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 ABC\triangle{ABC}, let II be the incentre and let X,Y,ZX, Y, Z be the points of tangency of the incircle with sides BC,AC,ABBC, AC, AB, respectively. Line AXAX cuts the incircle again in PP, and line AIAI cuts YZYZ in QQ. Prove that X,I,Q,PX, I, Q, P lie on a circle. (Costa Rican OMCC TST 3, 2014)

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

  3. Let cc be a positive integer. We define the sequence xnx_n as follows: x1=cx_1=c, and for n2n\geq{2}, xn=xn1+2xn1(n+2)nx_n=x_{n-1}+\lfloor{\dfrac{2x_{n-1}-(n+2)}{n}}\rfloor. Find a closed expression for xnx_n in terms of nn and cc. (Costa Rican OIM TST 1, 2014)

Note by José Marín Guzmán
7 years ago

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