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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, 3 months ago