The goal of this set of notes is to improve our problem solving and proof writing skills. You are encouraged to submit a solution to any of these problems, and join in the discussion in #imo-discussion. For more details, see IMO Problems Discussion Group.

So the wait for the IMO problems is over.....

To change from our rigorous yearly format, I have now switched to selected problems over the years! These are some really beautiful problems you should try your hand at! Post in your progress(even if it is little!) and we might find some out of the box Solutions! So here goes some problems from 1990:

**Problem 1:** Let \(\mathbb{Q}^+ \) be the set of positive rational numbers. Construct a function \(f: \mathbb{Q}^+ \rightarrow \mathbb{Q}^+ \) such that
\[f(xf(y))=\dfrac{f(x)}{y}\]
, for all \(x,y\) in \(\mathbb{Q}^+ \)

**Problem 2:** Find all positive integers n having the property that \(\dfrac{2^n +1}{n^2}\) is an integer.

**Problem 3:** Given a circle with two chords \(AB,CD\) that meet at \(E\), let \(M\) be a point of chord \(AB\) other than \(E\). Draw the circle through \(D, E\), and \(M\). The tangent line to the circle \(DEM\) at \(E\) meets the lines \(BC,AC\) at \(F,G,\) respectively. Given \(\dfrac{AM}{ AB} = \lambda\), ﬁnd \(\dfrac{GE}{ EF}\) .

**Problem 4:** On a circle, \(2n−1( n \geq 3)\) diﬀerent points are given. Find the minimal natural number \(N\) with the property that whenever \(N\) of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly \(n\) of the given \(2n−1\) points.

## Comments

Sort by:

TopNewestIs the discussion on? – Aditya Narayan Sharma · 1 year, 1 month ago

Log in to reply

– Sayandeep Ghosh · 1 year ago

May b offLog in to reply

@Sharky Kesa @Xuming Liang @Abdur Rehman Zahid @Aareyan Manzoor

The next note in the series is up!

(I didn't forget to tag you this time Sharky) – Sualeh Asif · 1 year, 3 months ago

Log in to reply