MathNet 2004 — Part 01

Let $O$ be the circumcentre and $H$ the orthocentre of an acute triangle $A B C$. Prove that the area of one of the triangles $A O H$, $B O H$ and $C O H$ is equal to the sum of the areas of the other two.

Determine all finite nonempty sets $S$ of positive integers satisfying
$$
\frac{i+j}{(i, j)} \quad \text{is an element of} S \text{ for all } i, j \text{in} S,
$$
where $(i, j)$ is the greatest common divisor of $i$ and $j$.

Prove that
$$
\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9(ab+bc+ca)
$$
for all real numbers $a, b, c \gt 0$.

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that
$$
\left\lfloor\frac{(n-1)!}{n(n+1)}\right\rfloor
$$
is even for every positive integer $n$.

Let a set $S$ of 2004 points in the plane be given, no three of which are collinear. Let $\mathcal{L}$ denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of $S$ with at most two colours, such that for any points $p, q$ of $S$, the number of lines in $\mathcal{L}$ which separate $p$ from $q$ is odd if and only if $p$ and $q$ have the same colour.

Note: A line $\ell$ separates two points $p$ and $q$ if $p$ and $q$ lie on opposite sides of $\ell$ with neither point on $\ell$.

Determine the positive integers $n$ such that the set of all positive divisors of $30^n$ can be divided into groups of three so that the product of the three numbers in each group is the same.

On the pizarrón there was written a 100-digit number with the last three digits at the right equal to $999$. Under this number, using exactly the same digits, but in another order, Luciano writes a new 100-digit number: leave the last three $999$, and voluntarily exchange the first $97$ digits. This operation is repeated once and again, until numbers of $100$ digits are written on the $99$ icon. Next, add these $99$ numbers, and divide them by $72$. Calculate the remainder of the division that Luciano makes.

There is a square board of $8 \times 8$ divided into squares of $1 \times 1$. Write in each square a $1$ or a $2$ so that in each square of $3 \times 3$ the sum of the $9$ numbers is a multiple of $4$, but the sum of the $64$ numbers on the board is not a multiple of $4$.

Using cardboard equilateral triangles with side $1$, an equilateral triangle with side $2^{2004}$ is formed. From this triangle, the small triangle with side $1$ is extracted, whose center coincides with the center of the larger triangle.

Determine if it is possible to completely cover the remaining surface, without overlaps or gaps, if only pieces in the shape of an isosceles trapezoid are available, each of which is made up of three equilateral triangles with side $1$.

The integers $2003$ from $1$ to $2003$ are written on the board. Lucas must delete $90$ numbers. Next, Mauro must choose $37$ from the numbers that remain written. If the $37$ numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins.
Decide if Lucas can choose the $90$ numbers to clear so that victory is assured.

We consider the 2004 integers $n$, from 1 to 2004. Determine for how many of these values of $n$ it is verified that the number $n^3+3^n$ is a multiple of 5.

The $ABCD$ trapezium of bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$, has $\angle A=90^\circ$, $AB=6$, $CD=3$ and $AD=4$. Let $E$, $G$, $H$ be the circumcenters of the triangles $ABC$, $ACD$, $ABD$, respectively. Find the area of ​​triangle $EGH$.

**CLARIFICATION:** The circumcenter of a triangle is the point of intersection of the three bisectors of its sides.

Given a circle $C$ and a point $P$ outside it, the two tangents to the circle are drawn through $P$, with $A$ and $B$ being the points of tangency.
A point $Q$ is taken on the minor arc $AB$ of $C$. Let $M$ be the intersection of the line $AQ$ with the perpendicular to $AQ$ drawn by $P$ and let $N$ be the intersection of the line $BQ$ with the perpendicular to $BQ$ drawn by $P$.
Show that, by varying $Q$ in the arc $AB$, all the lines $MN$ pass through the same point.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$ and $\frac{CE}{AC} = \frac{3}{7}$, $\frac{DE}{BD} = \frac{4}{9}$. Let $P$ and $Q$ be the points that divide the segment $BE$ into three equal parts, with $P$ between $B$ and $Q$, and let $R$ be the midpoint of the segment $AE$. Calculate $\frac{\text{area}(APQR)}{\text{area}(ABCD)}$.

Given a circle with center $O$, four tangent lines are drawn to the circle so that these four lines determine the trapezoid $ABCD$, with bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$. If $AO = 2\sqrt{6}$, $BO = 4\sqrt{3}$ and $CO=4$, calculate the measurements of the sides and angles of the trapezoid.

Let $a \geq 4$ be a positive integer. Determine the smallest value of $n \geq 5$, such that $a$ can be represented in the form
$$
\sigma = \frac{x_1^2 + x_2^2 + \dots + x_n^2}{x_1 x_2 \dots x_n}
$$
for a proper choice of positive integers $n$ $x_1, x_2, \dots, x_n$.

Arnaldo chooses a number $a$, $a \ge 0$, and Bernaldo chooses a number $b$, $b \ge 0$. They both secretly tell their chosen number to Cernaldo, who writes on a blackboard the numbers $5$, $8$ and $15$, one of them being the sum $a + b$.
Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on separate pieces of paper whether or not they know which of the numbers on the board is the sum of $a$ and $b$ and hand them to Cernaldo.
If NO is written on both pieces of paper, Cernaldo rings the bell again and the process is repeated.
Arnaldo and Bernaldo are known to be sincere and intelligent.
What is the maximum number of times you could ring the bell until one of them writes that they know the value of the sum?

There is a rectangular board of $5 \times 50$, divided into squares of $1 \times 1$, and dominoes of $1 \times 2$, each with two numbers written on it: a $1$ and a $-1$, one in each half. Each domino covers exactly two neighboring squares on the board. Gabriel must cover the board with these pieces, without gaps or overlaps, and in such a way that in each row of the board the multiplication of the $50$ numbers is positive, and in each column of the board the multiplication of the $5$ numbers is positive.

Decide if Gabriel can make it. What if the board is from $5 \times 100$?

If yes, indicate how the board is covered. If not, explain why.

The positive integer $n$ has exactly 18 positive divisors, counting $1$ and $n$. The divisors of $n$ are numbered from smallest to largest (the first is $1$ and the eighteenth is $n$) and the sixth of these divisors is denoted $x$. It is known that the thirteenth divisor, multiplied by the sum of the first plus the second plus the fifth divisor, is equal to divisor number $x+1$. Find $n$.

Let $m$, $n$ be positive integers. On a $m+n$ board gridded into $1 \times 1$ squares, we consider all paths from the upper right vertex to the lower left vertex, traversing grid lines exclusively in the directions $\leftarrow$ and $\downarrow$.

The area of ​​a path is defined as the number of squares on the board below that path. If $p$ is a prime such as $r_p(m) + r_p(n) \ge p$, where $r_p(m)$ denotes the remainder from dividing $m$ by $p$ and $r_p(n)$ denotes the remainder from dividing $n$ by $p$.

How many roads have area multiple of $p$?