Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Saint Petersburg Mathematical Olympiad
Baron Munchhausen has a collection of stones, such that they are of $1000$ distinct whole weights, $2^{1000}$ stones of every weight. Baron states that if one takes exactly one stone of every weight, then the weight of all these $1000$ stones chosen will be less than $2^{1010}$, and there is no other way to obtain this weight by picking another set of stones of the collection.

Can this statement happen to be true?



(М. Антипов)



ThanksThanks to the user Vlados021 for translating the problem.
2019 Saint Petersburg Mathematical Olympiad
Supppose that there are roads $AB$ and $CD$ but there are no roads $BC$ and $AD$ between four cities $A$, $B$, $C$, and $D$. Define restructing to be the changing a pair of roads $AB$ and $CD$ to the pair of roads $BC$ and $AD$. Initially there were some cities in a country, some of which were connected by roads and for every city there were exactly $100$ roads starting in it. The minister drew a new scheme of roads, where for every city there were also exactly $100$ roads starting in it. It's known also that in both schemes there were no cities connected by more than one road.

Prove that it's possible to obtain the new scheme from the initial after making a finite number of restructings.



 (Т. Зубов)



ThanksThanks to the user Vlados021 for translating the problem.
2019 Saint Petersburg Mathematical Olympiad
Let $\omega$ and $O$ be respectively the circumcircle and the circumcenter of a triangle $ABC$. The line $AO$ intersects $\omega$ second time at $A'$. $M_B$ and $M_C$ are the midpoints of $AC$ and $AB$, respectively. The lines $A'M_B$ and $A'M_C$ intersect $\omega$ secondly at points $B'$ and $C$, and also intersect $BC$  at points $D_B$ and $D_C$, respectively. The circumcircles of $CD_BB'$ and $BD_CC'$ intersect at points $P$ and $Q$.

Prove that $O$, $P$, $Q$ are collinear.



 (М. Германсков)



ThanksThanks to the user Vlados021 for translating the problem.
2019 Saint Petersburg Mathematical Olympiad
For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.
2019 Saint Petersburg Mathematical Olympiad
Every two of the $n$ cities of Ruritania are connected by a direct flight of one from two airlines. Promonopoly Committee wants at least $k$ flights performed by one company. To do this, he can at least every day to choose any three cities and change the ownership of the three flights connecting these cities each other (that is, to take each of these flights from a company that performs it, and pass the other). What is the largest $k$ committee knowingly will be able to achieve its goal in no time, no matter how the flights are distributed hour?
2019 Saint Petersburg Mathematical Olympiad
Let $a, b$ and $c$ be non-zero natural numbers such that $c \geq b$ . Show that

$$a^b\left(a+b\right)^c>c^b a^c.$$
2019 Saint Petersburg Mathematical Olympiad
Given a convex quadrilateral $ABCD$. The medians of the triangle $ABC$ intersect at point $M$, and the medians of the triangle $ACD$ at point$ N$. The circle, circumscibed around  the triangle $ACM$, intersects the segment $BD$ at the point $K$ lying inside the triangle $AMB$ . It is known that $\angle MAN = \angle ANC = 90^o$. Prove that $\angle AKD = \angle MKC$.
2019 Saint Petersburg Mathematical Olympiad
A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less  get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes?
2019 Saint Petersburg Mathematical Olympiad
Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?
2019 Saint Petersburg Mathematical Olympiad
In a square $10^{2019} \times 10^{2019},  10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.
2019 Saint Petersburg Mathematical Olympiad
A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares,  there is one palindrome.
2019 Saint Petersburg Mathematical Olympiad
In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen?
2019 Saint Petersburg Mathematical Olympiad
Prove that the distance between the midpoint of side $BC$ of triangle $ABC$ and the midpoint of the arc $ABC$ of its circumscribed circle is not less than $AB / 2$
2019 Saint Petersburg Mathematical Olympiad
Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?
2019 Saint Petersburg Mathematical Olympiad
Call the improvement  of a positive number its replacement by a power of two. (i.e. one of the numbers $1, 2, 4, 8, ...$), for which it increases, but not more than than $3$ times. Given $2^{100}$ positive numbers with a sum of $2^{100}$. Prove that you can erase some of them, and improve  each of the other numbers so that the sum the resulting numbers were again $2^{100}$.
2019 Saint Petersburg Mathematical Olympiad
The bisectors $BB_1$ and $CC_1$ of the acute triangle $ABC$ intersect in point $I$. On the extensions of the segments $BB_1$ and $CC_1$, the points $B'$ and $C'$ are marked, respectively So, the quadrilateral $AB'IC'$ is a parallelogram. Prove that if $\angle BAC = 60^o$, then the straight line $B'C'$ passes through the intersection point of the circumscribed circles of the triangles $BC_1B'$ and $CB_1C'$.
2019 Saint Petersburg Mathematical Olympiad
In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of

check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?
2018 Saint Petersburg Mathematical Olympiad
Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is  $\leq d$
2018 Saint Petersburg Mathematical Olympiad
Vasya has $100$ cards of $3$ colors, and there are not more than $50$ cards of same color. Prove that he can create $10\times 10$ square, such that every cards of same color have not common side.
2018 Saint Petersburg Mathematical Olympiad
Point $T$ lies on the bisector of $\angle B$ of acuteangled $\triangle ABC$. Circle $S$ with diameter $BT$ intersects $AB$ and $BC$ at points $P$ and $Q$. Circle, that goes through point $A$ and tangent to $S$ at $P$ intersects line $AC$ at $X$.  Circle, that goes through point $C$ and tangent to $S$ at $Q$ intersects line $AC$ at $Y$. Prove, that $TX=TY$
2018 Saint Petersburg Mathematical Olympiad
$$(b+c)x^2+(a+c)x+(a+b)=0$$has not real roots. Prove that $$4ac-b^2 \leq 3a(a+b+c)$$
2018 Saint Petersburg Mathematical Olympiad
Regular hexagon is divided to equal rhombuses, with sides, parallels to hexagon sides. On the three sides of the hexagon, among which there are no neighbors, is set directions in order of traversing the hexagon against hour hand. Then, on each side of the rhombus, an arrow directed just as the side of the hexagon parallel to this side. Prove that  there is not a closed path going along the arrows.
2018 Saint Petersburg Mathematical Olympiad
$\alpha,\beta$ are positive irrational numbers and $[\alpha[\beta x]]=[\beta[\alpha x]]$ for every positive $x$. Prove that $\alpha=\beta$
2018 Saint Petersburg Mathematical Olympiad
Points $A,B$ lies on the circle $S$. Tangent lines to $S$ at $A$ and $B$ intersects at $C$. $M$ -midpoint of $AB$. Circle $S_1$ goes through $M,C$ and intersects $AB$ at $D$ and $S$ at $K$ and $L$. Prove, that tangent lines to $S$ at $K$ and $L$ intersects at point on the segment $CD$.
2018 Saint Petersburg Mathematical Olympiad
Misha came to country with $n$ cities, and every $2$ cities are connected by the road. Misha want visit some cities, but he doesn`t visit one city two time. Every time, when Misha goes from city $A$ to city $B$, president of country destroy $k$ roads from city $B$(president can`t destroy road, where Misha goes). What maximal number of cities Misha can visit, no matter how president does?
2018 Saint Petersburg Mathematical Olympiad
Color every vertex of $2008$-gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call $2008$-gon as interesting.

Convex $2009$-gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting $2008$-gon. Prove, that if we remove marked vertex, then we get interesting $2008$-gon too.
2018 Saint Petersburg Mathematical Olympiad
$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?
2018 Saint Petersburg Mathematical Olympiad
$f(x)$ is polynomial with integer coefficients, with module not exceeded $5*10^6$. $f(x)=nx$ has integer root for $n=1,2,...,20$. Prove that $f(0)=0$
2018 Saint Petersburg Mathematical Olympiad
$ABCD$ is inscribed quadrilateral. Line, that perpendicular to $BD$ intersects segments $AB$ and $BC$ and rays $DA,DC$ at $P,Q,R,S$ . $PR=QS$. $M$ is midpoint of $PQ$. Prove that $AM=CM$
2018 Saint Petersburg Mathematical Olympiad
Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$
2018 Saint Petersburg Mathematical Olympiad
The checker moves from the lower left corner of the board $100 \times 100$ to the right top corner, moving at each step one cell to the right or one cell up. Let $a$ be the number of paths in which exactly $70$ steps the checker take under the diagonal going from the lower left corner to the upper right corner, and $b$ is the number of paths in which such steps are exactly $110$. What is more: $a$ or $b$?
2018 Saint Petersburg Mathematical Olympiad
Prove, that for every natural $N$ exists $k$, such that $N=a_02^0+a_12^1+...+a_k2^k$, where $a_0,a_1,...a_k$ are $1$ or $2$
2018 Saint Petersburg Mathematical Olympiad
$n>1$ is odd number. There are numbers $n,n+1,n+2,...,2n-1$ on the blackboard. Prove that we can erase one number, such that the sum of all numbers will be not divided any number on the blackboard.
2018 Saint Petersburg Mathematical Olympiad
$ABC$ is acuteangled triangle. Variable point $X$ lies on segment $AC$, and variable point $Y$ lies on the ray $BC$ but not segment $BC$, such that $\angle ABX+\angle CXY =90$. $T$ is projection of $B$ on the $XY$. Prove that all points $T$ lies on the line.
2018 Saint Petersburg Mathematical Olympiad
On the round necklace there are $n> 3$ beads, each painted in red or blue. If a bead has adjacent beads painted the same color, it can be repainted (from red to blue or from blue to red). For what $n$ for any initial coloring of beads it is possible to make a necklace in which all beads are painted equally?
2018 Saint Petersburg Mathematical Olympiad
Can we draw $\triangle ABC$ and points $X,Y$, such that $AX=BY=AB$, $ BX = CY = BC$,

$CX = AY = CA$?
2018 Saint Petersburg Mathematical Olympiad
$a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$.
2018 Saint Petersburg Mathematical Olympiad
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle.  It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or

the distance between their ends is at least 2. What is the maximum possible value of $n$?
2017 Saint Petersburg Mathematical Olympiad
Sasha’s computer can do the following two operations: If you load the card with number $a$, it will return that card back and also prints another card with number $a+1$, and if you consecutively load the cards with numbers $a$ and $b$, it will return them back and also prints cards with all the roots of the quadratic trinomial $x^2+ax+b$ (possibly one, two, or none cards.) Initially, Sasha had only one card with number $s$. Is it true that, for any $s> 0$, Sasha can get a card with number $\sqrt{s}$?
2017 Saint Petersburg Mathematical Olympiad
Given a triangle $ABC$, there’s a point $X$ on the side $AB$ such that $2BX = BA + BC$. Let $Y$ be the point symmetric to the incenter $I$ of triangle $ABC$, with respect to point $X$. Prove that $YI_B\perp AB$ where $I_B$ is the $B$-excenter of triangle $ABC$.
2017 Saint Petersburg Mathematical Olympiad
Petya, Vasya and Tolya play a game on a $100\times 100$ board. They take turns (starting from Petya, then Vasya, then Tolya, then Petya, etc.) paint the boundary cells of the board (i.e., having a common side with the boundary of the board.) It is forbidden to paint the cell that is adjacent to the already painted one. In addition, it’s also forbidden to paint the cell which is symmetrical to the painted one, with respect to the center of the board. The player who can’t make the move loss. Can Vasya and Tolya, after agreeing, play so that Petya loses?
2017 Saint Petersburg Mathematical Olympiad
Each cell of a $3\times n$ table was filled by a number. In each of three rows, the number $1,2,…,n$ appear in some order. It is know that for each column, the sum of pairwise product of three numbers in it is a multiple of $n$. Find all possible value of $n$.
2017 Saint Petersburg Mathematical Olympiad
Given a scalene triangle $ABC$ with $\angle B=130^{\circ}$. Let $H$ be the foot of altitude from $B$. $D$ and $E$ are points on the sides $AB$ and $BC$, respectively, such that $DH=EH$ and $ADEC$ is a cyclic quadrilateral. Find $\angle{DHE}$.
2017 Saint Petersburg Mathematical Olympiad
Given three real numbers $a,b,c\in [0,1)$ such that $a^2+b^2+c^2=1$. Find the smallest possible value of

$$\frac{a}{\sqrt{1-a^2}}+\frac{b}{\sqrt{1-b^2}}+\frac{c}{\sqrt{1-c^2}}.$$
2017 Saint Petersburg Mathematical Olympiad
Divide the upper right quadrant of the plane into square cells with side length $1$. In this quadrant, $n^2$ cells are colored, show that there’re at least $n^2+n$ cells (possibly including the colored ones) that at least one of its neighbors are colored.
2017 Saint Petersburg Mathematical Olympiad
It’s allowed to replace any of three coefficients of quadratic trinomial by its discriminant. Is it true that from any quadratic trinomial that does not have real roots, we can perform such operation several times to get a quadratic trinomial that have real roots?
2017 Saint Petersburg Mathematical Olympiad
$(a_{n})$ is sequence with positive integer. $a_{1}>10$

$ a_{n}=a_{n-1}+GCD(n,a_{n-1})$,  n>1

For some i $a_{i}=2i$.

Prove  that these numbers are infinite in this sequence.
2017 Saint Petersburg Mathematical Olympiad
Let $ABC$ be an acute triangle, with median $AM$, height $AH$ and internal angle bisector $AL$. Suppose that $B, H, L, M, C$ are collinear in that order, and $LH<LM$. Prove that $BC>2AL$.
2017 Saint Petersburg Mathematical Olympiad
The numbers from $1$ to $2000^2$ were written on a board. Vasya choose $2000$ of them whose sum of them equal to two thousandth of the sum of all numbers. Proof that his friend, Petya, will be able to color each of the remaining numbers by one of other $1999$ colors so that the sum of numbers with each of total $2000$ colors are the same.
2017 Saint Petersburg Mathematical Olympiad
Let $x,y,z>0 $ and $\sqrt{xyz}=xy+yz+zx$. Prove that$$x+y+z\leq \frac{1}{3}.$$
2017 Saint Petersburg Mathematical Olympiad
In acute-angled triangle $ABC$, the height $AH$ and median $BM$ were drawn. Point $D$ lies on the circumcircle of triangle $BHM$ such that $AD \parallel BM$ and $B, D$ are on opposite sides of line $AC$. Prove that $BC=BD$.
2017 Saint Petersburg Mathematical Olympiad
In a country, some pairs of cities are connected by one-way roads. It turns out that every city has at least two out-going and two in-coming roads assigned to it, and from every city one can travel to any other city by a sequence of roads. Prove that it is possible to delete a cyclic route so that it is still possible to travel from any city to any other city.
2017 Saint Petersburg Mathematical Olympiad
A1,A2,...,Am are subsets of X and we have |Ai|=mk (m,k natural numbers)

prove that we can separate X into k sets such that every set has at least one member of each Ai.
2017 Saint Petersburg Mathematical Olympiad
A circle passing through vertices $A$ and $B$ of triangle $ABC$ intersects the sides $AC$ and $BC$ again at points $P$ and $Q$, respectively. Given that the median from vertex $C$ bisect the arc $PQ$ of the circle. Prove that $ABC$ is an isosceles triangle.
2017 Saint Petersburg Mathematical Olympiad
Given real numbers $x,y,z,t\in (0,\pi /2]$ such that

$$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$
2017 Saint Petersburg Mathematical Olympiad
A positive integer $n$ is called almost-square if $n$ can be represented as $n=ab$ where $a,b$ are positive integers that $a\leq b\leq 1.01a$. Prove that there exists infinitely many positive integers $m$ that there’re no almost-square positive integer among $m,m+1,…,m+198$.
2017 Saint Petersburg Mathematical Olympiad
Given a tetrahedron $PABC$, draw the height $PH$ from vertex $P$ to $ABC$. From point $H$, draw perpendiculars $HA’,HB’,HC’$ to the lines $PA,PB,PC$. Suppose the planes $ABC$ and $A’B’C’$ intersects at line $\ell$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $OH\perp \ell$.
2017 Saint Petersburg Mathematical Olympiad
In the country some mathematicians know each other and any division of them into two sets contain 2 friends from different sets.It is known that if you put any set of four or more mathematicians at a round table so that any two neighbours know each other , then at the table there are two friends not sitting next to each other.We denote by $c_i $ the number of sets of $i$ pairwise familiar mathematicians(by saying "familiar" it means know each other).Prove that

$c_1-c_2+c_3-c_4+...=1$
2017 Saint Petersburg Mathematical Olympiad
Given a convex polygon with vertices at lattice points on a plane containing origin $O$. Let $V_1$ be the set of vectors going from $O$ to the vertices of the polygon, and $V_2$ be the set of vectors going from $O$ to the lattice points that lie inside or on the boundary of the polygon (thus, $V_1$ is contained in $V_2$.) Two grasshoppers jump on the whole plane: each jump of the first grasshopper shift its position by a vector from the set $V_1$, and the second by the set $V_2$. Prove that there exists positive integer $c$ that the following statement is true: if both grasshoppers can jump from $O$ to some point $A$ and the second grasshopper needs $n$ jumps to do it, then the first grasshopper can use at most $n+c$ jumps to do so.
2016 Saint Petersburg Mathematical Olympiad
In the sequence of integers $(a_n)$, the sum $a_m +  a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.
2016 Saint Petersburg Mathematical Olympiad
The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the

continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?
2016 Saint Petersburg Mathematical Olympiad
In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.
2016 Saint Petersburg Mathematical Olympiad
$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.
2016 Saint Petersburg Mathematical Olympiad
Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$.

Prove that $I_1I_2 \perp $ bisector of $\angle ABC$
2016 Saint Petersburg Mathematical Olympiad
A polynomial $P$ with real coefficients is called great, if for some integer $a>1$ and for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all great polynomials.



Proposed by A. Golovanov
2016 Saint Petersburg Mathematical Olympiad
Sasha multiplied all the divisors of the natural number $n$. Fedya increased each divider by $1$, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can $n$ be equal to ?
2016 Saint Petersburg Mathematical Olympiad
Given the positive numbers $x_1, x_2,..., x_n$, such that $x_i \le 2x_j$ with $1 \le i < j \le n$.

Prove that there are positive numbers  $y_1\le y_2\le...\le y_n$, such that $x_k \le y_k \le 2x_k$ for all $k=1,2,..., n$
2016 Saint Petersburg Mathematical Olympiad
The circle inscribed in the triangle $ABC$ is tangent to side $AC$ at point $B_1$, and to side $BC$ at point $A_1$. On the side $AB$ there is a point $K$ such that $AK = KB_1, BK = KA_1$. Prove that $ \angle ACB\ge 60$
2016 Saint Petersburg Mathematical Olympiad
The cells of a square $100 \times 100$ table are colored in one of two colors, black or white. A coloring is called admissible if for any row or column, the number $b$ of black colored cells satisfies $50 \le b \le 60$. It is permitted to recolor a cell if by doing so the resulting configuration is still admissible. Prove that one can transition from any admissible coloring of the board to any other using a sequence of valid recoloring operations.
2016 Saint Petersburg Mathematical Olympiad
Points $A$ and $P$ are marked in the plane not lying on the line $\ell$. For all right triangles $ABC$ with hypotenuse on $\ell$, show that the circumcircle of triangle $BPC$ passes through a fixed point other than $P$.
2016 Saint Petersburg Mathematical Olympiad
The circle contains a closed $100$-part broken line, such that no three segments pass through one point. All its corners are obtuse, and their sum in degrees is divided by $720$. Prove that this broken line has an odd number of self-intersection points.
2016 Saint Petersburg Mathematical Olympiad
A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$.  Find all such pairs of $(P(x),a)$.