IN-TST 2002

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$ . Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$ . Let $\Gamma$ be the circle which is tangent to the line $l$ , tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$ . The diameter of $\Gamma$ through $D$ meets $l$ in $E$ . Show that $AB=DE$ .

Show that there is a set of $2002$ consecutive positive integers containing exactly $150$ primes. (You may use the fact that there are $168$ primes less than $1000$ )

Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$ . How many quadratics are there of the form $ax^2+2bx+c$ , with equal roots, and such that $a,b,c$ are distinct elements of $X$ ?

Determine the number of $n$ -tuples of integers $(x_1,x_2,\cdots ,x_n)$ such that $|x_i| \le 10$ for each $1\le i \le n$ and $|x_i-x_j| \le 10$ for $1 \le i,j \le n$ .

Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.

Let $\sigma(n)=\sum_{d|n} d$ , the sum of positive divisors of an integer $n>0$ . [list](a) Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ (b) Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$ .[/list]

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$ , and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

Let $a,b$ be integers with $0<a<b$ . A set $\{x,y,z\}$ of non-negative integers is olympic if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$ . Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$ . Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$ , and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$ . Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$

Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}$$

Given a prime $p$ , show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.