IN-TST 2010

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$ . Suppose $P(x)$ and $Q(x)$ take negative values on $I$ , and they take non-negative values outside $I$ . Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$ .

For any integer $n\ge 2$ , let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:

(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$ ; (b) $j\neq k$ , then $a_j\neq a_k$ , $b_j\neq b_k$ and $c_j\neq c_k$ .

Determine $N(n)$ for all $n\ge 2$ .

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$ . Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})$$

Given an integer $k>1$ , show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$ , all greater than $1$ , such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$ -th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$ .)

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$ s and $1'$ s such that there are equal number of $00$ and $11$ blocks in each string is equal to $$2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}$$

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$ . Suppose $AD$ and $BC$ meet in $F$ . Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma$ is the circumcircle of triangle $EGH$ , prove that $FE$ is tangent to $\Gamma$ .

Call a positive integer good if either $N=1$ or $N$ can be written as product of even number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers.

(a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$ . Prove that $a=b$ .

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$ . Show that there exists $j<k$ and $l<m$ such that $$a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}$$

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$