IMO Problems Weekly! (1990)

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

Problem 2: Find all positive integers n having the property that 2n+1n2\dfrac{2^n +1}{n^2} is an integer.

Problem 3: Given a circle with two chords AB,CDAB,CD that meet at EE, let MM be a point of chord ABAB other than EE. Draw the circle through D,ED, E, and MM. The tangent line to the circle DEMDEM at EE meets the lines BC,ACBC,AC at F,G,F,G, respectively. Given AMAB=λ\dfrac{AM}{ AB} = \lambda, find GEEF\dfrac{GE}{ EF} .

Problem 4: On a circle, 2n1(n3)2n-1( n \geq 3) different points are given. Find the minimal natural number NN with the property that whenever NN of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly nn of the given 2n12n-1 points.

This is part of the set International Mathematical Olympiads

Note by Sualeh Asif
5 years, 6 months ago

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Is the discussion on?

Aditya Narayan Sharma - 5 years, 3 months ago

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May b off

Sayandeep Ghosh - 5 years, 2 months ago

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@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 - 5 years, 6 months ago

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