MathNet 2011

Find the square root of $25 \cdot 26 \cdot 27 \cdot 28 + 1$.

Find all polynomials $P$, $Q$ with real coefficients, such that, for infinitely many positive integers $n$, $P(1)P(2)\dots P(n) = Q(n!)$.

Let $f(m) = 2^{2^{2 \cdots}}$ ($m$ times). Find the least $m$ so that $\log_{10} f(m)$ exceeds $6$.

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that $x f(y)=y f(x)$ for all $x, y \in \mathbb{R}$. Find the intersection of the graphs of $y=f(x)$ and $y=x^{2}+1$ if $f(1)=-1$.

Consider $A=\{1,2,3,\ldots,2011\}$. How many subsets of $A$ have element sum $2023060$?

There is a triple $k, m, n$ of positive integers without common factors such that $k \log_{400} 5 + m \log_{400} 2 = n$. Find the sum $k + m + n$.

Let $a$ and $b$ be two complex numbers. Prove that the following statements are equivalent:
1) The absolute values of the roots of the equation $x^2 - a x + b = 0$ are respectively equal to the absolute values of the roots of the equation $x^2 - b x + a = 0$.
2) $a^3 = b^3$ or $b = \bar{a}$.

An urn contains one black ball, four white balls, and some blue balls. A ball is drawn at random, its colour observed, and it is returned to the urn. Then another ball is drawn at random. For which numbers of blue balls is the probability that the two drawn balls have the same colour equal to $1/2$?

a) Show that if $a,b\gt 1$ are distinct real numbers then
$$\log_a(\log_a b)\gt \log_b(\log_a b).$$
b) Let $a_1\gt a_2\gt \dots\gt a_n\gt 1$, $n\ge2$. Prove that
$$\begin{gathered}\log_{a_1}(\log_{a_1}(a_2)+\log_{a_2}(\log_{a_2}(a_3)+\dots\\+\log_{a_{n-1}}(\log_{a_{n-1}}(a_n)+\log_{a_n}(\log_{a_n}(a_1))))\gt 0.\end{gathered}$$

A tournament consists of matches between exactly three players, each, respectively, garnering 2 points, 1 point, and a zero score. The ones who obtained no score are eliminated and the rest are grouped into threes to engage again in matches, with possibly one or two players having a bye. If there are 999 players in the tournament, how many matches would have been played by the time a champion, the one who earned two points in the last match, is declared?

Find all nonnegative integer solutions of the system
$$
\begin{aligned}
& 5x + 7y + 5z = 37 \\
& 6x - y - 10z = 3
\end{aligned}
$$

Let $m$ be a positive integer and $p$ a prime. Suppose $A$ is a unitary ring having exactly $m$ invertible elements and such that $\underbrace{1+1+\cdots+1}_{p \text{ times}} = 0$.
Prove that $A$ has non-zero nilpotent elements if and only if $p$ divides $m$.

Give three real roots of $\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1$.

Let $A$ be a ring and let $a$ be an element of $A$. Prove that

a) If $A$ is commutative and $a$ is nilpotent, then $a + x$ is invertible for any invertible element $x \in A$.

b) If $A$ is finite and $a + x$ is invertible for any invertible element $x \in A$, then $a$ is nilpotent.

(An element $a$ of a ring is called nilpotent if there exists some positive integer $n$ such that $a^n = 0$.)

Tio Mané has two boxes, one containing seven distinct balls numbered $1$ to $7$, and the other containing eight distinct balls labelled with all primes less than $20$. He draws one ball from each box at random.

Hint: Calculate the probability that the product is odd. What is the probability that the product of the numbers on the drawn balls is even?

In $\triangle A B C$, let $X$ and $Y$ be the midpoints of $A B$ and $A C$, respectively. On segment $B C$, there is a point $D$, different from its midpoint, such that $\angle X D Y = \angle B A C$. Prove that $A D$ is perpendicular to $B C$.

A convex polyhedron has $100$ edges. All its vertices are cut off close to the vertices with a sharp flat knife, so that the cutting planes do not intersect inside or on the boundary of the polyhedron. For the resulting polyhedron, calculate:
(a) the number of vertices;
(b) the number of edges.

How many faces does this new polyhedron have?

Find all positive integers $r$ with the property that there exists positive prime numbers $p$ and $q$ so that $p^2 + pq + q^2 = r^2$.

Find the sum $\sum_{k=1}^{19} k\binom{19}{k}$.

What are the last two digits of $5^{2011}$?