SMO 2005 Open Round 1

Find the last three digits of $9^{100}-1$.

Circles $C_1$ and $C_2$ have radii $3$ and $7$ respectively. The circles intersect at distinct points $A$ and $B$. A point $P$ outside $C_2$ lies on the line determined by $A$ and $B$ at a distance of $5$ from the center of $C_1$. Point $Q$ is chosen on $C_2$ so that $PQ$ is tangent to $C_2$ at $Q$. Find the length of the segment $PQ$.

Find the largest positive integer $n$ such that $n!$ ends with exactly $100$ zeros.

Find the largest value of $3k$ for which the following equation has a real root:

$$\sqrt{x^2-k}+2\sqrt{x^2-1}=x.$$

Find the last two digits (in order) of $7^{3^{2005}}$.

From the first $2005$ natural numbers, $k$ of them are arbitrarily chosen. What is the least value of $k$ to ensure that there is at least one pair of numbers such that one of them is divisible by the other?

Let $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$, where $a_i$ are nonnegative integers for $i=0,1,\ldots,n$. If $f(1)=21$ and $f(25)=78357$, find the value of $f(10)$.

If $x$ is a real number that satisfies

$$\left\lfloor x+\frac{11}{100}\right\rfloor+\left\lfloor x+\frac{12}{100}\right\rfloor+\cdots+\left\lfloor x+\frac{99}{100}\right\rfloor=765,$$

find the value of $\lfloor100x\rfloor$. Here $\lfloor a\rfloor$ denotes the largest integer $\le a$.

The function $f(n)$ is defined for all positive integers $n$ and takes nonnegative integer values such that $f(2)=0$, $f(3)>0$ and $f(9999)=3333$. Also, for all positive integers $m,n$,

$$f(m+n)-f(m)-f(n)\in\{0,1\}.$$

Determine $f(2005)$.

It is known that the three sides of a triangle are consecutive positive integers and the largest angle is twice the smallest angle. Find the perimeter of this triangle.

A triangle $ABC$ is inscribed in a circle of radius $1$ with $\angle BAC=60^\circ$. Altitudes $AD$ and $BE$ of $ABC$ intersect at $H$. Find the smallest possible value of the length of the segment $AH$.

Let $\mathbb N$ be the set of all positive integers. A function $f:\mathbb N\to\mathbb N$ satisfies $f(m+n)=f(f(m)+n)$ for all $m,n\in\mathbb N$, and $f(6)=2$. Also, no two of the values $f(6),f(9),f(12),f(15)$ coincide. How many three-digit positive integers $n$ satisfy $f(n)=f(2005)$?

Let $f$ be a real-valued function such that

$$f(x,y)=f(x,z)-2f(y,z)-2z$$

for all real numbers $x,y,z$. Find $f(2005,1000)$.

Let $a_1=2006$, and for $n\ge2$,

$$a_1+a_2+\cdots+a_n=n^2a_n.$$

What is the value of $2005a_{2005}$?

Find the smallest three-digit number $n$ such that if the three digits are $a,b,c$, then

$$n=a+b+c+ab+bc+ac+abc.$$

Let $a_1=21$ and $a_2=90$, and for $n\ge3$, let $a_n$ be the last two digits of $a_{n-1}+a_{n-2}$. What is the remainder of $a_1^2+a_2^2+\cdots+a_{2005}^2$ when it is divided by $8$?

Find the smallest two-digit number $N$ such that the sum of digits of $10^N-N$ is divisible by $170$.

Find the least $n$ such that whenever the elements of the set $\{1,2,\ldots,n\}$ are coloured red or blue, there always exist $x,y,z,w$ (not necessarily distinct) of the same colour such that $x+y+z=w$.

Let $x$ and $y$ be positive integers such that $\frac{100}{151}\lt \frac yx\lt \frac{200}{251}$. What is the minimum value of $x$?

Find the maximum positive integer $n$ such that $n^2\le160\times170\times180\times190$.