digits of .
Hint 1

Last two digits = remainder on division by 100. Try small powers of 7.

Hint 2
Solution

The last two digits are 49. ∎

Divisibility · Foundation

Prove that is divisible by 6 for every integer .
Hint 1

Factorise the expression completely.

Hint 2
Solution

The expression is a product of three consecutive integers. Among any three consecutive integers one is divisible by 3 and at least one is even, so the product is divisible by 2 × 3 = 6. ∎

Algebra · Class 9+ / Olympiad

If , prove that .
Hint 1
Solution

By the identity in the hint, the left side minus 3abc has the factor a + b + c = 0, so it equals zero. Hence a³ + b³ + c³ = 3abc. ∎

Limits · EAMCET / JEE Main

Evaluate
Hint 1
Solution

The limit is 1/2. ∎

Definite integrals · JEE

Evaluate
Hint 1
Hint 2

Replace x by π/2 − x and add the two forms of I.

Solution

∎

Combinatorics · JEE / Quant

In how many ways can 6 people sit around a round table?
Hint 1

Rotations of the same arrangement are identical. Fix one person’s seat.

Solution

Fixing one person, the remaining 5 can be arranged in 5! = 120 ways. ∎

Linear algebra · IIT JAM / GATE

Find the eigenvalues of
Hint 1
Solution

The eigenvalues are 1 and 3. ∎

Inequalities · Olympiad / ISI

For positive reals , prove .
Hint 1
Solution

Dividing by ab > 0 gives the result, with equality exactly when a = b. ∎

From the founder

Read more
on the blogs.

Blog · Research mathematics

Mathematics in Research

Problems and discussions in the Mathematics behind research — calculus, differential equations, probability and stochastic processes, linear algebra, optimisation and signal analysis, with a focus on wireless communications and signal processing.

Visit blog Blog · Classes & lectures

Ekaveera Math Lectures

Class announcements from Ekaveera Gouribhatla: IIT Foundation, Vedic Maths, Olympiads (IOQM to EGMO), CBSE, ICSE, IGCSE and IB, B.Sc and B.Tech Mathematics, and TIFR, IIT JAM and GATE.

Visit blog

Study guide

How to study
Mathematics well.

  1. Derive, then use

    Before using a formula, try to derive it once. It turns memory into understanding.

  2. Struggle productively

    Give a problem real time before reading the solution — the struggle is where learning happens.

  3. Keep an error log

    Write down every mistake and why it happened. Review it weekly.

  4. Mix your practice

    Exams combine chapters. Practise mixed sets, not only chapter-wise exercises.

  5. Explain it aloud

    If you can explain a solution to someone else, you have truly understood it.

  6. Test under real conditions

    Timed practice builds the calm and pace that examination day requires.

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We’ll recommend the route.

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