SMO 2006 Open Round 1

How many integers are there between $0$ and $10^5$ having the digit sum equal to $8$?

Given that $p$ and $q$ are integers that satisfy the equation $36x^2-4(p^2+11)x+135(p+q)+576=0$, find the value of $p+q$.

A function $f:\mathbb R\to\mathbb R$ satisfies $f(xy+1)=f(x)f(y)-f(y)-x+2$ for all $x,y\in\mathbb R$. Find $10f(2006)+f(0)$.

Three people $A,B,C$ play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with $A$ and reaching $A$ again on the $11$th pass. For example, one possible sequence is
$$A\to B\to A\to B\to C\to A\to B\to C\to B\to C\to B\to A.$$

There are $10!$ permutations $s_0s_1\cdots s_9$ of $0,1,\ldots,9$. How many satisfy $s_k\ge k-2$ for $k=0,1,\ldots,9$?

A triangle $ABC$ has its vertices lying on a circle $\mathcal C$ of radius $1$, with $\angle BAC=60^\circ$. A circle with centre $I$ is inscribed in $ABC$. The line $AI$ meets $\mathcal C$ again at $D$. Find the length of $ID$.

Find the number of consecutive zeros at the end of the base-$10$ representation of $2006!$.

For any nonempty finite set $A$ of real numbers, let $s(A)$ be the sum of its elements. There are exactly $61$ three-element subsets $A$ of $\{1,\ldots,23\}$ with $s(A)=36$. Find the number of three-element subsets of $\{1,\ldots,23\}$ with $s(A)\lt 36$.

Suppose $f$ satisfies $f(x+x^{-1})=x^6+x^{-6}$ for all $x\ne0$. Determine $f(3)$.

Points $A,B,C$ lie on a circle centred at $O$ with radius $7$. The perpendicular bisector of $AB$ meets the segment $BC$ at $P$ and the extension of $AC$ at $Q$. Determine $OP\cdot OQ$.

In triangle $ABC$, $AB=AC=1$, and $D,E$ are the midpoints of $AB,AC$, respectively. Let $P$ be a point on $DE$. The extensions of $BP$ and $CP$ meet the sides $AC$ and $AB$ at $G$ and $F$, respectively. Find $\dfrac1{BF}+\dfrac1{CG}$.

In triangle $ABC$, $AB=14$, $BC=16$, $AC=26$. Let $M$ be the midpoint of $BC$, and $D$ the point on $BC$ such that $AD$ bisects $\angle BAC$. Let $P$ be the foot of the perpendicular from $B$ onto $AD$. Determine $PM$.

In triangle $ABC$, $\angle A=20^\circ$, $\angle C=90^\circ$, $O$ is a point on $AB$, and $D$ is the midpoint of $OB$. Suppose the circle centred at $O$ with radius $OD$ touches the side $AC$ at $T$. Determine $\angle BCD$ in degrees.

In triangle $ABC$, $D,E$ are the midpoints of $AB,AC$, respectively. The lines $CD,BE$ intersect at $P$, with $\angle BPC=90^\circ$. Suppose $BD=1829$ and $CE=1298$. Find $BC$.

Let $X=\{1,2,\ldots,17\}$. Find the number of subsets $Y$ of $X$ with odd cardinalities.

Find $400(\cos^5 15^\circ+\sin^5 15^\circ)\div(\cos15^\circ+\sin15^\circ)$.

Find the number of real solutions of $x^2+\dfrac1{x^2}=2006+\dfrac1{2006}$.

Find the largest integer $n$ such that $n$ is a divisor of $a^5-a$ for every integer $a$.

Given $A=\{1,2,\ldots,15\}$ and $B=\{0,1\}$, find the number of mappings $f:A\to B$ with $1$ being the image of at least two elements of $A$.

Let $a_1,a_2,\ldots$ satisfy $a_1=1$ and $a_n=10a_{n-1}-1$ for all $n\ge2$. Find the minimum $n$ such that $a_n\gt 10^{100}$.