Polish Mathematical Olympiad 2020

Question

Recently, a user has posted questions from the Polish Mathematical Olympiad $2020$, which is an ongoing contest. If one of these questions appears, please flag it for moderator attention, and request it to be deleted until the contest is over.

Series $1$: until September $30$

1. Let $a$ and $b$ be real numbers. Suppose the inequality $|(ax+by)(ay+bx)| \le x^2+y^2$ holds for all real numbers $x, y$. Prove that $a^2+b^2 ≤ 2$.

2. The triangle $ABC$ is given with $AB>AC$. Let $\ell$ be the tangent line at point $A$ to the circumcircle of $ABC$. Point $X$ lies on the line segment $AB$, point $Y$ lies on the line $\ell$, where $AX = AY = AC$, and points $X$ and $Y$ lie on opposite sides of the straight line containing the angle bisector $BAC$. Prove that the incenter of $ABC$ lies on the line $XY$.

3. Suppose a positive integer $n$ has no divisor $d$ satisfying the inequality $\sqrt n \le d \le \sqrt[3] {n^2}$. Prove that the number $n$ has a divisor $p > \sqrt[3] {n^2}$ which is prime.

4. Among the points of the plane with both coordinates in the set $\{1, \ldots, 106\}$ some points were marked, and for every two marked points $(x, y)$ and $(x', y')$, at least one of the following conditions is met:
   (1) $x> x' - 10$ and $y> y' - 10$;
   (2) $x' > x - 10$ and $y' > y - 10$.
   Determine the biggest possible number of marked points.

The full list of questions can be found here. Google Translate will easily give you the translated versions of series $2$ and $3$.

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I am now posting the rest of the questions as the deadline for series $1$ is fast approaching.

Series $2$: until November $2$

5. It is given an isosceles triangle $ABC$ where $AB = AC$. Point $I$ is the incenter of the triangle $ABC$. Line $BI$ intersects side $AC$ at point $D$. Point $D$ is the midpoint of segment $IX$. The point $O$ is the circumcenter of $BCX$. Prove that the lines $OD$ and $AC$ are perpendicular.

6. Given positive real numbers $a, b, c, d$ with $a, c> 1$ and $b, d <1$, prove that:

$$\frac{a}{ab+c+1} + \frac{b}{bc+d+1} + \frac{c}{cd+a+1} + \frac{d}{da+b+1} > 1.$$

7. A real number has been entered in each cell of a $2020$ by $2020$ board. The following condition must be fulfilled: for any four cells with a common vertex, if by $a, b, c, d$ we denote the numbers entered in these cells as in the figure ($a$ is in the upper-left cell, $b$ is in the upper-right cell, $c$ is in the lower-left cell, and $d$ is in the lower-right corner), then the following system of equations holds: \begin{align}a+b+2c+3d&= 0\\2a+b+3c+4d &= 0.\end{align} Prove that there exist at least $500$ different cells that contain the same number.

8. Polynomials $f_1, f_2, f_3, f_4$ are given with real coefficients such that the sum of any two of them has no real root. Prove that if the polynomial $f_1 + f_2 + f_3 + f_4$ has a real root, then at least one of the polynomials $f_1, f_2, f_3, f_4$ does not have a real root.

Series $3$: until December $2$

9. Let $n\ge 2$ be an integer. Real numbers $a_{ij}$, where $1 ≤ i,j ≤ n$, are given. For any $x_1, \ldots, x_n \in \{-1,1\}$, the following condition holds: $$\sum_{1 ≤ i,j ≤ n} a_{ij}x_i x_j \in\{-1,1\}.$$ Determine, in terms of $n$, the greatest number of pairs $i,j$ such that $a_{ij} \ne 0$ and $i < j$.

10. A sphere inscribed in the tetrahedron $ABCD$ is tangent to the faces $BCD, CDA, DAB$, and $ABC$, at the points $A', B', C', D'$ respectively. Prove that if the lines $AA'$ and $BB'$ intersect then $CC'$ and $DD'$ intersect as well.

11. A prime number is given with $p \ge 2$. Let us call a positive integer $n$ pretty, if and only if the sum of the remainders when $n, n^2, n^3, \ldots, n^{p-1}$ is divided by $p$ is equal to $\frac{1}{2}p(p-1)$. Prove that in the set $\{1, \ldots, p- 1\}$, there are an odd number of pretty numbers.
    Attention: We call $r$ the remainder from dividing an integer $a$ by a positive integer $b$, where $r \in {0, 1, 2, \cdots, b-1}$, and that the number $a-r$ is divisible by $b$.

12. Let $A_{m,n}$ denote the set of vectors $(k, l)$, where $0 \le k \le m-1$ and $0 \le l \le n-1$, and $m,n$ are integers. We call the function $f:A_{m,n} \to A_{m,n}$ good if and only if both conditions are met:
    (1) $f$ is an injective function.
    (2) if $v,w \in A_{m,n}$ and $v +f(v)-w-f(w) =(am,bn)$ for certain integers $a,b$, then $v = w$.
    Find all pairs of positive integers $m, n$ for which good functions $f:A_{m,n} \to A_{m,n}$ exist.