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Problem Set 2

Hello! My intention for this week's problem set was a set about combinatorics. However, last week I couldn't study as much as I wanted, so I wasn't able to compile three nice enough problems in combinatorics. Sorry about that. Anyway, this set contains two problems in algebra (number 3 is particularly hard!!) and an easy geometry one. Please reshare the note if you liked it. Enjoy!

  1. In a \(\triangle{ABC}\), let \(I\) be the incentre and let \(X, Y, Z\) be the points of tangency of the incircle with sides \(BC, AC, AB\), respectively. Line \(AX\) cuts the incircle again in \(P\), and line \(AI\) cuts \(YZ\) in \(Q\). Prove that \(X, I, Q, P\) lie on a circle. (Costa Rican OMCC TST 3, 2014)

  2. Let \(P(x)\) be a polynomial with integer coefficients such that there exist four distinct, positive integers \(a, b, c, d\) which satisfy \(P(a)=P(b)=P(c)=P(d)=5\). Show that there does not exist an integer \(k\) which satisfies \(P(k)=8\). (Canada, 1970)

  3. Let \(c\) be a positive integer. We define the sequence \(x_n\) as follows: \(x_1=c\), and for \(n\geq{2}\), \(x_n=x_{n-1}+\lfloor{\dfrac{2x_{n-1}-(n+2)}{n}}\rfloor\). Find a closed expression for \(x_n\) in terms of \(n\) and \(c\). (Costa Rican OIM TST 1, 2014)

Note by José Marín Guzmán
3 years ago

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