MathNet 2013

If at the cinema three box offices are open, the visitors have to wait 15 min to buy a ticket. By how many minutes is the waiting time reduced if two more box offices are opened?
(A) 3
(B) 5
(C) 6
(D) 7
(E) 10

Prove that, if $a, b, c$ are the complex coordinates of the vertices of a triangle and $|a + 2b - 3c| = |a - 4b + 3c|$, then the triangle is right.

Prove that the product of the first 2013 terms of the sequence
$$
a_n = 1 + \frac{1}{n^3}
$$
is less than 3.

Find all functions $f:\mathbb R\setminus\{-1\}\to\mathbb R$ such that
$$f(x)+f(y)=(x+y+2)f(x)f(y)$$
for all $x,y\in\mathbb R\setminus\{-1\}$.

Determine all integers $m \ge 2$ such that every integer $n, \frac{m}{3} \le n \le \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

Let $D$ be the midpoint of the side $AB$, let $E$ be the point of intersection of the side $BC$ and the bisector of the angle $\angle BAC$, and let $F$ be the orthogonal projection of the point $E$ onto the side $AB$ of the triangle $ABC$. Suppose $\angle CDA = \angle ACB$ and $|CE| = |BF|$. Find the sizes of the angles of the triangle $ABC$.

Suppose $f: \mathbb{R} \to \mathbb{R}$ is a monotonic function.

a. Prove that $f$ has one-sided limits at any point $x_0 \in \mathbb{R}$.

b. Define the function $g: \mathbb{R} \to \mathbb{R}$, $g(x) = \lim_{t \to x} f(t)$, i.e. $g(x)$ is the left-sided limit at $x$ of the function $f$. Prove that $g$ is a continuous function, then $f$ is also continuous.

In acute triangle $ABC$, points $D, E, F$ are the altitude feet from $A, B, C$ respectively. Let $I_1, I_2$ be the incenters of $AEF$ and $BDF$, and $O_1, O_2$ the circumcenters of $ACI_1$ and $BCI_2$ respectively. Prove lines $I_1I_2$ and $O_1O_2$ parallel.

Find all natural numbers $n$ of the form $n = \overline{23ab16c}$, such that all their digits are different and they are divisible by $9$ and $11$. Here, $a$, $b$ and $c$ are digits.

A table tennis tournament is conducted in the following way. In each round, if the number of contestants is odd, one contestant is drawn to proceed automatically to the next round. After that, pairs are drawn from the other contestants. The contestants from each pair compete against each other, and the winner proceeds to the next round. Let $f(n)$ denote the number of rounds in the tournament with $n$ contestants. (For example, $f(5) = 3$.) Determine $f(2013)$ and find the least positive integer $n$ such that $f(n) = f(2013)$.

What is the largest possible greatest common divisor of $11n+4$ and $7n+2$ when $n$ is a natural number?

Find the two integer values of $x$ nearest to $2013^\circ$, one below and one above, satisfying
$$
2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}
$$

Let $ABC$ be a triangle with $D$ a point on the side $AB$ and $E$ a point on the side $AC$ such that $|AE| = |ED| = |DB|$ and $|AD| = |DC| = |CB|$. Determine the sizes of the angles of the triangle $ABC$.

Let $a_1, a_2, \dots$ be a sequence of integers defined recursively by $a_1 = 2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the $2013$th power of the digits of $a_n$. Do there exist distinct positive integers $i, j$ such that $a_i = a_j$?

Prove that every tangential quadrilateral whose diagonals intersect at right angles is a deltoid.

Starting with $37$, successively prefix the digit $5$ to obtain
$37$, $537$, $5537$, $55537$, $555537$, ...
How many terms are prime?

Let $P$ be inside acute triangle $ABC$. The distances from $P$ to the three vertices are $d_A, d_B, d_C$, and its perpendicular distances to the sides are $d_1, d_2, d_3$ respectively. Prove
$$
d_A + d_B + d_C \geq 2(d_1 + d_2 + d_3).
$$

Let $M,N,P$ be the midpoints of $BC,AC,AB$, respectively, and let $O$ be the circumcenter of acute triangle $ABC$. The circumcircles of triangles $BOC$ and $MNP$ intersect at distinct points $X,Y$ inside triangle $ABC$. Prove that $\angle BAX=\angle CAY$.

Calculate the sum of the reciprocals of the first 2013 terms of the sequence with general term
$$
a_n = 1 - \frac{1}{4n^2}
$$

Let $k$ be a positive integer. Let $f:\mathbb Z\to\mathbb Z$ be a bijection such that for any integers $i,j$ satisfying $|i-j|\le k$, we have $|f(i)-f(j)|\le k$. Prove that $|f(i)-f(j)|=|i-j|$ for all $i,j\in\mathbb Z$.