JBMO-SL 2020

Find all triples $(a, b, c)$ of real numbers such that the following system holds:

$$\left\{\begin{array}{l} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\ a^{2}+b^{2}+c^{2}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}} \end{array}\right.$$

Consider the sequence $a_{1}, a_{2}, a_{3}, \ldots$ defined by $a_{1}=9$ and

$$a_{n+1}=\frac{(n+5) a_{n}+22}{n+3}$$

for $n \geqslant 1$.

Find all natural numbers $n$ for which $a_{n}$ is a perfect square of an integer.

Find all triples of positive real numbers $(a, b, c)$ so that the expression

$$M=\frac{(a+b)(b+c)(a+b+c)}{a b c}$$

gets its least value.

Alice and Bob play the following game: starting with the number 2 written on a blackboard, each player in turn changes the current number $n$ to a number $n+p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than $\underbrace{2 \ldots 2}_{2020}$. Assuming perfect play, who will win the game.

Viktor and Natalia bought 2020 buckets of ice-cream and want to organize a degustation schedule with 2020 rounds such that:

• In every round, each one of them tries 1 ice-cream, and those 2 ice-creams tried in a single round are different from each other.
• At the end of the 2020 rounds, each one of them has tried each ice-cream exactly once.

We will call a degustation schedule fair if the number of ice-creams that were tried by Viktor before Natalia is equal to the number of ice creams tried by Natalia before Viktor.

Prove that the number of fair schedules is strictly larger than $2020!\left(2^{1010}+(1010!)^{2}\right)$.

Alice and Bob play the following game: Alice begins by picking a natural number $n \geqslant 2$. Then, with Bob starting first, they alternately choose one number from the set $A=\{1,2, \ldots, n\}$ according to the following condition: The number chosen at each step should be distinct from all the already chosen numbers, and should differ by 1 from an already chosen number. (At the very first step Bob can choose any number he wants.) The game ends when all numbers from the set $A$ are chosen.

For example, if Alice picks $n=4$, then a valid game would be for Bob to choose 2, then Alice to choose 3, then Bob to choose 1, and then Alice to choose 4 .

Alice wins if the sum $S$ of all of the numbers that she has chosen is composite. Otherwise Bob wins. (In the above example $S=7$, so Bob wins.)

Decide which player has a winning strategy.

Let $\triangle A B C$ be an acute triangle. The line through $A$ perpendicular to $B C$ intersects $B C$ at $D$. Let $E$ be the midpoint of $A D$ and $\omega$ the the circle with center $E$ and radius equal to $A E$. The line $B E$ intersects $\omega$ at a point $X$ such that $X$ and $B$ are not on the same side of $A D$ and the line $C E$ intersects $\omega$ at a point $Y$ such that $C$ and $Y$ are not on the same side of $A D$. If both of the intersection points of the circumcircles of $\triangle B D X$ and $\triangle C D Y$ lie on the line $A D$, prove that $A B=A C$.

Problem: Let $\triangle A B C$ be a right-angled triangle with $\angle B A C=90^{\circ}$, and let $E$ be the foot of the perpendicular from $A$ on $B C$. Let $Z \neq A$ be a point on the line $A B$ with $A B=B Z$. Let (c), $\left(c_{1}\right)$ be the circmucircles of the triangles $\triangle A E Z$ and $\triangle B E Z$, respectively. Let ( $c_{2}$ ) be an arbitrary circle passing through the points $A$ and $E$. Suppose $\left(c_{1}\right)$ meets the line $C Z$ again at the point $F$, and meets $\left(c_{2}\right)$ again at the point $N$. If $P$ is the other point of intesection of $\left(c_{2}\right)$ with $A F$, prove that the points $N, B, P$ are collinear.

Let $\triangle A B C$ be a right-angled triangle with $\angle B A C=90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ on $B C$. Let $Z \neq A$ be a point on the line $A B$ with $A B=B Z$. Let (c) be the circumcircle of the triangle $\triangle A E Z$. Let $D$ be the second point of intersection of $(c)$ with $Z C$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $F E$ and $C Z$. If the tangent to $(c)$ at $Z$ meets $P A$ at $T$, prove that the points $T, E, B, Z$ are concyclic.