MathNet 2018-part-03 — 181 problems (EN–VI)

Let $n \geqslant 1$ be an integer. For every nonempty subset $A$ of $\{1,2, \ldots, n\}$, denote by $P(A)$ the product of all the elements of $A$. For example, for $A=\{2,4,7\}$, we have $P(A)=56$. Determine the sum of the numbers $\frac{1}{P(A)}$ as $A$ ranges over all nonempty subsets of $\{1,2, \ldots, n\}$.

A country has $2018 n+1$ cities, where $n$ is a nonzero natural number. Some pairs of cities are joined by direct railway lines, in such a way that there is at most one line between two cities; each line runs in both directions. The distance between two cities $A$ and $B$ is then the minimal number of lines one must take to go from $A$ to $B$.

Find the set of integers $n$ for which it is possible to build a railway network satisfying the following criterion:

For every city $C$ and every $i \in \{1,2, \ldots, 2018\}$, there are exactly $n$ cities at distance $i$ from $C$.

In the plane, 16 pairwise distinct points are given, denoted $A_{i, j}$ for $i, j \in \{1,2,3,4\}$. These points satisfy the following collinearity and concyclicity relations:

$\triangleright$ for every $i \in \{1,2,3,4\}$, the points $A_{i, 1}, A_{i, 2}, A_{i, 3}$ and $A_{i, 4}$ are collinear;

$\triangleright$ for every $j \in \{1,2,3,4\}$, the points $A_{1, j}, A_{2, j}, A_{3, j}$ and $A_{4, j}$ are collinear;

$\triangleright$ the quadrilaterals $A_{1,1} A_{1,2} A_{2,2} A_{2,1}, A_{2,1} A_{2,2} A_{3,2} A_{3,1}, A_{3,1} A_{3,2} A_{4,2} A_{4,1}, A_{1,2} A_{1,3} A_{2,3} A_{2,2}$, $A_{1,3} A_{1,4} A_{2,4} A_{2,3}, A_{1,1} A_{2,2} A_{2,3} A_{1,4}$ and $A_{1,1} A_{2,2} A_{3,2} A_{4,1}$ are cyclic.

Prove that the quadrilateral $A_{4,1} A_{3,2} A_{3,3} A_{4,4}$ is also cyclic.

Note: a quadrilateral is called cyclic if its four vertices lie on one circle.

An integer $k$ is called olympic if there exist four integers $a, b, c$ and $d$, all coprime to $k$, such that $k$ divides $a^{4}+b^{4}+c^{4}+d^{4}$. Let $n$ be any integer.

Prove that $n^{2}-2$ is olympic.

Given that $a$, $b$ and $c$ are positive real numbers such that $ab + bc + ca \ge 1$, prove that
$$
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \ge \frac{\sqrt{3}}{abc}.
$$

Find the number of nonnegative integers $k$, $0 \le k \le 2188$, such that $\binom{2188}{k}$ is divisible by $2188$.
(Note that the binomial coefficient is defined by $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.)

The incircle of $\triangle ABC$, with incentre $I$, meets $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. The line $EF$ cuts the lines $BI$, $CI$, $BC$ and $DI$ at points $K$, $L$, $M$ and $Q$ respectively. The line through the midpoint of $CL$ and $M$ meets $CK$ at $P$.

a. Determine $\angle BKC$.

b. Show that the lines $PQ$ and $CL$ are parallel.

Find all integers $n \ge 3$ with the following property: there exist $n$ distinct points on the plane such that each point is the circumcentre of a triangle formed by 3 of the points.

Find all positive integer(s) $n$ such that $n^2 + 32n + 8$ is a perfect square.

Find all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f(1) = 7$ and $f(2) = 2017$.

In a school there are $1200$ students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every $23$ students, but there is no common club joined by all $1200$ students, find the smallest possible value of $k$.

Find infinitely many positive integers $m$ such that for each such $m$, the number $\frac{2^{m-1}-1}{8191m}$ is an integer.

Let $ABC$ be a triangle with $AB = AC$. A circle $\Gamma$ lies outside triangle $ABC$ and is tangent to line $AC$ at $C$. Point $D$ lies on $\Gamma$ such that the circumcircle of triangle $ABD$ is internally tangent to $\Gamma$. Segment $AD$ meets $\Gamma$ again at $E$. Prove that $BE$ is tangent to $\Gamma$.

There are three piles of coins, with $a$, $b$ and $c$ coins respectively, where $a$, $b$, $c \ge 2015$ are positive integers. The following operations are allowed:
(1) Choose a pile with an even number of coins and remove all coins from this pile. Add coins to each of the remaining two piles with amount equal to half of that removed; or
(2) Choose a pile with an odd number of coins and at least $2017$ coins. Remove $2017$ coins from this pile. Add $1009$ coins to each of the remaining two piles.
Suppose there are sufficiently many spare coins. Find all ordered triples $(a, b, c)$ such that after some finite sequence of allowed operations, there exists a pile with at least $2017^{2017}$ coins.

Given triangle $ABC$, let $D$ be an inner point of the segment $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K = BP \cap AC$, $L = CP \cap AB$, $E = BQ \cap AC$, $F = CQ \cap AB$. Given that $KL \parallel EF$, find all possible values of the ratio $BD : DC$.

The altitudes $AD$ and $BE$ of acute triangle $ABC$ intersect at $H$. Let $F$ be the intersection of $AB$ and a line that is parallel to the side $BC$ and goes through the circumcentre of $ABC$. Let $M$ be the midpoint of $AH$. Prove that $\angle CMF = 90^\circ$.

For which natural number $n$ is it possible to place natural numbers from $1$ to $3n$ on the edges of a right $n$-angled prism (on each edge there is exactly one number placed and each one is used exactly 1 time) in such a way that the sum of all the numbers that surround each face is the same?

Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x = \sum_{x \in G} 1 / x$;
(2) There exists a positive rational number $r \lt 1$ such that $\sum_{x \in F} 1 / x \neq r$ for all finite subsets $F$ of $S$.

Determine all functions $f:(0, \infty) \rightarrow \mathbb{R}$ satisfying
$$
\left(x+\frac{1}{x}\right) f(y)=f(x y)+f\left(\frac{y}{x}\right) \tag{1}
$$
for all $x, y \gt 0$.

Let $m, n \geqslant 2$ be integers. Let $f\left(x_{1}, \ldots, x_{n}\right)$ be a polynomial with real coefficients such that
$$
f\left(x_{1}, \ldots, x_{n}\right)=\left\lfloor\frac{x_{1}+\ldots+x_{n}}{m}\right\rfloor \text{ for every } x_{1}, \ldots, x_{n} \in\{0,1, \ldots, m-1\} .
$$
Prove that the total degree of $f$ is at least $n$.