MathNet 2019-part-07 — 184 problems (EN–VI)

Let $ABC$ be a triangle with $\angle CAB = 2 \cdot \angle ABC$. Assume that there is a point $D$ in the interior of triangle $ABC$ such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \cdot \angle DCB$.

Let $k$ be a circle with centre $O$ and let $A$, $B$ and $C$ be three points on $k$ with $\angle ABC \gt 90^{\circ}$. The bisector of $\angle AOB$ meets the circumcircle of triangle $BOC$ a second time at $D$. Show that $D$ lies on the line $AC$.

Let $k_{1}$ be a circle and $l$ a line that meets $k_{1}$ at two distinct points $A$ and $B$. Let $k_{2}$ be another circle outside $k_{1}$ that touches $k_{1}$ at $C$ and $l$ at $D$. Let $T$ be the second intersection point of $k_{1}$ and the line $CD$. Show that $AT = TB$.

Let $n$ be a positive integer. Maurice writes on one line all the $2^{n}-1$ non-empty subsets of the set $\{1,2, \ldots, n\}$. Then, below each subset, he writes the product of its elements. Finally, he writes the reciprocals of the numbers in the second line and computes their sum. What will be the value of the sum (as a function of $n$) that Maurice obtains?

Example: for $n=3$, Maurice obtains

| $\{1\}$ | $\{2\}$ | $\{3\}$ | $\{1,2\}$ | $\{1,3\}$ | $\{2,3\}$ | $\{1,2,3\}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 2 | 3 | $1 \cdot 2=2$ | $1 \cdot 3=3$ | $2 \cdot 3=6$ | $1 \cdot 2 \cdot 3=6$ |
| $\frac{1}{1}+$ | $\frac{1}{2}+$ | $\frac{1}{3}+$ | $\frac{1}{2}+$ | $\frac{1}{3}+$ | $\frac{1}{6}+$ | $\frac{1}{6}=3$. |

Let $n$ be a natural number. A volleyball team consisting of $n$ women and $n$ men lines up for a game. Each team member occupies one of the positions $1,2, \ldots, 2 n$, where exactly the positions $1$ and $n+1$ are outside the court. During the game all team members rotate, each moving from position $i$ to position $i+1$ (respectively from $2 n$ to $1$). How many possibilities are there for the starting line-up such that there are always at least $n-1$ women on the court, no matter how often the team rotates?

Remark: Two starting line-ups are different if at least one team member occupies a different position.

Determine all pairs of positive integers $(a, b)$ such that
$$
a b+2=a^{3}+2 b
$$

Determine all natural numbers $n \geq 2$ that have a representation of the form
$$
n = k^{2} + d^{2}
$$
where $k$ is the smallest divisor of $n$ greater than 1 and $d$ is an arbitrary divisor of $n$.

Let $Q_{\gt 0}$ denote the set of all positive rational numbers. Find all functions $f: Q_{\gt 0} \to Q_{\gt 0}$ satisfying
$$
f(x^2 f(y)^2) = f(x)^2 f(y), \text{ for all } x, y \in Q_{\gt 0}.
$$

Given a set $S$ of positive integers, prove that at least one of the following two statements holds:

(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that
$$
\sum_{x \in F} \frac{1}{x} = \sum_{x \in G} \frac{1}{x};
$$
(2) There exists a positive rational number $r \lt 1$ such that for every finite subset $F$ of $S$,
$$
\sum_{x \in F} \frac{1}{x} \neq r.
$$

In the plane a circle $\omega$ of radius $1$ is given. A set $T$ of triangles is called good if it satisfies the following two conditions:
(i) every triangle in $T$ is inscribed in $\omega$,
(ii) no two triangles in $T$ have a common interior point.
Determine all positive real numbers $t$ such that, for every positive integer $n$, there exists a good set $T$ consisting of $n$ triangles, each of which has perimeter greater than $t$.

Let $a, b, c, d$ be positive real numbers satisfying $a + b + c + d = 4$. Prove that
$$
\sum_{\text{cyc}} \frac{3a^3}{a^2 + ab + b^2} + \sum_{\text{cyc}} \frac{2ab}{a+b} \geq 8.
$$

Consider the sequence $a_0, a_1, a_2, \dots$, where $a_n = 2^n + 2^{\lfloor n/2 \rfloor}$. Prove that infinitely many terms of the sequence can be written as a sum of two or more distinct terms of the sequence, and that infinitely many terms cannot be written as such a sum.

Let $n$ be a positive integer greater than 3. There are $n$ people in a room, and some pairs of them are enemies (enmity is mutual). Suppose we know that this group of people satisfies both of the following properties:

a. Among any 4 people, there are always two who are not enemies of each other.

b. For any positive integer $m \ge 1$, if we can find $m$ of the people who are pairwise not enemies, then among the remaining $n-m$ people there must be 3 people any two of whom are enemies of each other. (Note: nobody is an enemy of himself or herself.)

Find the smallest possible value of $n$.

Let the incircle of the acute triangle $ABC$ be $\omega$ and its circumcircle be $\Omega$, and let $M$ be the midpoint of side $BC$. The incircle $\omega$ touches $CA$ and $AB$ at the points $E$ and $F$, respectively, and the line $EF$ meets the circumcircle $\Omega$ at the two points $P$ and $Q$. On the circumcircle $\Gamma$ of $\triangle MPQ$ take a point $R$ such that $MR$ is perpendicular to $EF$. Prove that the line $AR$, the circle $\Gamma$ and the circle $\omega$ pass through a common point.

Alice and Bob team up to play a game. At the start of the game, the two of them are dropped at two positions on a train whose total length is $1$ kilometre. The train is closed and completely dark, so unless they are at the front or the rear of the train, they do not know their own position, and they have no way of knowing their teammate's position either. The game gives each of them a device, and the left half of the device displays:
- which direction the holder is currently facing (so he/she can choose to move towards the front or the rear of the train);
- the total distance the holder has moved so far;
- whether the holder is currently touching the front of the train;
- whether the holder is currently touching the rear of the train.
The right half of the device displays the information shown on the left half of the teammate's device. The game ends at the instant the two of them meet (their positions coincide).
Assume that before the game starts Alice and Bob are fully informed about the train and all functions of the device, and can discuss their strategy together beforehand. Find the smallest real number $x$ such that Alice and Bob have a strategy guaranteeing that, no matter where the two of them are placed at the start of the game, the sum of the distances the two of them have moved when the game ends does not exceed $x$ kilometres.

Old Zhao is so bored during his military service that he draws $n + 1$ cells in a row on the ground ($n$ is a positive integer), numbered from left to right as cell 0 to cell $n$. Initially, cell 0 contains $n$ stones and all other cells are empty. In each turn, Old Zhao first chooses a non-empty cell; suppose it contains $k$ stones. Then he takes one stone from the chosen cell and moves it to the right by at most $k$ cells (the stone may not go beyond the rightmost cell). Old Zhao's goal is to put all the stones into cell $n$.
Prove that the number of turns Old Zhao needs to achieve his goal is at least
$$
\left\lceil \frac{n}{1} \right\rceil + \left\lceil \frac{n}{2} \right\rceil + \left\lceil \frac{n}{3} \right\rceil + \cdots + \left\lceil \frac{n}{n} \right\rceil,
$$
where $\lceil x \rceil$ denotes the smallest integer not less than $x$.

Find all positive integers $x, y, z, t$ satisfying
$$
xy - zt = x + y = z + t
$$
such that $xy$ and $zt$ are both perfect squares.

Let $I$ be the incentre of triangle $ABC$ and $J$ the excentre opposite $A$. Let $\overline{AA'}$ be a diameter of the circumcircle of $\triangle ABC$, and let $H_1, H_2$ be the orthocentres of $\triangle BIA'$ and $\triangle CJA'$, respectively. Prove that $H_1H_2$ is parallel to $BC$.

In the plane, a triangle $ABC$ and a point $P$ are given. Let the circumcentres of $\triangle ABC$, $\triangle BPC$, $\triangle CPA$, $\triangle APB$ be the points $O$, $D$, $E$, $F$, respectively. Let the lines $BC$ and $EF$ meet at the point $T$, and let $X$ be the reflection of the point $O$ in the line $EF$. Prove that $PT \perp DX$.

Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f : (0, \infty) \to \mathbb{R}$ satisfying
$$ \left(x + \frac{1}{x}\right) f(y) = f(xy) + f\left(\frac{y}{x}\right), \text{ for all } x, y \gt 0. $$