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 $$\sqrt n \le d \le \sqrt {n^2}$$. Prove that the number $$n$$ has a divisor $$p > \sqrt {n^2}$$ 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 $$3$$.

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

Series $$2$$: until November $$2$$

1. 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.

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

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

1. 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.

2. 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 $$2$$

1. 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 $$i < j$$.

2. 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.

3. 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 $$b$$.

4. 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.

• As somebody who likes languages, it was fun to find out which Polish word stands for which English word. Thank you for that. – Teresa Lisbon Sep 28 at 4:16
• @TeresaLisbon I agree. I found it interesting as well, and the last few problems were challenging to translate word-for-word. I definitely could not have translated them without the help of Wiktionary since I don't know any Polish. – Toby Mak Sep 28 at 9:34
• Thanks for making this post. I have reported a few weeks ago two or three posts with problems from the Polish MO. It turns out that since then more cheaters appeared and other people reported them, hence your post. I have corrected the translation as some problems were translated inaccurately (e.g. "różnowartościowy" means "injective", not "multivalued" as the original post said). – timon92 Sep 30 at 19:28
• What's the general procedure for dealing with cheating on ongoing contests? There are so many that happen that I don't imagine that each one should be upvoted like this on meta. I myself am on a committee for a Problem of the Month type thing, but I wouldn't want to clog up meta with posts about it. – Favst Oct 1 at 20:37
• If one of these questions appears, please flag it for moderator attention, and request it to be deleted. If you find the question to be particularly interesting, you can request for it to be undeleted in this thread after the contest ends. – Toby Mak Oct 2 at 3:10
• I believe the last question says $k$ and $l$ are integers, instead of $m$ and $n$. – L. F. Oct 4 at 5:46