SMO 2007 Open Round 1

Let $A$ be any $k$-element subset of $\{1,2,3,\ldots,100\}$. Determine the minimum $k$ such that there must be two numbers $a,b$ in $A$ with $|a-b|\le4$.

Determine the number of binary sequences consisting of ten zeros and ten ones which do not contain three consecutive zeros.

Let $A$ be a set of any $20$ points on the circumference of a circle. Joining any two points of $A$ produces one chord. Suppose every three such chords are not concurrent. Find the number of regions inside the circle divided by all these chords.

In each of the seven-digit natural numbers $1001011$, $5550000$, $3838383$, $7777777$, every digit appearing in the number appears at least three times. Find the number of such seven-digit natural numbers.

Let $A=\{1,2,\ldots,1000\}$. Let $m$ be the number of two-element subsets $\{a,b\}$ of $A$ such that $ab$ is divisible by $6$. Find $\lfloor m/10\rfloor$. Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

Find the number of nonnegative integer solutions of $x+y+z+u\le20$.

In how many ways can seven different prizes be awarded to five students so that each student receives at least one prize?

Let $ABC$ be any triangle. Let $D,E$ lie on the segments $AB,BC$, respectively, with $AD=7DB$ and $BE=10EC$. Suppose $AE$ and $CD$ meet at $F$. Determine $\lfloor k\rfloor$, where $AF=k\cdot FE$.

Let $S=\{1,2,\ldots,50\}$. A three-element subset $\{a,b,c\}$ is good if $a+b+c$ is divisible by $3$. Find the number of good three-element subsets of $S$.

Let $x_1,\ldots,x_{1970}$ be positive integers with $x_1+\cdots+x_{1970}=2007$. Determine the largest possible value of $x_1^3+\cdots+x_{1970}^3$.

Determine the largest real $a$ for which $a^2-bc-8a+7=0$ and $b^2+c^2+bc-6a+6=0$ hold for some real $b,c$.

Determine the number of distinct integers among $\left\lfloor1^2/2007\right\rfloor,\left\lfloor2^2/2007\right\rfloor,\ldots,\left\lfloor2007^2/2007\right\rfloor$.

Determine the number of pairs $(a,b)$ of integers with $1\le b\lt a\le200$ such that $(a+b)+(a-b)+ab+a/b$ is a square of a number.

In an acute-angled triangle $ABC$, $D,E,F$ are the feet of the perpendiculars from $A,B,C$ onto $BC,AC,AB$, respectively. Suppose $\sin A=3/5$ and $BC=39$. Find $AH$, where $H=AD\cap BE$.

Let $O$ be the circumcentre of triangle $ABC$, and $P,Q$ the midpoints of $AO,BC$, respectively. Suppose $\angle CBA=4\angle OPQ$ and $\angle ACB=6\angle OPQ$. Find $\angle OPQ$ in degrees.

In triangle $ABC$, $AC\gt AB$. The internal angle bisector of $A$ meets $BC$ at $D$, and $E$ is the perpendicular foot from $B$ onto $AD$. Suppose $AB=5$, $BE=4$, $AE=3$. Find $\dfrac{AC+AB}{AC-AB}\,ED$.

Find $\displaystyle\prod_{k=1}^{45}\tan(2k-1)^\circ$.

Find the radius of the circle inscribed in a triangle of side lengths $50,120,130$.

Suppose $0\lt a\lt b\lt c\lt d=2a$ and
$$(d-a)\left(\frac{a^2}{b-a}+\frac{b^2}{c-b}+\frac{c^2}{d-c}\right)=(a+b+c)^2.$$
Find $bcd/a^3$.

Let $f$ satisfy $f(x)-\frac12f(1/x)=\log x$ for all $x\gt 0$, where the logarithm has base $10$. Find $f(1000)$.