Navvye/Olympiad-Problems-CSV huggingface dataset:
1984 Tournament Of Towns
Consider the $4(N-1)$ squares on the boundary of an $N$ by $N$ array of squares. We wish to insert in these squares $4 (N-1)$ consecutive integers (not necessarily positive) so that the sum of the numbers at the four vertices of any rectangle with sides parallel to the diagonals of the array (in the case of a “degenerate” rectangle, i.e. a diagonal, we refer to the sum of the two numbers in its corner squares) are one and the same number.

Is this possible? Consider the cases

(a) $N = 3$

(b) $N = 4$

(c) $N = 5$



(VG Boltyanskiy, Moscow)
1984 Tournament Of Towns
The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?
1984 Tournament Of Towns
An infinite squared sheet is given, with squares of side length $1$. The “distance” between two squares is defined as the length of the shortest path from one of these squares to the other if moving between them like a chess rook (measured along the trajectory of the centre of the rook). Determine the minimum number of colours with which it is possible to colour the sheet (each square being given a single colour) in such a way that each pair of squares with distance between them equal to $6$ units is given different colours. Give an example of such a colouring and prove that using a smaller number of colours we cannot achieve this goal.



(AG Pechkovskiy, IV Itenberg)
1984 Tournament Of Towns
In a ballroom dance class $15$ boys and $15$ girls are lined up in parallel rows so that $15$ couples are formed. It so happens that the difference in height between the boy and the girl in each couple is not more than $10$ cm. Prove that if the boys and the girls were placed in each line in order of decreasing height, then the difference in height in each of the newly formed couples would still be at most $10$ cm.



(AG Pechkovskiy, Moscow)
1984 Tournament Of Towns
Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size.



(AV Savkin)
1984 Tournament Of Towns
The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs.



(A Andjans, Riga)
1984 Tournament Of Towns
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane.



(IF Sharygin, Moscow)
1984 Tournament Of Towns
From a squared sheet of paper of size $29 \times 29, 99$ pieces, each a $2\times 2$ square, are cut off (all cutting is along the lines bounding the squares). Prove that at least one more piece of size $2\times 2$ may be cut from the remaining part of the sheet.



(S Fomin, Leningrad)
1984 Tournament Of Towns
Prove that, for any natural number $n$, the graph of any increasing function $f : [0,1] \to [0, 1]$ can be covered by $n$ rectangles each of area whose sides are parallel to the coordinate axes. Assume that a rectangle includes both its interior and boundary points.

(a) Assume that $f(x)$ is continuous on $[0,1]$.

(b) Do not assume that $f(x)$ is continuous on $[0,1]$.



(A Andjans, Riga)



PS. (a) for O Level, (b) for A Level
1984 Tournament Of Towns
(a) On each square of a squared sheet of paper of size $20 \times  20$ there is a soldier. Vanya chooses a number $d$ and Petya moves the soldiers to new squares in such a way that each soldier is moved through a distance of at least $d$ (the distance being measured between the centres of the initial and the new squares) and each square is occupied by exactly one soldier. For which $d$ is this possible?

(Give the maximum possible $d$, prove that it is possible to move the soldiers through distances not less than $d$ and prove that there is no greater $d$ for which this procedure may be carried out.)

(b) Answer the same question as (a), but with a sheet of size $21 \times 21$.



(SS Krotov, Moscow)
1984 Tournament Of Towns
An infinite (in both directions) sequence of rooms is situated on one side of an infinite hallway. The rooms are numbered by consecutive integers and each contains a grand piano. A finite number of pianists live in these rooms. (There may be more than one of them in some of the rooms.) Every day some two pianists living in adjacent rooms (the Arth and ($k +1$)st) decide that they interfere with each other’s practice, and they move to the ($k - 1$)st and ($k + 2$)nd rooms, respectively. Prove that these moves will cease after a finite number of days.



(VG Ilichev)
1984 Tournament Of Towns
Let $p(n)$ be the number of partitions of the natural number $n$ into natural summands. The diversity of a partition is by definition the number of different summands in it. Denote by $q(n)$ the sum of the diversities of all the $p(n) $ partitions of $n$.

(For example, $p(4) = 5$, the five distinct partitions of $4$ being $4, 3 + 1, 2+2, 2 + 1 + 1, 1 + 1 + 1 + 1,$ and $g(4) =1 + 2+1+ 2+1 = 7$.)

Prove that, for all natural numbers $n$,

(a) $q(n)= 1 + P(1) + P(2) + p(3) + ...+ p(n -1)$,

(b) $q(n) < \sqrt{2n}  p(n)$.



(AV Zelevinskiy, Moscow)
1984 Tournament Of Towns
In triangle $ABC$ the bisector of the angle at $B$ meets $AC$ at $D$ and the bisector of the angle at $C$ meets $AB$ at $E$. These bisectors intersect at $O$ and the lengths of $OD$ and $OE$ are equal. Prove that either $\angle BAC = 60^o$ or triangle $ABC$ is isosceles.
1984 Tournament Of Towns
A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner?



(Muscovite folklore)
1984 Tournament Of Towns
Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.
1984 Tournament Of Towns
Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle.



(V. V . Proizvolov , Moscow)
1984 Tournament Of Towns
Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.
1984 Tournament Of Towns
On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely?



(V.E. Kolosov)
1984 Tournament Of Towns
Six musicians gathered at a chamber music festival . At each scheduled concert some of these musicians played while the others listened as members of the audience . What is the least number of such concerts which would need to be scheduled in order to enable each musician to listen , as a member of the audience, to all the other musicians?



(Canadian origin)
1984 Tournament Of Towns
On the Island of Camelot live $13$ grey, $15$ brown and $17$ crimson chameleons . If two chameleons of different colours meet , they both simultaneously change colour to the third colour (e .g . if a grey and brown chameleon meet each other they both change to crimson) . Is it possible that they will eventually all be the same colour?



(V . G . Ilichev)
1984 Tournament Of Towns
In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .
1984 Tournament Of Towns
In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$.

Prove that $BD + DA = BC$.
1984 Tournament Of Towns
A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers

$b_1=a_1$

$b_2= a_1 + a_2$

$b_3=a_1 + a_2 + a_3$

...

$b_{100}=a_1 + a_2 + ...+a_{100}$

Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different .



( L . D . Kurlyandchik , Leningrad)
1984 Tournament Of Towns
We are given a regular decagon with all diagonals drawn. The number "$+ 1$ " is attached to each vertex and to each point where diagonals intersect (we consider only internal points of intersection). We can decide at any time to simultaneously change the sign of all such numbers along a given side or a given diagonal . Is it possible after a certain number of such operations to have changed all the signs to negative?
1984 Tournament Of Towns
A $7 \times  7$ square is made up of $16$ $1 \times  3$ tiles and $1$ $1 \times  1$ tile. Prove that the $1 \times  1$ tile lies either at the centre of the square or adjoins one of its boundaries .
1985 Tournament Of Towns
A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .
1985 Tournament Of Towns
There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam.



(S. Fomin, Leningrad)
1985 Tournament Of Towns
Find all real solutions of the system of equations $\begin{cases} 

(x + y) ^3 = z \\ (y + z) ^3 = x \\ ( z+ x) ^3 = y \end{cases} $



(Based on an idea by A . Aho , J. Hop croft , J. Ullman )
1985 Tournament Of Towns
Three grasshoppers are on a straight line. Every second one grasshopper jumps. It jumps across one (but not across two) of the other grasshoppers . Prove that after $1985$ seconds the grasshoppers cannot be in the initial position .



(Leningrad Mathematical Olympiad 1985)
1985 Tournament Of Towns
Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ?



(S. Fomin , Leningrad)
1985 Tournament Of Towns
$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$.

Prove that $c \ge  (a + b) \sin \frac{\gamma}{2}$



(V. Prasolov )
1985 Tournament Of Towns
The integer part $I (A)$ of a number $A$ is the greatest integer which is not greater than $A$ , while the fractional part $F(A)$ is defined as $A - I(A)$ .

(a) Give an example of a positive number $A$ such that $F(A) + F( 1/A) = 1$ .

(b) Can such an $A$ be a rational number?



(I. Varge, Romania)
1985 Tournament Of Towns
A certain class of $32$ pupils is organised into $33$ clubs , so that each club contains $3$ pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.
1985 Tournament Of Towns
A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square.
1985 Tournament Of Towns
The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times .

(a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits?

(b) Prove that there must be a row or column containing more than $3$ different digits .



{ L . D . Kurlyandchik , Leningrad)
1985 Tournament Of Towns
In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.
1985 Tournament Of Towns
From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .
1985 Tournament Of Towns
Three real numbers $a, b$ and $c$ are given . It is known that $a + b + c >0 , bc+ ca + ab > 0$ and $abc > 0$ . Prove that $a > 0 , b > 0$ and $c > 0$ .
1985 Tournament Of Towns
Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.
1985 Tournament Of Towns
The radius $OM$ of a circle rotates uniformly at a rate of $360/n$ degrees per second , where $n$ is a positive integer . The initial radius is $OM_0$. After $1$ second the radius is $OM_1$ , after two more seconds (i.e. after three seconds altogether) the radius is $OM_2$ , after $3$ more seconds (after $6$ seconds altogether) the radius is $OM_3$, ..., after $n - 1$ more seconds its position is $OM_{n-1}$. For which values of $n$ do the points $M_0, M_1 , ..., M_{n-1}$ divide the circle into $n$ equal arcs?

(a) Is it true that the powers of $2$ are such values?

(b) Does there exist such a value which is not a power of $2$?



(V. V. Proizvolov , Moscow)
1985 Tournament Of Towns
The convex set $F$ does not cover a semi-circle of radius $R$.

Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ?

What if $F$ is not convex?



( N . B . Vasiliev , A. G . Samosvat)
1985 Tournament Of Towns
A square is divided into rectangles.

A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$.

(a) Prove that every two rectangles in such a division are members of a certain "chain".

(b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") .



(A.I . Golberg, V.A. Gurevich)
1985 Tournament Of Towns
Eight football teams participate in a tournament of one round (each team plays each other team once) . There are no draws. Prove that it is possible at the conclusion of the tournament to be able to find $4$ teams , say $A, B, C$ and $D$ so that $A$ defeated $B, C$ and $D, B$ defeated $C$ and $D$ , and $C$ defeated $D$ .
1985 Tournament Of Towns
In the game "cat and mouse" the cat chases the mouse in either labyrinth $A, B$ or $C$ .



The cat makes the first move starting at the point marked "$K$" , moving along a marked line to an adjacent point . The mouse then moves , under the same rules, starting from the point marked "$M$" . Then the cat moves again, and so on . If, at a point of time , the cat and mouse are at the same point the cat eats the mouse.

Is there available to the cat a strategy which would enable it to catch the mouse , in cases $A, B$ and $C$?



(A. Sosinskiy, Moscow)
1985 Tournament Of Towns
A teacher gives each student in the class the following task in their exercise book .

"Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$."

Prove that each student would obtain the same result .



( A . K . Tolpygo, Kiev)
1985 Tournament Of Towns
Two chess players play each other at chess using clocks (when a player makes a move , the player stops his clock and starts the clock of his opponent) . It is known that when both players have just completed their $40$th move , both of their clocks read exactly $2$ hr $30$ min . Prove that there was a moment in the game when the clock of one player registered $1$ min $51$ sec less than that of the other . Furthermore , can one assert that the difference between the

two clock readings was ever equal to $2$ minutes?



(S . Fomin , Leningrad)
1985 Tournament Of Towns
Two people throw coins. One throws his coin $10$ times, the other throws his $11$ times . What is the probability that the second coin fell showing "heads" a greater number of times than the first?



(For those not familiar with Probability Theory this question could have been reformulated thus : Consider various arrangements of a $21$ digit number in which each digit must be a " $1$ " or a "$2$" . Among all such numbers what fraction of them will have more occurrences of the digit "$2$" among the last $11$ digits than among the first $10$?)



(S. Fomin , Leningrad)
1985 Tournament Of Towns
The numerical sequence $x_1 , x_2 ,.. $ satisfies $x_1 = \frac12$ and $x_{k+1} =x^2_k+x_k$ for all natural integers $k$ . Find the integer part of the sum $\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}$



{A. Andjans, Riga)
1985 Tournament Of Towns
(a)The game of "super- chess" is played on a $30 \times 30$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g. $m$ squares forward and $n$ squares to the right) does not depend on the piece 's starting square . Prove that

(i) A piece cannot cap ture a piece on a given square from more than $20$ starting squares.

(ii) It is possible to arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move.



(b) The game of "super-chess" is played on a $100 \times 100$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares. A piece "captures" another piece which is on a square to which it has moved. It is possible that a permitted move (e.g. $m$ squares forward and $n$ squares to the right) may vary, depending on the piece's position .

Prove that one can arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move.



( A . K . Tolpygo, Kiev)



PS. (a) for Juniors , (b) for Seniors
1985 Tournament Of Towns
We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ .



(V. Prasolov)
1985 Tournament Of Towns
(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least  $n- 1$ of these angles are not acute .

(b) Same problem for a convex polyhedron with $n$ vertices.



(V. Boltyanskiy, Moscow)
1985 Tournament Of Towns
In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ .



(I. Sharygin , Moscow)
1986 Tournament Of Towns
Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral .



(G . Galperin , A . Savin, Moscow)
1986 Tournament Of Towns
A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$?



(S . Fomin, Leningrad)
1986 Tournament Of Towns
The streets of a town are arranged in three directions , dividing the town into blocks which are equilateral triangles of equal area. Traffic , when reaching an intersection, may only go straight ahead, or turn left or right through $120^0$ , as shown in the diagram.



No turns are permitted except the ones at intersections . One car departs for a certain nearby intersection and when it reaches it a second car starts moving toward it. From then on both cars continue travelling at the same speed (but do not necessarily take the same turns). Is it possible that there will be a time when they will encounter each other somewhere?



( N . N . Konstantinov , Moscow )
1986 Tournament Of Towns
We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points$ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second point of intersection (i.e. the one other than $M$) of the points of intersection of circles circumscribed about triangles $AKM$ and $MLC$ lies on the diagonal $AC$.



(V . N . Dubrovskiy)
1986 Tournament Of Towns
$20$ football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select $10$ teams, so that no two of them have yet played each other.



( S . A . Genkin)
1986 Tournament Of Towns
( "Sisyphian Labour" )

There are $1001$ steps going up a hill , with  rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are $500$ rocks, originally located on the first $500$ steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step.

Can Aid stop him?



( S . Yeliseyev)
1986 Tournament Of Towns
Thirty pupils from the same class decided to exchange visits. Any pupil may make several visits during one evening, but must stay home if he is receiving guests that evening. Prove that in order that each pupil visit each of his classmates

(a) four evenings are not enough

(b) five evenings are not enough

(c) ten evenings are enough

(d) even seven evenings are enough
1986 Tournament Of Towns
For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?
1986 Tournament Of Towns
Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector?



(Problem from Leningrad)
1986 Tournament Of Towns
The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous.



(A.I. Plotkin, Leningrad)
1986 Tournament Of Towns
The bisector of angle $BAD$ in the parallelogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. It is known that $ABCD$ is not a rhombus. Prove that the centre of the circle passing through the points $C, K$ and $L$ lies on the circle passing through the points $B, C$ and $D$.
1986 Tournament Of Towns
Given the nonincreasing sequence of non-negative numbers in which $a_1 \ge a_2 \ge a_3 \ge ... \ge a_{2n-1}\ge 0$.



Prove that $a^2_1 -a^2_2 + a^2_3 - ... + a^2_{2n- l} \ge  (a_1 - a_2 + a_3 -  ... + a_{2n- l} )^2$ .



( L . Kurlyandchik , Leningrad )
1986 Tournament Of Towns
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.