MathNet 2012

Determine all pairs of prime numbers $p$ and $q$ for which there exists an integer $a$ such that $a^4 = p a^3 + q$.

Ana is four times as old as Petar was when Ana was as old as Petar is now. When Petar becomes as old as Ana is now, the sum of their ages will be $95$. How old is Ana, and how old is Petar?

For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$.
Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences
$t(n+a) - t(n)$, $t(n+a+1) - t(n+1)$, ..., $t(n+2a-1) - t(n+a-1)$ are divisible by $4$.

Let $a$, $b$ be two odd natural numbers. Suppose that $a$ is a divisor of $b^2 + 2$ and $b$ is a divisor of $a^2 + 2$. Prove that $a$ and $b$ belong to the sequence $(v_n)$ defined by
$$
v_1 = v_2 = 1 \quad \text{and} \quad v_n = 4v_{n-1} - v_{n-2} \quad \forall n \ge 3.
$$

Given two arithmetic progressions $(a_n)$, $(b_n)$ and an integer $m \gt 2$. Consider $m$ quadratics $P_k(x) = x^2 + a_k x + b_k$, $k = 1, 2, \dots, m$. Suppose that neither $P_1(x)$ nor $P_m(x)$ has a real root. Prove that none of the given quadratics contains a real root.

One of the numbers $x^2$ and $(1-x)^2$ is smaller, and the other is greater than $1$.
Prove that $0 \lt x^2 - x \lt 2$.

The figure consisting of five unit squares is called a “plus”. In how many ways can we place a plus on a board of the same shape that contains $5 \cdot 5^2$ unit squares, so that the plus covers exactly five unit squares of the board?

Prove that for every integer $k \ge 2$ there exist $k$ positive integers whose sum is equal to their product.

Let $ABCDE$ be a regular pentagon. The lines $BC$ and $DE$ intersect at point $F$. Determine the angles of triangle $BEF$.

A triangle with an area of $1.5 \text{ cm}^2$ is inscribed in a circle of radius $1.25$ cm, with one side of the triangle as the diameter of the circle. What are the lengths of the sides of the triangle?

For which $n \in \mathbb{N}$ exist an angle $\alpha$ and a convex $n$-gon with angles $\alpha, 2\alpha, \dots, n\alpha$?

There is a person standing in each square of a $2012 \times 2012$ checkerboard; each one can be a truth-teller, someone who always tells the truth, or a liar, someone who always lies. Each person states the same: "In my row, there are as many liars as in my column." Determine the minimum amount of truth-tellers that there can be on the board.

There are $n$ cities ($n \ge 3$) and two airline companies in a country. Between any two cities, there is exactly one 2-way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$.

Suppose that $1000$ students are standing in a circle.
Prove that there exists an integer $k$ with $100 \le k \le 300$ such that in the circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half.

Let $\triangle ABC$ be an acute triangle, and let $M$ and $N$ be the feet of the altitudes from $A$ and $B$, respectively. If $|AN| = |NM|$, prove that the incentre of $\triangle ABC$ lies on the altitude $\overline{BN}$.

Ante wrote all integers from $1$ to $40$ in a row without spacings, thus obtaining the multi-digit number $12345\ldots383940$. Then he decided to erase $60$ digits of this number. What is the largest number Ante can get by doing this?

For each natural number $n$, let $a_n$ be the greatest perfect square number lower than or equal to $n$ and $b_n$ the smallest perfect square number greater than $n$. For instance, $a_9 = 3^2$, $b_9 = 4^2$, $a_{20} = 4^2$ and $b_{20} = 5^2$.
Calculate the sum of the 600 terms
$$
\frac{1}{a_1b_1} + \frac{1}{a_2b_2} + \dots + \frac{1}{a_{600}b_{600}}
$$

There are 100 metal balls which look exactly the same; 50 of them are radioactive. There are also three radiation detectors. For any group of balls, each detector is supposed to establish if there are radioactive balls in it or not. But it is known that one detector always gives the right answer, another one always gives the wrong answer, and the third one sometimes gives a right answer and sometimes a wrong one. However, it is not known which detector behaves in which way. Give a procedure by which the 50 radioactive balls can be sorted out with all certainty. The detectors can be used infinitely and with any amount of balls.

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcenter and circumradius of triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ similarly. Prove that
$$\begin{aligned}
&\frac1{O_1A_1^2-r_1^2}+\frac1{O_2A_2^2-r_2^2}\\
&\qquad+\frac1{O_3A_3^2-r_3^2}+\frac1{O_4A_4^2-r_4^2}=0.
\end{aligned}$$

Illustrative configuration only.

Let $D$ be a point on the side $\overline{BC}$ of triangle $ABC$. Denote $\alpha_1 = \angle DAB$ and $\alpha_2 = \angle CAD$. Prove the equality
$$
\frac{\sin(\alpha_1 + \alpha_2)}{|AD|} = \frac{\sin \alpha_1}{|AC|} + \frac{\sin \alpha_2}{|AB|}.
$$