After seeing this post about problems from USAMTS I thought it might be a good idea to post here about problems from Polish Mathematical Olympiad. The first stage of this olympiad is involves high-schoolers solving a number of problems and submitting solutions within given deadlines. I have seen problems from this contest here both last year and two years ago, so I thought it is reasonable to get people informed about this. The problems and deadlines can be found here, below I am posting translations of problem statements and the deadlines.

Per our ongoing contest problem policy, any questions about these problems should be flagged or otherwise reported.

Thank you very much for your help.

Part 1, deadline: September 30th (we are past deadline, now we can discuss these four problems)

1. Determine whether there exist three distinct, nonzero real numbers $a,b,c$ such that among the numbers $$\frac{a+b}{a^2+ab+b^2},\frac{b+c}{b^2+bc+c^2},\frac{c+a}{c^2+ca+a^2}$$ some two are equal and the third one is different from them.

2. In a chest there are 2017 balls. On each ball an integer is written. We are randomly choosing two balls in order (the same ball may be drawn twice) and add the numbers appearing on them. Show that the probability of the sum being even is greater than $\frac{1}{2}$.

3. Line segments $AD,BE$ are two altitudes in an acute triangle $ABC$. $M$ is the midpoint of $AB$. Points $P,Q$ are symmetric to point $M$ with respect to lines, respectively, $AD,BE$. Show that the midpoint of $DE$ lies on line $PQ$.

4. $t$ is a real number from the interval $(0,1)$. Show that for arbitrary real $a,b$ the following inequality holds: $$|a+(1+t)b|+|a+(1-t)b|\geq\frac{2t}{2+t}(|a|+|b|)$$

Part 2, deadline: October 31st (we are past deadline, now we can discuss these four problems)

5. Show that there exists a natural number $n$ which has over $2017$ divisors $d$ satisfying $$\sqrt{n}\leq d<1.01\sqrt{n}$$

6. At a ball there were $20$ men and $20$ women. In each of $99$ dances exactly one pair was dancing, each time a different one. In each pair a woman was dancing with a man. Show that there are two men and two women such that each of these men danced with both these women.

7. A trapezoid $ABCD$ is given with $AB\|CD$. Perpendicular bisector of $AD$ intersects $BC$ at $E$. Line parallel to $AE$ through $C$ intersects $AD$ at $F$. Show that $$\angle AFB=\angle CFD$$

8. Integers $a,b,c$ are given. Show that there is a positive integer $n$ such that $n^3+an^2+bn+c$ is not a square of an integer.

9. Show that the equation $$(x^2+2y^2)^2-2(z^2+2t^2)^2=1$$ has infinitely many integer solutions $x,y,z,t$.

10. For a fixed positive integer $n$ consider the equation $$x_1+2x_2+\dots+nx_n=n$$ in which $x_1,\dots,x_n$ can take nonnegative integer values. Show that there are as many solutions $(x_1,\dots,x_n)$ satisfying

1. for each $k=1,\dots,n-1$ either $x_k>0$ or $x_{k+1}=0$,

as there are solutions $(x_1,\dots,x_n)$ satisfying

1. for each $k=1,\dots,n$ either $x_k=0$ or $x_k=1$.

11. $AD$ is an altitude in an acute triangle $ABC$. Points $E,F$ are orthogonal projections of $D$ on, respectively, $AB,AC$. $M,N$ are midpoints of, respectively, $AB,AC$. Lines $MF,EN$ intersect at point $S$. Show that circumcenter of $ABC$ lies on line $SD$.

12. Let $\alpha$ be a real number such that $\tan(\alpha\cdot\pi)=\sqrt{2}$. Determine whether $\alpha$ has to be rational.

• @Joel, I don't think it matters in the least which integers are written there. Sep 19 '16 at 2:55
• @GerryMyerson, I see, thanks. Sep 19 '16 at 4:45
• @JoelReyesNoche These can be arbitrary integers. Sep 19 '16 at 7:36
• I think there is one (general) problem with this contests policy. If one asks any of these problems on the main site, it is likely (though this somewhat depends on the astuteness of the moderators) that they would answered before they can be closed. Sep 20 '16 at 1:42
• @MathematicsStudent1122 That's true. But the point here is, the more users are aware of the fact these problems are "illegal questions", the more likely it is that someone will attract moderators' attention quickly enough for the question to be locked before anyone provides an answer. In either case, if it wasn't for the policy, then these questions would stay open together with their answers freely available to everyone. Sep 20 '16 at 7:00
• Referring to (6) we need to use the preposition "at a ball, " and not "on a ball." Sep 20 '16 at 19:55
• Can students belonging to any nation join? Oct 12 '16 at 10:32
• @N.S.JOHN I'm afraid not. I think only Polish citizens can take part in the contest. Oct 12 '16 at 11:30
• I'm guessing that the part 2 deadline has now expired? math.stackexchange.com/questions/1991353/proof-in-19x19-square Nov 1 '16 at 1:11
• @Joffan I agree. The question can now be unlocked. Nov 2 '16 at 10:14