PMO 2021

In convex quadrilateral $ABCD$ , $\angle CAB = \angle BCD$ . $P$ lies on line $BC$ such that $AP = PC$ , $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$ , and $S$ is the point of intersection of lines $AC$ and $QR$ . Line $AD$ meets the circumcircle of $AQS$ again at $T$ . Prove that $AB$ and $QT$ are parallel.

Let $n$ be a positive integer. Show that there exists a one-to-one function $\sigma : \{1,2,...,n\} \to \{1,2,...,n\}$ such that $$\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^2} < \frac{1}{2}.$$

Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies

• $f(p) = 1$ for all primes $p$ , and

• $f(ab) = af(b) + bf(a)$ for all $a,b \in \mathbb{Q}^+$ .

For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$ ?

Determine the set of all polynomials $P(x)$ with real coefficients such that the set $\{P(n) | n \in \mathbb{Z}\}$ contains all integers, except possibly finitely many of them.

A positive integer is called $\emph{lucky}$ if it is divisible by $7$ , and the sum of its digits is also divisible by $7$ . Fix a positive integer $n$ . Show that there exists some lucky integer $l$ such that $\left|n - l\right| \leq 70$ .

A certain country wishes to interconnect $2021$ cities with flight routes, which are always two-way, in the following manner:

• There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights.

• For every pair $(A, B)$ of cities that are connected by a direct flight, there is another city $C$ such that $(A, C)$ and $(B, C)$ are connected by direct flights.

Show that at least $3030$ flight routes are needed to satisfy the two requirements.

Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and $$a+b+c+d = 13$$ $$a^2+b^2+c^2+d^2=43.$$ Show that $ab \geq 3 + cd$ .

In right triangle $ABC$ , $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$ . $M$ is the midpoint of $AB$ , $P$ is the foot of the altitude from $C$ , and $N$ is the midpoint of $CP$ . Line $AB$ meets the circumcircle of $CNB$ again at $Q$ . $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$ lies on ray $CA$ past $A$ such that $BR = RS$ , and $V$ lies on segment $SP$ such that $AV = VP$ . Line $SP$ meets the circumcircle of $CPB$ again at $T$ . $W$ lies on ray $VA$ past $A$ such that $2AW = ST$ , and $O$ is the circumcenter of $SPM$ . Prove that lines $OM$ and $BW$ are perpendicular.