We have a cyclic pentagon, such that AE is a diameter and equals 10. We know and . Now F, G, H, I are the feet of the perpendiculars from C to AB, EB, AD, ED. If FG and HI concur at J, find FJI. Now, let K be the foot of the perpendicular from […]
500 Miles High
First of all, R.I.P. Chick Corea. He’s recently passed away, and I cannot explain how sad it’s been for me to hear such terrible news. He’s been my primary source of inspiration in my path through jazz music. His memory will live forever, through the beautiful pieces and performances he’s left to us. So, I […]
Problem 7, Combinatorics
I proposed this problem in the last mathematical competition hosted on the Italian ‘gasmatematica’ website. The numbers from 000 to 999 must be put into some boxes, numbered from 00 to 99. If the label of a box is equal to a number with one digit removed, then the number can be put in that […]
Hidden Telescoping Sum
Here’s an exercise I proposed in an online competition. You have the following sequence: Find the integer part of: Now, notice that Hence By substitution, we have that the sum telescopes and reduces to Since is a very large number (the sequence has a quadratic increasing rate), we have that the integer part of the […]
The Earthsea Saga
The Earthsea saga, by Ursula K. Le Guin, is a collection of fantasy books I particularly enjoyed. It talks about a world of magical creatures that live together in harmony. Some humans have the power to cast magic spells and control nature. The most gifted and powerful ones gather on the island of Roke, where […]
The Sicilian Defense May Be Nasty
Here’s a game I played with white in the Sicilian Defense which really shows how the Sicilian is a double-edge defense. 1. e4 c5 2. Nf3 Nc6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 e6 6. f3 a6 7. Bc4 Bb4 8. Nxc6 bxc6 9. Bd2 d5 Here I should have gone for e5 […]
IMO 2007/2
Consider five points A, B, C, D, E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let l be a line passing through A. Suppose that l intersects the interior of the segment DC at F and intersects line BC at G. Suppose also that EF = EG = EC. Prove […]
Vishwanathan Anand
Vishwanathan ‘Vishy’ Anand is one of my favourite chess players. He was the world champion before Magnus took the throne in 2013. Vishy is a very aggressive and creative player. Here are some of my favourite games he played. He became world champion in 2007 and held it against Kramnik (2008), Topalov (2010) and Gelfand(2012). […]
An Old Birthday Gift
This is an old birthday gift for a special person. It was the first time I tried to plot something using python, and I wasn’t very familiar with matplotlib, so I used the turtle library. import turtleimport timedef auguri(): colors = [‘blue’, ‘red’, ‘green’, ‘yellow’, ‘orange’, ‘purple’] turtle.pensize(5) turtle.bgcolor(‘black’) turtle.speed(1000) for x in range(360): turtle.pencolor(colors[x […]
Chick Corea
Chick Corea is by far my favourite pianist. He’s won 23 grammy awards and he’s written 2 of my 3 favourite pieces of all time. He can literally play anything, but I think that his genius is expressed by latin music. In fact, the 2 pieces I love are Spain and Armando’s Rhumba. These are […]