MathNet 2011

Find all positive integers $n$ for which there exists a set $S \subset \mathbb{C}$ with $n$ elements, such that

i) each element of $S$ has modulus $1$,
ii) $\sum_{z \in S} z = 0$, and
iii) $z + w \neq 0$, for each $z, w \in S$.

Let $\triangle ABC$ be a triangle. Angle bisector of $\angle CAB$ intersects the segment $\overline{BC}$ at $D$, and angle bisector of $\angle ABC$ intersects the segment $\overline{AC}$ at $E$. If $\angle ACB \ge 60^\circ$, prove that $|AE| + |BD| \le |AB|$.

The difference in reciprocal values of two consecutive positive integers is $0.0\dot{a} = 0.0aaa...$. Determine all possible values of the digit $a$.

Let $\overline{AB}$ be a chord of a circle, parallel to its diameter $\overline{MN}$. Let $t$ be the tangent of that circle in $M$ and let $C, D$ be the intersections of the lines $NA, NB$ with $t$, respectively. Prove
$$
|MC| \cdot |MD| = |MN|^2.
$$

Find all pairs of positive integers $m$ and $n$, such that $\frac{m^2 n}{m^2 + n}$ is a prime number.

A sequence of positive real numbers contains at least 5 different terms. For any two terms of this sequence we can find two more terms with the same product. At least how many terms does the progression have?

Determine all $a \in \mathbb{R}$ such that for every $x \in \mathbb{R}$ the following inequality holds:
$$
\frac{x}{x^2 + 2x + 3} \gt \frac{x+a}{1+x+x^2}.
$$

Let $A$ be a finite set of positive reals, let $B = \{\frac{x}{y}: x, y \in A\}$ and let $C = \{xy: x, y \in A\}$. If we denote $cardX = |X|$, show that: $|A| \cdot |B| \le |C|^2$.

Determine all integers $a$ such that $\log_2(1 + a + a^2 + a^3)$ is also an integer.

A football tournament with $57$ teams follows these rules: a. No match may end in a draw. b. A team is eliminated after losing two matches. c. The tournament ends when only one team remains, the champion. If the champion loses once, how many matches are played?

Given $\log_a x = 3$, $\log_{ab} x = 2$, determine $\log_b x$.

Given nine positive integers, none of which has a prime divisor greater than $6$, prove that there are two of them whose product is a perfect square.

Bisector of an acute angle in the right triangle divides the opposite side into parts of lengths $4$ and $5$. What is the area of that triangle?

Let $a, b \gt 1$ be real numbers such that $\log_b a$, $\log_{2b}(2a)$ and $\log_{4b}(4a)$, in that order, form an arithmetic progression. Prove that $a = b$. (Tonći Kokan)

Given integers $n \ge 3$ and $k \ge 1$, find all positive integers $a_1 \lt a_2 \lt \dots \lt a_n$ such that $\frac{1}{a_1^k} + \frac{1}{a_2^k} + \dots + \frac{1}{a_n^k} = 1$.

Put one of the four letters $A, B, C, D$ and one of the four numbers $1, 2, 3, 4$ into each cell of the $4 \times 4$ table, so that:
(a) in each row and in each column, each letter and each number appears exactly once,
(b) each combination (pair) of one letter and one number appears in exactly one cell of the table.

$a_0=1$, $a_1=3$ and $a_{n+2}=1+\left\lfloor a_{n+1}^2/a_n\right\rfloor$ for all $n\ge0$. Show that $a_{n+2}a_n-a_{n+1}^2=2^n$ for all integers $n$.

At a volleyball tournament any two of the $n$ teams played against each other exactly once. For any two different teams $A$ and $B$ there were exactly $k$ teams that lost against both of them. Prove that $n = 4k + 3$.

A regular tetrahedron $ABXY$ is given inside the unit cube $ABCD'A'B'C'D'$ so that the point $X$ lies in the plane $ABCD$. Determine the distance between the points $Y$ and $A'$.

Find all the functions $f : [-\frac{\pi}{2}; \frac{\pi}{2}] \to \mathbb{R}$ such that:
a. $f$ is differentiable on $[-\frac{\pi}{2}, \frac{\pi}{2}]$;
b. there exists an antiderivative $F$ of $f$ such that $F(x) + f'(x) \le 0, \forall x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ and $F(-\frac{\pi}{2}) = F(\frac{\pi}{2}) = 0$.