Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 Silk Road
Let $0<a<b<1$ be reals numbers and

$$g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0<x<b\\b-a, & \mbox{ if } x=a \\x-a, & \mbox{ if } a<x<b\\1-a ,&\mbox{ if } x=b \\ x-a  ,&\mbox{ if } b<x<1 \end{array}\right.$$

Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$, for which $g^{[n]}(x_i)=x_i  \ (0 \leq i \leq n)$. Prove that there exists a positive integer $N$, such that $g^{[N]}(x)=x$ for all $0<x<1$.



($g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$)



Official solution here
2003 Silk Road
Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $.



Problem was post earlier here , but solution not gives and olympiad doesn't indicate, so I post it again  :blush:



Official solution here
2002 Silk Road
Let $ \triangle ABC$ be a triangle with incircle $ \omega(I,r)$and circumcircle $ \zeta(O,R)$.Let $ l_{a}$ be the angle bisector of $ \angle BAC$.Denote $ P=l_{a}\cap\zeta$.Let $ D$ be the point of tangency $ \omega$ with $ [BC]$.Denote $ Q=PD\cap\zeta$.Show that $ PI=QI$ if $ PD=r$.
2002 Silk Road
I tried to search SRMC problems,but i didn't find them(I found only SRMC 2006).Maybe someone know where on this site i could find SRMC problems?I have all SRMC problems,if someone want i could post them, :wink:

Here is one of them,this is one nice inequality from first SRMC:

Let $ n$ be an integer with $ n>2$ and $ a_{1},a_{2},\dots,a_{n}\in R^{+}$.Given any positive integers $ t,k,p$ with $ 1<t<n$,set $ m=k+p$,prove the following inequalities:

a)

$ \frac{a_{1}^{p}}{a_{2}^{k}+a_{3}^{k}+\dots+a_{t}^{k}}+\frac{a_{2}^{p}}{a_{3}^{k}+a_{4}^{k}+\dots+a_{t+1}^{k}}+\dots+\frac{a_{n-1}^{p}}{a_{n}^{k}+a_{1}^{k}+\dots+a_{t-2}^{k}}+\frac{a_{n}^{p}}{a_{1}^{k}+a_{2}^{k}+\dots+a_{t-1}^{k}}\geq\frac{(a_{1}^{p}+a_{2}^{p}\dots+a_{n}^{p})^{2}}{(t-1) ( a_{1}^{m}+a_{2}^{m}+\dots+a_{n}^{m})}$

b)$ \frac{a_{2}^{k}+a_{3}^{k}\dots+a_{t}^{k}}{a_{1}^{p}}+\frac{a_{3}^{k}+a_{4}^{k}\dots+a_{t+1}^{k}}{a_{2}^{p}}+\dots+\frac{a_{1}^{k}+a_{2}^{k}\dots+a_{t-1}^{k}}{a_{n}^{p}}\geq\frac{(t-1)(a_{1}^{k}+a_{2}^{k}\dots+a_{n}^{k})^{2}}{( a_{1}^{m}+a_{2}^{m}+\dots+a_{n}^{m})}$ :wink:
2002 Silk Road
In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by $2002$. Prove that every number $\alpha$ can be replaced by a certain number $\alpha'$ , divisible by $2002$  so that $|\alpha-\alpha'| <2002$  and the sum of the numbers in all rows, and in all columns will not change.
2002 Silk Road
Observe that the fraction $ \frac{1}{7}=0,142857$ is a pure periodical decimal with period $ 6=7-1$,and in one period one has $ 142+857=999$.For $ n=1,2,\dots$ find a sufficient and necessary

condition that the fraction $ \frac{1}{2n+1}$ has the same properties as above and find two such fractions other than $ \frac{1}{7}$.
1980 Tournament Of Towns
On the circumference of a circle there are red and blue points. One may add a red point and change the colour of both its neighbours (to the other colour) or remove a red point and change the colour of both its previous neighbours. Initially there are two red points. Prove that there is no sequence of allowed operations which leads to the configuration consisting of two blue points.



(K Kazarnovskiy, Moscow)
1980 Tournament Of Towns
In a $N \times N$ array of numbers, all rows are different (two rows are said to be different even if they differ only in one entry). Prove that there is a column which can be deleted in such a way that the resulting rows will still be different.



(A Andjans, Riga)
1980 Tournament Of Towns
If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.
1980 Tournament Of Towns
We are given convex quadrilateral $ABCD$. Each of its sides is divided into $N$ line segments of equal length. The points of division of side $AB$ are connected with the points of division of side $CD$ by straight lines (which we call the first set of straight lines), and the points of division of side BC are connected with the points of division of side $DA$ by straight lines (which we call the second set of straight lines) as shown in the diagram, which illustrates the case $N = 4$.

This forms $N^2$ smaller quadrilaterals. From these we choose $N$ quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of $ABCD$ divided by $N$.



(A Andjans, Riga)
1980 Tournament Of Towns
A finite set of line segments, of total length $18$, belongs to a square of unit side length (we assume that the square includes its boundary and that a line segment includes its end points). The line segments are parallel to the sides of the square and may intersect one another. Prove that among the regions into which the square is divided by the line segments, at least one of these must have area not less than $0.01$.



(A Berzinsh, Riga)
1980 Tournament Of Towns
We are given $30$ non-zero vectors in $3$ dimensional space.

Prove that among these there are two such that the angle between them is less than $45^o$.
1981 Tournament Of Towns
Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.
1981 Tournament Of Towns
$M$ is a finite set of points in a plane. Point $O$ in the plane is called an “almost centre of symmetry” of set $M$ if it is possible to remove from $M$ one point in such a way that among the remaining members $O$ is the centre of symmetry in the usual sense. How many such “almost centres of symmetry” may a finite point set in a plane have? Indicate all such points.



(V Prasolov, Moscow)
1981 Tournament Of Towns
$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area.



(V Varvarkin)
1981 Tournament Of Towns
Each of $K$ friends simultaneously learns one different item of news. They begin to phone one another to tell them their news. Each conversation lasts exactly one hour, during which time it is possible for two friends to tell each other all of their news. What is the minimum number of hours needed in order for all of the friends to know all of the news? Consider in this problem

(a) $K = 64$.

(b) $K = 55$.

(c) $K = 100$.



(A Andjans, Riga)



PS. (a) was the junior problem, (a),(b),(c) the senior one
1981 Tournament Of Towns
a) A game is played on an infinite plane. There are fifty one pieces, one “wolf” and $50$ “sheep”. There are two players. The first commences by moving the wolf. Then the second player moves one of the sheep, the first player moves the wolf, the second player moves a sheep, and so on. The wolf and the sheep can move in any direction through a distance of up to one metre per move. Is it true that for any starting position the wolf will be able to capture at least one sheep?

b) A game is played on an infinite plane. There are two players. One has a piece known as a “wolf”, while the other has $K$ pieces known as “sheep”. The first player moves the wolf, then the second player moves a sheep, the first player moves the wolf again, the second player moves a sheep, and so on. The wolf and the sheep can move in any direction, with a maximum distance of one metre per move. Is it true that for any value of $K$ there exists an initial position from which the wolf can not capture any sheep?



PS. (a) was the junior version, (b) the senior one
1981 Tournament Of Towns
We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces?



(A Andjans, Riga)
1981 Tournament Of Towns
Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$.



(E Turkevich)
1981 Tournament Of Towns
On an infinite “squared” sheet six squares are shaded as in the diagram. On some squares there are pieces. It is possible to transform the positions of the pieces according to the following rule: if the neighbour squares to the right and above a given piece are free, it is possible to remove this piece and put pieces on these free squares.

The goal is to have all the shaded squares free of pieces. Is it possible to reach this goal if

(a) In the initial position there are $6$ pieces and they are placed on the $6$ shaded squares?

(b) In the initial position there is only one piece, located in the bottom left shaded square?



(M Kontsevich, Moscow)
1982 Tournament Of Towns
Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors.



(M Levin)
1982 Tournament Of Towns
The lengths of all sides and both diagonals of a quadrilateral are less than $1$ metre.

Prove that it may be placed in a circle of radius $0.9$ metres.
1982 Tournament Of Towns
a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$, one can find $100$ members for which $a_k > k$.

b) Prove that in an infinite sequence  ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infinitely many such numbers $a_k$ such that $a_k > k$.



(A Andjans, Riga)



PS. (a) for juniors  (b) for seniors
1982 Tournament Of Towns
In a certain country there are more than $101$ towns. The capital of this country is connected by direct air routes with $100$ towns, and each town, except for the capital, is connected by direct air routes with $10$ towns (if $A$ is connected with $B, B$ is connected with $A$). It is known that from any town it is possible to travel by air to any other town (possibly via other towns). Prove that it is possible to close down half of the air routes connected with the capital, and preserve the capability of travelling from any town to any other town within the country.



(IS Rubanov)
1982 Tournament Of Towns
Consider the sequence $1, \frac12, \frac13, \frac14 ,...$

Does there exist an arithmetic progression composed of terms of this sequence

(a) of length $5$,

(b) of length greater than $5$ (if so, what possible length)?



(G Galperin, Moscow)
1982 Tournament Of Towns
(a) Prove that for any positive numbers $x_1,x_2,...,x_k$ ($k > 3$),

$$\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2$$(b) Prove that for every $k$ this inequality cannot be sharpened, i.e. prove that for every given $k$ it is not possible to change the number $2$ in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers $x_1,x_2,...,x_k$.



(A Prokopiev)
1982 Tournament Of Towns
A square is subdivided into $K^2$ equal smaller squares. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). Find the minimum number of links in this broken line.



(A Andjans, Riga)
1982 Tournament Of Towns
A polynomial $P(x)$ has unity as the coefficient of its highest power, and has the property that with natural number arguments, it can take all values of form $2^M$ , where $M$ is a natural number. Prove that the polynomial is of degree $1$.
1982 Tournament Of Towns
There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits.



(A Merkov, Moscow)
1982 Tournament Of Towns
A number of objects, each coloured in one of two given colours, are arranged in a line (there is at least one object having each of the given colours). It is known that each two objects, between which there are exactly $10$ or $15$ other objects, are of the same colour. What is the greatest possible number of such pieces?
1982 Tournament Of Towns
Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .
1982 Tournament Of Towns
(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords.

Is it necessary that two of these chords are of equal length?

(b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords.

Prove that among these $10$ chords there are two chords of equal length.



(VV Proizvolov, Moscow)
1982 Tournament Of Towns
Prove that for all natural numbers greater than $1$

$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$.



(VV Kisil)
1982 Tournament Of Towns
Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges?

$AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH$.
1982 Tournament Of Towns
$60$ symbols, each of which is either $X$ or $O$, are written consecutively on a strip of paper. This strip must then be cut into pieces with each piece containing symbols symmetric about their centre, e.g. $O, XX, OXXXXX, XOX$, etc.

(a) Prove that there is a way of cutting the strip so that there are no more than $24$ such pieces.

(b) Give an example of such an arrangement of the signs for which the number of pieces cannot be less than $15$.

(c) Try to improve the result of (b).
1982 Tournament Of Towns
(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$  equals $\frac{n}{2}\overrightarrow{MO}$,  where $O$ is the centre of the $n$-gon.

(b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron.



(VV Prasolov, Moscow)
1982 Tournament Of Towns
The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.
1983 Tournament Of Towns
A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour?



(NN Konstantinov, Moscow)
1983 Tournament Of Towns
(a) A regular $4k$-gon is cut into parallelograms. Prove that among these there are at least $k$ rectangles.

(b) Find the total area of the rectangles in (a) if the lengths of the sides of the $4k$-gon equal $a$.



(VV Proizvolov, Moscow)
1983 Tournament Of Towns
In Shvambrania there are $N$ towns, every two of which are connected by a road. These roads do not intersect. If necessary, some of them pass over or under others via bridges. An evil magician establishes one-way rules along the roads in such a way that if someone goes out of a certain town he is unable to come back. Prove that

(a) It is possible to establish such rules.

(b) There exists a town from which it is possible to reach any other town, and there exists a town from which it is not possible to go out.

(c) There is one and only one route passing through all towns.

(d) The magician can realise his intention in $N!$ ways.



(LM Koganov, Moscow)



PS. (a),(b),(c) for Juniors, (a),(b),(d) for Seniors
1983 Tournament Of Towns
The natural numbers $M$ and $K$ are represented by different permutations of the same digits. Prove that

(a) The sum of the digits of $2M$ equals the sum of the digits of $2K$.

(b) The sum of the digits of $M/2$ equals the sum of the digits of $K/2$ ($M, K$ both even).

(c) The sum of the digits of $5M$ equals the sum of the digits of $5 K$.



(AD Lisitskiy)
1983 Tournament Of Towns
A version of billiards is played on a right triangular table, with a pocket in each of the three corners, and one of the acute angles being $30^o$. A ball is played from just in front of the pocket at the $30^o$. vertex toward the middle of the opposite side. Prove that if the ball is played hard enough, it will land in the pocket of the $60^o$  vertex after $8$ reflections.
1983 Tournament Of Towns
(a) An infinite sheet is divided into squares by two sets of parallel lines. Two players play the following game: the first player chooses a square and colours it red, the second player chooses a non-coloured square and colours it blue, the first player chooses a non-coloured square and colours it red, the second player chooses a non-coloured square and colours it blue, and so on. The goal of the first player is to colour four squares whose vertices form a square with sides parallel to the lines of the two parallel sets. The goal of the second player is to prevent him. Can the first player win?



(b) What is the answer to this question if the second player is permitted to colour two squares at once?



(DG Azov)



PS. (a) for Juniors, (a),(b) for Seniors
1983 Tournament Of Towns
Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other.



(An established theorem)
1983 Tournament Of Towns
Numbers from $1$ to $1000$ are arranged around a circle. Prove that it is possible to form $500$ non-intersecting line segments, each joining two such numbers, and so that in each case the difference between the numbers at each end (in absolute value) is not greater than $749$.



(AA Razborov, Moscow)
1983 Tournament Of Towns
On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM},  \overrightarrow{BK}$ and  $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.
1983 Tournament Of Towns
There are $K$ boys placed around a circle. Each of them has an even number of sweets. At a command each boy gives half of his sweets to the boy on his right. If, after that, any boy has an odd number of sweets, someone outside the circle gives him one more sweet to make the number even. This procedure can be repeated indefinitely. Prove that there will be a time at which all boys will have the same number of sweets.



(A Andjans, Riga)
1983 Tournament Of Towns
A point is chosen inside a regular $k$-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into $2k$ segments. Let these segments be enumerated consecutively by the numbers $1,2, 3, ... ,2k$. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers.



(A Andjans, Riga)
1983 Tournament Of Towns
$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied:

If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$.

Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation.



(M Kontsevich, Moscow)
1983 Tournament Of Towns
Inside square $ABCD$ consider a point $M$. Prove that the points of intersection of the medians of triangles $ABM, BCM, CDM$ and $DAM$ form a square.



(V Prasolov)
1983 Tournament Of Towns
Find all natural numbers $k$ which can be represented as the sum of two relatively prime numbers not equal to $1$.
1983 Tournament Of Towns
Construct a quadrilateral given its side lengths and the length of the segment joining the midpoints of its diagonals.



(IZ Titovich)
1983 Tournament Of Towns
$a_1,a_2,a_3,...$ is a monotone increasing sequence of natural numbers. It is known that for any $k, a_{a_k} = 3k$.

a) Find $a_{100}$.

b) Find $a_{1983}$.



(A Andjans, Riga)



PS. (a) for Juniors, (b) for Seniors
1983 Tournament Of Towns
$N^2$ pieces are placed on an $N \times N$ chessboard. Is it possible to rearrange them in such a way that any two pieces which can capture each other (when considered to be knights) after the rearrangement are on adjacent squares (i.e. squares having at least one common boundary point)? Consider two cases:

(a) $N = 3$.

(b) $N = 8$



(S Stefanov)
1983 Tournament Of Towns
On sides $CB$ and $CD$ of square $ABCD$ are chosen points $M$ and $K$ so that the perimeter of triangle $CMK$ equals double the side of the square. Find angle $MAK$.
1983 Tournament Of Towns
Consider all nine-digit numbers, consisting of non-repeating digits from $1$ to $9$ in an arbitrary order. A pair of such numbers is called “conditional” if their sum is equal to $987654321$.

(a) Prove that there exist at least two conditional pairs (noting that ($a,b$) and ($b,a$) is considered to be one pair).

(b) Prove that the number of conditional pairs is odd.



(G Galperin, Moscow)
1983 Tournament Of Towns
The centre $O$ of the circumcircle of $\vartriangle ABC$ lies inside the triangle. Perpendiculars are dropped from $O$ to the sides. When produced beyond the sides they meet the circumcircle at points $K, M$ and $P$. Prove that $\overrightarrow{OK} + \overrightarrow{OM} + \overrightarrow{OP} = \overrightarrow{OI}$, where $I$ is the centre of the inscribed circle of $\vartriangle  ABC$.



(V Galperin, Moscow)
1983 Tournament Of Towns
A set $A$ of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to $A$ there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number $k$ and such a way of moving the kings that after $k$ moves the kings will occupy all squares of the chessboard? Consider the following cases:

(a) $A$ is the set of all squares, both of whose coordinates are multiples of $100$.

(There is a horizontal line numbered by the integers from $-\infty$ to $+\infty$, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.)

(b) $A$ is the set of all squares which are covered by $100$ fixed arbitrary queens

(i.e. each square covered by at least one queen).



Remark:

If $A$ consists of just one square, then $k = 1$ and the required way is the following:

all kings to the left of the square of $A$ make one move to the right.
1984 Tournament Of Towns
The price of $175$ Humpties is more than the price of $125$ Dumpties but less than that of $126$ Dumpties.

Prove that you cannot buy three Humpties and one Dumpty for

(a) $80$ cents.

(b) $1$ dollar.



(S Fomin, Leningrad)



PS. (a) for Juniors , (a),(b) for Seniors
1984 Tournament Of Towns
In the convex pentagon $ABCDE, AE = AD, AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.