SMO 2008 Open Round 1

Determine the number of three-element subsets of $\{1,2,3,\ldots,120\}$ for which the sum of the three elements is a multiple of $3$.

There are $10$ students taking part in a mathematics competition. After the competition, they discover that each solves exactly $3$ problems and any two solve at least one common problem. What is the minimum possible number of students solving a problem solved by the most students?

Evaluate $\displaystyle\sum_{n=1}^{6237}\lfloor\log_2 n\rfloor$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

Determine the number of positive integer divisors of $998^{49999}$ that are not divisors of $998^{49998}$.

Let $p(x)$ be a polynomial with real coefficients such that, for all real $x$, $2(1+p(x))=p(x-1)+p(x+1)$, and $p(0)=8$, $p(2)=32$. Determine $p(40)$.

In triangle $ABC$, $AC=2BC$, $\angle C=90^\circ$, and $D$ is the foot of the altitude from $C$ onto $AB$. A circle with diameter $AD$ intersects the segment $AC$ at $E$. Find $AE:EC$.

In triangle $ABC$, $AB=8$, $BC=7$, and $CA=6$. Let $E$ be the point on $BC$ such that $\angle BAE=3\angle EAC$. Find $4AE^2$.

In triangle $ABC$, the bisectors of $\angle A$ and $\angle B$ meet at the incentre $I$. The extension of $AI$ meets the circumcircle of $ABC$ at $D$. Let $P$ be the foot of the perpendicular from $B$ onto $AD$, and $Q$ a point on the extension of $AD$ such that $ID=DQ$. Determine $\dfrac{BQ\cdot IB}{BP\cdot ID}$.

In a convex quadrilateral $ABCD$, $\angle BAC=\angle CAD$ and $\angle ABC=\angle ACD$. The extensions of $AD$ and $BC$ meet at $E$, and the extensions of $AB$ and $DC$ meet at $F$. Determine $\dfrac{AB\cdot DE}{BC\cdot CE}$.

For a positive integer $n$, let $N_n=\{1,2,3,\ldots,n\}$. Assume $n\ge10$. Determine the maximum $n$ such that $\displaystyle\min_{\substack{a,b\in A\\a\ne b}}|a-b|\le10$ for every $A\subseteq N_n$ with $|A|\ge10$.

How many four-digit numbers greater than $5000$ can be formed from the digits $0,1,2,3,4,5,6,7,8,9$ if only the digit $4$ may be repeated?

Three girls $A,B,C$ and nine boys are to be lined up in a row. Let $n$ be the number of ways to do this if $B$ must lie between $A$ and $C$, and $A,B$ must be separated by exactly four boys. Determine $\lfloor n/7!\rfloor$.

Determine the number of four-element subsets $\{a,b,c,d\}$ of $\{1,2,3,\ldots,20\}$ such that $a+b+c+d$ is divisible by $3$.

How many three-digit numbers between $100$ and $999$ inclusive have two and only two consecutive digits identical?

Find the maximum natural number that is divisible by $30$ and has exactly $30$ different positive divisors.

Determine the number of zeros at the end of the product $1\cdot2\cdot3\cdot4\cdots2008$.

Let $a_k$ be the coefficient of $x^k$ in $(1+2x)^{100}$, for $0\le k\le100$. Find the number of integers $r$ with $0\le r\le99$ such that $a_r\lt a_{r+1}$.

Let $a_k$ be the coefficient of $x^k$ in $(x+1)+(x+1)^2+\cdots+(x+1)^{99}$. Determine $\lfloor a_4/a_3\rfloor$.

Let $a,b,c,d,e$ be five numbers satisfying $a+b+c+d+e=0$ and $abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde=2008$. Find $a^3+b^3+c^3+d^3+e^3$.

Let $a_1,a_2,\ldots$ be rational numbers with $a_1=2$ and $a_{n+1}=(1+a_n)/(1-a_n)$ for $n\ge1$. Determine $30a_{2008}$.