MathNet 2011

Determine, with proof, whether there is any odd integer $n \ge 3$ and $n$ distinct prime numbers $p_1, p_2, \dots, p_n$ such that all $p_i + p_{i+1}$ ($i = 1, 2, \dots, n$, and $p_{n+1} = p_1$) are perfect squares? (posed by Tao Pingsheng)

Suppose that a line passing the circumcentre $O$ of $\triangle ABC$ intersects $AB$ and $AC$ at points $M$ and $N$, respectively, and $E$ and $F$ are the midpoints of $BN$ and $CM$, respectively. Prove that $\angle EOF = \angle A$. (posed by Tao Pingsheng)

Every two of $n$ towns in a country are connected by one way or two way road. It is known that for every $k$ towns there exists round trip passing through each of these $k$ towns exactly once. Find the maximal possible number of one way roads.

Determine all $a$ such that there exists unique $(x, y) \in \mathbb{R}^2$ satisfying
$$
2^{|x|} + |x| = x^2 + y + a, \quad x^2 + y^2 = 1.
$$

Let $a_1, a_2, \dots$ be a permutation of all positive integers. Prove that there exist infinite positive integers $i$'s, such that $(a_i, a_{i+1}) \le \frac{3}{4}i$. (posed by Chen Yonggao)

Let $a$, $b$, $c$ be positive real numbers satisfying $a^2 \lt bc$. Prove that
$$
b^3 + ac^2 \gt ab(a + c)
$$

Let $3n^2$ be the vertex number of a simple graph $G$ (integer $n \ge 2$). If the degree of each vertex is not greater than $4n$, there exists at least one vertex with degree $1$, and there exists a route with length not greater than $3$ between any two vertices. Prove that the minimum number of edges of $G$ is $\frac{7}{2}n^2 - \frac{3}{2}n$.

Remark. A route between two distinct vertices $u$ and $v$ with length $k$ is a sequence of vertices $u = v_0, v_1, \dots, v_k = v$, where $v_i$ and $v_{i+1}$, $i = 0, 1, \dots, k-1$, are adjacent. (posed by Leng Gangsong)

Let $n \ge 3$ be an integer. Determine the minimum number of points one has to mark inside a convex $n$-gon in order for the interior of any triangle with the vertices at vertices of the $n$-gon to contain at least one of the marked points.

We call a point sequence $(A_0, A_1, \dots, A_n)$ interesting, if the abscissa and ordinate are equal for each $A_i$, and the slopes of segment $OA_0, OA_1, \dots, OA_n$ strictly increase ($O$ is the origin), and the area of each $\triangle OA_iA_{i+1}$ ($0 \le i \le n-1$) is $\frac{1}{2}$.

For a point sequence $(A_0, A_1, \dots, A_n)$, insert a point $A$ adjacent to two points $A_i, A_{i+1}$ satisfying $\overrightarrow{OA} = \overrightarrow{OA_i} + \overrightarrow{OA_{i+1}}$, then we call thus obtained new point sequence $(A_0, \dots, A_i, A, A_{i+1}, \dots, A_n)$ an expansion of $(A_0, A_1, \dots, A_n)$.

Let $(A_0, A_1, \dots, A_n)$ and $(B_0, B_1, \dots, B_m)$ be any two interesting point sequences. Prove that if $A_0 = B_0$ and $A_n = B_m$, then we can expand both point sequences to some same point sequence $(C_0, C_1, \dots, C_k)$.

(posed by Qu Zhenhua)

Let $n \ge 2$ be a given integer.
(1) Prove that one can arrange all the subsets of the set $\{1, 2, \dots, n\}$ as a sequence of subsets $A_1, A_2, \dots, A_{2^n}$, such that $|A_{i+1}| = |A_i| + 1$ or $|A_i| - 1$, where $i = 1, 2, \dots, 2^n$ and $A_{2^n+1} = A_1$.
(2) Determine, with proof, all possible values of the sum $\sum_{i=1}^{2^n} (-1)^i S(A_i)$, where $S(A_i) = \sum_{x \in A_i} x$ and $S(\emptyset) = 0$, for any subset sequence $A_1, A_2, \dots, A_{2^n}$ satisfying the condition in (1). (posed by Liang Yingde)

Suppose $f(x) = |\lg(x+1)|$ and real numbers $a, b$ ($a \lt b$) satisfy $f(a) = f\left(-\frac{b+1}{b+2}\right)$, $f(10a + 6b + 21) = 4\lg 2$. Find the values of $a, b$.

Prove that for any given positive integers $m$, $n$, there exist infinitely many pairs of coprime positive integers $a$, $b$, such that $a + b \mid am^a + bn^b$.

Let set $M = \{1, 2, 3, ..., 50\}$. Find all positive integer $n$, such that there are at least two different elements $a$ and $b$ in any subset with 35 elements of $M$, such that $a + b = n$ or $a - b = n$. (posed by Li Shenghong)

Determine all pairs $(m, n)$ of positive integers so that $n \mid 2m-1$ and $m \mid 2n-1$.

Given real number $\alpha$, please find the minimum real number $\lambda = \lambda(\alpha)$, such that for any complex numbers $z_1, z_2$ and real number $x \in [0, 1]$, if $|z_1| \le \alpha |z_1 - z_2|$ then $|z_1 - xz_2| \le \lambda |z_1 - z_2|$. (posed by Li Shenghong)

If
$$
\frac{x_1}{x_1 + 1} = \frac{x_2}{x_2 + 3} = \frac{x_3}{x_3 + 5} = \dots = \frac{x_{1006}}{x_{1006} + 2011},
$$
$$
x_1 + x_2 + \dots + x_{1006} = 503^2,
$$
determine $x_{1006}$.

Let $a$, $b$, $c$ be real numbers and let $a \neq 0$. Prove that at least one of the equations
$$
a \sin x + b \cos x + c = 0,
$$
$$
a \operatorname{tgg} y + b \operatorname{ctg} y + 2c = 0
$$
has real solutions.

If $\min_{x \in \mathbb{R}} \frac{a x^2 + b}{\sqrt{x^2 + 1}} = 3$, find
(1) the range of $b$;

Isosceles $\triangle ABC$ ($AC = BC$) is inscribed in a circle $k$. A point $M$ lies on the side $BC$. A point $N$ from the ray $AM$ ($M$ lies between $A$ and $N$) is such that $AN = AC$. The circumcircle of $\triangle MCN$ intersects $k$ at $C$ and $P$, where $P$ is from the arc $BC$, not containing $A$. The lines $AB$ and $CP$ intersect at $Q$. Prove that $\angle QMB = \angle QMN$.

Let $a$ be a positive integer. Denote by $\tau(a)$ and $\varphi(a)$ respectively the number of all positive integers that divide $a$ and the number of all positive integers not greater than $a$ and relatively prime to $a$. Find all positive integers $n$ having only two prime divisors and such that $\varphi(\tau(n)) = \tau(\varphi(n))$.