# 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
4 years ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$