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
5 years, 6 months ago

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