MathNet 2013

A finite set of distinct positive integers is written on a blackboard. A *move* consists in choosing two numbers and write their lowest common multiple, given that is not already written. The set is called *closed* if no moves are allowed – e.g., the set $\{2, 3, 4, 6\}$ will be closed after number $12$ is added. Determine the maximum number of elements in a closed set given that the initial set contains $10$ numbers.

Find all pairs of real numbers $(a, b)$ such that the equality $|ax + by| + |bx + ay| = 2|x| + 2|y|$ holds for all reals $x$ and $y$.

How many permutations $\left(s_{1}, s_{2}, \cdots, s_{n}\right)$ of $(1,2, \ldots, n)$ are there satisfying the condition $s_{i}\gt s_{j}$ for all $i \geq j+3$ when $n=5$ and when $n=7$?

A set S of positive integers is called canalero if for any three distinct $a,b,c\in S$, a divides bc, b divides ca, and c divides ab.

a. Prove that for every finite set of positive integers $\{c_1,\ldots,c_n\}$ there are infinitely many positive integers k such that $\{kc_1,\ldots,kc_n\}$ is canalero.

b. Prove that for every integer $n\ge3$ there is a canalero set with exactly n elements, no integer greater than 1 dividing all its elements.

The first 50 natural numbers will walk down a corridor containing 50 doors numbered 1 to 50, all initially locked. Number 1 goes first, then 2, then 3, and so on, with 50 last. Number n carries the keys to doors whose numbers are multiples of n. Thus 1 carries every key, 2 carries only keys to even-numbered doors, and 25 carries only keys to doors 25 and 50. Each number uses its keys to lock doors that are open and unlock doors that are closed.

a) Which doors will number 15 unlock?

b) Show that after number 50 has walked through, door 10 will be unlocked whereas door 9 will be locked.

c) After number 50 has walked through, which doors will be unlocked?

a. Show that $x^4 - x^3 - x + 1 \ge 0$, for all real numbers $x$.

b. Find all real numbers $x_1, x_2$ and $x_3$ given that $x_1 + x_2 + x_3 = 3$ and $x_1^3 + x_2^3 + x_3^3 = x_1^4 + x_2^4 + x_3^4$.

Let $ABCD$ be a square. Point $E$ lies in the interior of the angle $\angle CAB$ such that angle $\angle BAE$ is $15^\circ$, and lines $BE$ and $BD$ are perpendicular. Show that $AE = BD$.

$\triangle ABC$ is a triangle with $AB \lt BC$, $\mathcal{C}$ its circumcircle, $K$ the midpoint of the minor arc $\widehat{CA}$ of the circle $\mathcal{C}$ and $T$ a point on $\mathcal{C}$ such that $KT$ is perpendicular to $BC$. If $A'$, $B'$ are the intouch points of the incircle of $\triangle ABC$ with the sides $BC$, $AC$, prove that the lines $AT$, $BK$, $A'B'$ are concurrent.

Let $ABC$ be a triangle and let points $M$ and $N$ on the sides $AB$ and $AC$ respectively such that $\angle ABC \equiv \angle ANM$. Let $D$ be the mirror image of point $A$ across $B$, and let $P$ and $Q$ be the midpoints of the line segments $MN$ and $CD$ respectively. Show that points $A$, $P$ and $Q$ are collinear if and only if $AC = AB\sqrt{2}$.
Petru Braica

Find all real $x \gt 0$ and integer $n \gt 0$ so that $\lfloor x \rfloor + \left\{ \frac{1}{x} \right\} = 1.005 \cdot n$.

What is the largest positive integer $k$ such that $27!$ is divisible by $2^{k}$?

Suppose $A$ is a singular matrix of order $n$ with complex entries, all of which having absolute value equal to $1$.

a. Let $n = 3$. Show that two lines or two columns of the matrix $A$ are proportional.

b. Find, with proof, if the above claim holds for $n = 4$.

A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on.
a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves?
b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

Determine all injective functions $f: \mathbb{Z} \to \mathbb{Z}$ that satisfy
$$
|f(x) - f(y)| \le |x - y|,
$$
for all $x, y \in \mathbb{Z}$.

a) Consider $f: [0, \infty) \to [0, \infty)$ a differentiable and convex function such that $f(x) \le x$, for $x \ge 0$. Then $f'(x) \le 1$, for all $x \ge 0$.

b) Determine all differentiable and convex functions $f: [0, \infty) \to [0, \infty)$ with the properties $f(0) = 0$ and $f'(x) \cdot f(f(x)) = x$, for all $x \ge 0$.

Suppose $a$ and $b$ are complex numbers. Prove that $|az + b\bar{z}| \le 1$ for all $z \in \mathbb{C}$ with $|z| = 1$ if and only if $|a| + |b| \le 1$.
Liviu Vlaicu

If $p$ is a real constant such that the roots of the equation $x^{3}-6 p x^{2}+5 p x+88=0$ form an arithmetic sequence, find $p$.

Let $P$ be a point in the interior of $\triangle ABC$. Extend $AP$, $BP$, and $CP$ to meet $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. If $\triangle APF$, $\triangle BPD$, and $\triangle CPE$ have equal areas, prove that $P$ is the centroid of $\triangle ABC$.

Let $n$ be a positive integer. The numbers $1, 2, 3, \ldots, 2n$ are randomly assigned to $2n$ distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints.

Show that one can choose $n$ pairwise non-intersecting chords such that the sum of the values assigned to them is $n^{2}$.

The right prism $ABCA'B'C'$, with $AB = AC = BC = a$, has the property that there exists a unique point $M \in (BB')$ so that $AM \perp MC'$. Find the measure of the angle of the straight line $AM$ and the plane $(ACC')$.
Mircea Fianu, Cristian Mangra