US-TST 2010

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and $$\gcd(P(0), P(1), P(2), \ldots ) = 1.$$ Show there are infinitely many $n$ such that $$\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.$$

Let $a, b, c$ be positive reals such that $abc=1$ . Show that $$\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.$$

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that $$\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.$$

Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$ . Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$ . The incircle of triangle $CMN$ touches segment $AC$ at $E$ . The incircle of triangle $BPQ$ touches segment $AB$ at $F$ . Line $EN$ and $AB$ meet at $R$ , and lines $FQ$ and $AC$ meet at $S$ . Given that $AE = AF$ , prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$ .

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and, for $n > 1$ , $$a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.$$ Prove that there are infinitely many $n$ such that $a_n \equiv n \pmod{2^{2010}}$ .

Let $T$ be a finite set of positive integers greater than 1. A subset $S$ of $T$ is called good if for every $t \in T$ there exists some $s \in S$ with $\gcd(s, t) > 1$ . Prove that the number of good subsets of $T$ is odd.

In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$ . Point $D$ lies on segment $BC$ . Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$ .

Let $m,n$ be positive integers with $m \geq n$ , and let $S$ be the set of all $n$ -term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$ . Show that $$\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} = {n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots + (-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.$$

Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and $$\binom{3k}{k} \equiv 1 \pmod{p}.$$