Navvye/Olympiad-Problems-CSV huggingface dataset:
2001 Romania National Olympiad
Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $|f(x)-f(y)|\le |x-y|$ for every $x,y\in\mathbb{R}$.



Show that:



a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$.



b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.
2001 Romania National Olympiad
The continuous function $f:[0,1]\rightarrow\mathbb{R}$ has the property:

$$\lim_{x\rightarrow\infty}\ n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)=0 $$

for every $x\in [0,1)$.



Show that:



a) For every $\epsilon >0$ and $\lambda\in (0,1)$, we have:

$$ \sup\ \{x\in[0,\lambda )\mid |f(x)-f(0)|\le \epsilon x \}=\lambda $$

b) $f$ is a constant function.
2001 Romania National Olympiad
a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots.



b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.
2001 Romania National Olympiad
Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.
2001 Romania National Olympiad
Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that:



a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$.



b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.
2001 Romania National Olympiad
Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$.



a) Show that:

$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx$$

b) Show that:

$$ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx$$
2000 Romania National Olympiad
Let be two natural primes $ 1\le q \le  p. $ Prove that $ \left( \sqrt{p^2+q} +p\right)^2 $ is irrational and its fractional part surpasses $ 3/4. $
2000 Romania National Olympiad
Let $ A,B $ be two points in a plane and let two numbers $ a,b\in (0,1) . $ For each point $ M $ that is not on the line $ AB $ consider $ P $ on the segment $ AM $ and $ N $ on $ BM $ (both excluding the extremities) such that $ BN=b\cdot BM $ and $ AP=a\cdot AM. $ Find the locus of the points $ M $ for which $ AN=BP. $
2000 Romania National Olympiad
Let be a natural number $ n\ge 2 $ and an expression of $ n $ variables

$$ E\left( x_1,x_2,...,x_n\right) =x_1^2+x_2^2+\cdots +x_n^2-x_1x_2-x_2x_3-\cdots -x_{n-1}x_n -x_nx_1. $$Determine $ \sup_{x_1,...,x_n\in [0,1]} E\left( x_1,x_2,...,x_n\right)  $ and the specific values at which this supremum is attained.
2000 Romania National Olympiad
Let $ I $ be the center of the incircle of a triangle $ ABC. $ Shw that, if for any point $ M $ on the segment $ AB $ (extremities excluded) there exist two points $ N,P $ on $ BC, $ respectively, $ AC $ (both excluding the extremities) such that the center of mass of $ MNP $ coincides with $ I, $ then $ ABC $ is equilateral.
2000 Romania National Olympiad
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence having $ x_1=3 $ and defined as $ x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor , $ for every natural number $ n. $ Find all values $ m $ for which the terms $ x_m,x_{m+1},x_{m+2} $ are in arithmetic progression, where $ \lfloor\rfloor $ denotes the integer part.
2000 Romania National Olympiad
Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation:

$$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$then $ z_1^6=z_2^6 $
2000 Romania National Olympiad
Let be a tetahedron $ ABCD, $ and $ E $ be the projection of $ D $ on the plane formed by $ ABC. $ If $ \mathcal{A}_{\mathcal{R}} $ denotes the area of the region $ \mathcal{R}, $ show that the following affirmations are equivalent:



a) $ C=E\vee CE\parallel AB $

b) $ M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2  $
2000 Romania National Olympiad
Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.
2000 Romania National Olympiad
Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that:

$$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$
2000 Romania National Olympiad
Study the convergence of a sequence $ \left( x_n\right)_{n\ge 0} $ for which $ x_0\in\mathbb{R}\setminus\mathbb{Q} , $ and $ x_{n+1}\in \left\{ \frac{x_n+1}{x_n} , \frac{x_n+2}{2x_n-1}\right\} , $ for all $ n\ge 1. $
2000 Romania National Olympiad
A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is olympic if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon.



Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.
2000 Romania National Olympiad
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions:

$ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $

$ \text{(ii)}\quad f $ has Darboux’s property



a) Prove that the limits of $ f $ at $ \pm\infty $ exist.

b) Is possible for the limits from a) to be finite?
2000 Romania National Olympiad
Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property:

$$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$

a) Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal.



b) Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $
2000 Romania National Olympiad
For any partition $ P $ of $ [0,1] $ , consider the set

$$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that

$$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$

Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $ 

[0,1] . $
2000 Romania National Olympiad
We say that the abelian group $ G $ has property (P) if, for any commutative group $ H, $ any $ H’\le H $ and any momorphism $ \mu’:H\longrightarrow G, $ there exists a morphism $ \mu :H\longrightarrow G $ such that $ \mu\bigg|_{H’} =\mu’ . $ Show that:



a) the group $ \left( \mathbb{Q}^*,\cdot \right) $ hasn’t property (P).

b) the group $ \left( \mathbb{Q}, +\right) $ has property (P).
2000 Romania National Olympiad
Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $
2021 Saint Petersburg Mathematical Olympiad
Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.



A. Khrabov
2021 Saint Petersburg Mathematical Olympiad
The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?



S. Berlov
2021 Saint Petersburg Mathematical Olympiad
In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.
2021 Saint Petersburg Mathematical Olympiad
The following functions are written on the board, $$F(x) = x^2 + \frac{12}{x^2}, G(x) = \sin(\pi x^2), H(x) = 1.$$If functions $f,g$ are currently on the board, we may write on the board the functions $$f(x) + g(x), f(x) - g(x), f(x)g(x), cf(x)$$(the last for any real number $c$). Can a function $h(x)$ appear on the board such that $$|h(x) - x| < \frac{1}{3}$$for all $x \in [1,10]$ ?
2021 Saint Petersburg Mathematical Olympiad
A natural number $n$ is given. Prove that $$\sum_{n \le p \le n^2} \frac{1}{p} < 2$$where the sum is across all primes $p$ in the range $[n, n^2]$
2021 Saint Petersburg Mathematical Olympiad
Point $M$ is the midpoint of base $AD$ of an isosceles trapezoid $ABCD$ with circumcircle $\omega$. The angle bisector of $ABD$ intersects $\omega$ at $K$. Line $CM$ meets $\omega$ again at $N$. From point $B$, tangents $BP, BQ$ are drawn to $(KMN)$. Prove that $BK, MN, PQ$ are concurrent.



A. Kuznetsov
2021 Saint Petersburg Mathematical Olympiad
A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.
2021 Saint Petersburg Mathematical Olympiad
Solve the following system of equations $$\sin^2{x} + \cos^2{y} = y^4. $$$$\sin^2{y} + \cos^2{x} = x^2. $$

A. Khrabov
2021 Saint Petersburg Mathematical Olympiad
Given are $2021$ prime numbers written in a row. Each number, except for those in the two ends, differs from its two adjacent numbers with $6$ and $12$. Prove that there are at least two equal numbers.
2021 Saint Petersburg Mathematical Olympiad
Given a convex pentagon $ABCDE$, points $A_1, B_1, C_1, D_1, E_1$ are such that $$AA_1 \perp BE, BB_1 \perp AC, CC_1 \perp BD, DD_1 \perp CE, EE_1 \perp DA.$$In addition, $AE_1 = AB_1, BC_1 = BA_1, CB_1 = CD_1$ and $DC_1 = DE_1$. Prove that $ED_1 = EA_1$
2021 Saint Petersburg Mathematical Olympiad
Stierlitz wants to send an encryption to the Center, which is a code containing $100$ characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads:



i) when transmitting encryption over the radio, exactly $49$ characters should be replaced with their opposites;



ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it.



Prove that Stierlitz can send $10$ encryptions, each time choosing some $49$ characters to flip, such that when the Center receives these $10$ ciphers, it may unambiguously restore the original code.
2021 Saint Petersburg Mathematical Olympiad
The vertices of a convex $2550$-gon are colored black and white as follows: black, white, two black, two white, three black, three white, ..., 50 black, 50 white. Dania divides the polygon into quadrilaterals with diagonals that have no common points. Prove that there exists a quadrilateral among these, in which two adjacent vertices are black and the other two are white.



D. Rudenko
2021 Saint Petersburg Mathematical Olympiad
A line $\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB, AD$ at points $X,Y$. Lines $DX, BY$ meet $(AXY)$ for the second time at $P,Q$. Prove that the circumcircle of $\triangle PCQ$ is tangent to $\ell$



A. Kuznetsov
2021 Saint Petersburg Mathematical Olympiad
Kolya found several pairwise relatively prime integers, each of which is less than the square of any other. Prove that the sum of reciprocals of these numbers is less than $2$.
2021 Saint Petersburg Mathematical Olympiad
There are $2021$ points on a circle. Kostya marks a point, then marks the adjacent point to the right, then he marks the point two to its right, then three to the next point's right, and so on. Which move will be the first time a point is marked twice?



K. Kokhas
2021 Saint Petersburg Mathematical Olympiad
Misha has a $100$x$100$ chessboard and a bag with $199$ rooks.  In one move he can either put one rook from the bag on the lower left cell of the grid, or remove two rooks which are on the same cell, put one of them on the adjacent square which is above it or right to it, and put the other in the bag. Misha wants to place exactly $100$ rooks on the board, which don't beat each other.  Will he be able to achieve such arrangement?
2021 Saint Petersburg Mathematical Olympiad
Given is cyclic quadrilateral $ABCD$ with∠$A = 3$∠$B$.  On the $AB$ side is chosen point $C_1$, and on side $BC$ - point $A_1$ so that $AA_1 = AC = CC_1$.  Prove that $3A_1C_1>BD$.
2021 Saint Petersburg Mathematical Olympiad
Given are $n$ points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with  leading coefficient equal to $1$. A parabola is called $good$ if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it).  What is the greatest number of $good$ parabolas?
2021 Saint Petersburg Mathematical Olympiad
Given is an isosceles trapezoid $ABCD$, such that $AD$ and $BC$ are bases and $AD=2AB$, and it is inscribed in a circle $c$.  Points $E$ and $F$ are selected on a circle $c$ so that $AC$ || $DE$ and $BD$ || $AF$. The line $BE$ intersects lines $AC$ and $AF$ at points $X$ and $Y$, respectively.  Prove that the circumcircles of triangles $BCX$ and $EFY$ are tangent to each other.
2021 Saint Petersburg Mathematical Olympiad
A school has $450$ students. Each student has at least $100$ friends among the others and among any $200$ students, there are always two that are friends. Prove that $302$ students can be sent on a kayak trip such that each of the $151$ two seater kayaks contain people who are friends.



D. Karpov
2021 Saint Petersburg Mathematical Olympiad
For a positive integer $n$, prove that $$\sum_{n \le p \le n^4} \frac{1}{p} < 4$$where the sum is taken across primes $p$ in the range $[n, n^4]$



N. Filonov
2020 Saint Petersburg Mathematical Olympiad
A positive integer is called hypotenuse if it can be represented as a sum of two squares of non-negative integers.

Prove that any natural number greater than $10$ is the difference of two hypotenuse numbers.
2020 Saint Petersburg Mathematical Olympiad
A short-sighted rook is a rook that beats all squares in the same column and in the same row for which he can not go more than $60$-steps.

What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a $100\times 100$ chessboard.
2020 Saint Petersburg Mathematical Olympiad
$BB_1$ is the angle bisector of $\triangle ABC$, and $I$ is its incenter. The perpendicular bisector of segment $AC$ intersects the circumcircle of $\triangle AIC$ at $D$ and $E$. Point $F$ is on the segment $B_1C$ such that $AB_1=CF$.Prove that the four points $B, D, E$ and $F$ are concyclic.
2020 Saint Petersburg Mathematical Olympiad
The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point.
2020 Saint Petersburg Mathematical Olympiad
The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$. The circle centered at $O_b$ passes through points $A,C_1$, and the midpoint of $BH$. The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$. Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$
2020 Saint Petersburg Mathematical Olympiad
The sequence $a_n$ is given as $$a_1=1, a_2=2 \;\;\; \text{and} \;\;\;\; a_{n+2}=a_n(a_{n+1}+1) \quad \forall n\geq 1$$Prove that $a_{a_n}$ is divisible by $(a_n)^n$ for $n\geq 100$.
2020 Saint Petersburg Mathematical Olympiad
$N$ oligarchs built a country with $N$ cities with each one of them owning one city. In addition, each oligarch built some roads such that the maximal amount of roads an oligarch can build between two cities is $1$ (note that there can be more than $1$ road going through two cities, but they would belong to different oligarchs).

A total of $d$ roads were built. Some oligarchs wanted to create a corporation by combining their cities and roads so that from any city of the corporation you can go to any city of the corporation using only corporation roads (roads can go to other cities outside corporation) but it turned out that no group of less than $N$ oligarchs can create a corporation. What is the maximal amount that $d$ can have?
2020 Saint Petersburg Mathematical Olympiad
Andryusha has $100$ stones of different weight and he can distinguish the stones by appearance, but does not know their weight. Every evening, Andryusha can put exactly $10$ stones on the table and at night the brownie will order them in increasing weight. But, if the drum also lives in the house then surely he will in the morning change the places of some $2$ stones.Andryusha knows all about this but does not know if there is a drum in his house. Can he find out?
2020 Saint Petersburg Mathematical Olympiad
Find all positive integers $n$  such that the sum of the squares of the five smallest divisors of $n$ is a square.
2020 Saint Petersburg Mathematical Olympiad
On the side $AD$ of the convex quadrilateral $ABCD$ with an acute angle at $B$, a point $E$ is marked.

It is known that $\angle CAD = \angle ADC=\angle ABE =\angle DBE$.

(Grade 9 version) Prove that $BE+CE<AD$.

(Grade 10 version) Prove that $\triangle BCE$ is isosceles.(Here the condition that $\angle B$ is acute is not necessary.)
2020 Saint Petersburg Mathematical Olympiad
Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds

$$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$
2020 Saint Petersburg Mathematical Olympiad
Rays $\ell, \ell_1, \ell_2$ have the same starting point $O$, such that the angle between $\ell$ and $\ell_2$ is acute and the ray $\ell_1$ lies inside this angle. The ray $\ell$ contains a fixed point of $F$ and an arbitrary point $L$. Circles passing through $F$ and $L$ and tangent to $\ell_1$ at $L_1$, and passing through $F$ and $L$ and tangent to $\ell_2$ at $L_2$. Prove that the circumcircle of $\triangle FL_1L_2$ passes through a fixed point other than $F$ independent on $L$.
2020 Saint Petersburg Mathematical Olympiad
On a social network, no user has more than ten friends ( the state "friendship" is symmetrical). The network is connected: if, upon learning interesting news a user starts sending it to its friends, and these friends to their own friends and so on, then at the end, all users hear about the news.

Prove that the network administration can divide users into groups so that the following conditions are met:



each user is in exactly one group

each group is connected in the above sense

one of the groups contains from $1$ to $100$ members and the remaining from $100$ to $900$.
2020 Saint Petersburg Mathematical Olympiad
The exam has $25$ topics, each of which has $8$ questions. On a test, there are $4$ questions of different topics.

Is it possible to make $50$ tests so that each question was asked exactly once, and for any two topics there is a test where are questions of both topics?
2020 Saint Petersburg Mathematical Olympiad
What is the maximal number of solutions can the equation have $$\max  \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$.
2020 Saint Petersburg Mathematical Olympiad
For the triple $(a,b,c)$ of positive integers we say it is interesting if $c^2+1\mid (a^2+1)(b^2+1)$ but none of the $a^2+1, b^2+1$ are divisible by $c^2+1$.

Let $(a,b,c)$ be an interesting triple, prove that there are positive integers $u,v$ such that $(u,v,c)$ is interesting and $uv<c^3$.
2020 Saint Petersburg Mathematical Olympiad
On a table with $25$ columns and $300$ rows, Kostya painted all its cells in three colors. Then, Lesha, looking at the table, for each row names one of the three colors and marks in that row all cells of that color (if there are no cells of that color in that row, he does nothing). After that, all columns that have at least a marked square will be deleted.

Kostya wants to be left as few as possible columns in the table, and Lesha wants there to be as many as possible columns in the table. What is the largest number of columns Lesha can guarantee to leave?
2020 Saint Petersburg Mathematical Olympiad
Point $I_a$ is the $A$-excircle center of $\triangle ABC$ which is tangent to $BC$ at $X$. Let  $A'$ be diametrically opposite point of $A$ with respect to the circumcircle of $\triangle ABC$. On the segments $I_aX, BA'$ and $CA'$ are chosen respectively points $Y,Z$ and $T$ such that $I_aY=BZ=CT=r$ where $r$ is the inradius of $\triangle ABC$.

Prove that the points $X,Y,Z$ and $T$ are concyclic.
2020 Saint Petersburg Mathematical Olympiad
The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way :



If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked

If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$.



Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.
2020 Saint Petersburg Mathematical Olympiad
Let $G$ be a graph with $400$ vertices. For any edge $AB$ we call a cuttlefish the set of all edges from $A$ and $B$ (including $AB$). Each edge of the graph is assigned a value of $1$ or $-1$. It is known that the sum of edges at any cuttlefish is greater than or equal to $1$.

Prove that the sum of the numbers at all edges is at least $-10^4$.
2019 Saint Petersburg Mathematical Olympiad
A polynomial $f(x)$ of degree $2000$ is given. It's known that $f(x^2-1)$ has exactly $3400$ real roots while $f(1-x^2)$ has exactly $2700$ real roots. Prove that there exist two real roots of $f(x)$ such that the difference between them is less that $0.002$.



(А. Солынин)



ThanksThanks to the user Vlados021 for translating the problem.
2019 Saint Petersburg Mathematical Olympiad
On the blackboard there are written $100$ different positive integers . To each of these numbers is added the $\gcd$ of the $99$ other numbers . In the new $100$ numbers , is it possible for $3$ of them to be equal.



 (С. Берлов)
2019 Saint Petersburg Mathematical Olympiad
Kid and Karlsson play a game. Initially they have a square piece of chocolate $2019\times 2019$ grid with $1\times 1$ cells . On every turn Kid divides an arbitrary piece of chololate  into three rectanglular pieces by cells, and then Karlsson chooses one of them and eats it. The game finishes when it's impossible to make a legal move. Kid wins if there was made an even number of moves, Karlsson wins if there was made an odd number of moves.

Who has the winning strategy?



 (Д. Ширяев)



ThanksThanks to the user Vlados021 for translating the problem.
2019 Saint Petersburg Mathematical Olympiad
A non-equilateral triangle $\triangle ABC$  of perimeter $12$ is inscribed in circle $\omega$ .Points $P$ and $Q$ are arc midpoints of arcs $ABC$ and $ACB$ , respectively. Tangent to $\omega$ at $A$ intersects line $PQ$ at $R$.

It turns out that the midpoint of segment $AR$ lies on line $BC$ . Find the length of the segment $BC$.



 (А. Кузнецов)