Navvye/Olympiad-Problems-CSV huggingface dataset:
1977 All Soviet Union Mathematical Olympiad
Seven elves are sitting at a round table. Each elf has a cup. Some cups are filled with some milk. Each elf in turn and clockwise divides all his milk between six other cups. After the seventh has done this, every cup was containing the initial amount of milk.  How much milk did every cup contain, if there was three litres of milk total?
1977 All Soviet Union Mathematical Olympiad
Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also.

a) Find all two- and four-digit fine numbers.

b) Is there any six-digit fine number?

c) Prove that there exists $20$-digit fine number.

d) Prove that there exist at least ten $100$-digit fine numbers.

e) Prove that there exists $30$-digit fine number.
1977 All Soviet Union Mathematical Olympiad
Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers.

Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.
1977 All Soviet Union Mathematical Olympiad
There are $1000$ tickets with the numbers $000, 001, ... , 999$, and $100$ boxes with the numbers $00, 01, ... , 99$. You may put a ticket in a box, if you can obtain the box number from the ticket number by deleting one digit. Prove that:

a) You can put all the tickets in $50$ boxes;

b) $40$ boxes is not enough for that;

c) it is impossible to use less than $50$ boxes.

d) Consider $10000$ $4$-digit tickets, and you are allowed to delete two digits. Prove that $34$ boxes is enough for storing all the tickets.

e) What is the minimal used boxes set in the case of $k$-digit tickets?
1977 All Soviet Union Mathematical Olympiad
Given a square $100\times 100$ on the sheet of cross-lined paper. There are several broken lines drawn inside the square. Their links consist of the small squares sides. They are neither pairwise- nor self-intersecting (have no common points). Their ends are on the big square boarder, and all the other vertices are in the big square interior.

Prove that there exists (in addition to four big square angles) a node (corresponding to the cross-lining family, inside the big square or on its side) that does not belong to any broken line.
1977 All Soviet Union Mathematical Olympiad
Given natural numbers $x_1,x_2,...,x_n,y_1,y_2,...,y_m$.

The following condition is valid: $(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m)<mn$. (*)



Prove that it is possible to delete some terms from (*) (not all and at least one) and to obtain another valid condition.
1977 All Soviet Union Mathematical Olympiad
Given $1000$ squares on the plane with their sides parallel to the coordinate axes. Let $M$ be the set of those squares centres.  Prove that you can mark some squares in such a way, that every point of $M$ will be contained not less than in one and not more than in four marked squares
1977 All Soviet Union Mathematical Olympiad
Given scales and a set of $n$ different weights. We take weights in turn and add them on one of the scales sides. Let us denote "$L$" the scales state with the left side down, and "$R$" -- with the right side down.

a) Prove that you can arrange the weights in such an order, that we shall obtain the sequence $LRLRLRLR...$ of the scales states. (That means that the state of the scales will be changed after putting every new weight.)

b) Prove that for every $n$-letter word containing $R$'s and $L$'s only you can arrange the weights in such an order, that the sequence of the scales states will be described by that word.
1977 All Soviet Union Mathematical Olympiad
Let us consider one variable polynomials with the senior coefficient equal to one. We shall say that two polynomials $P(x)$ and $Q(x)$ commute, if $P(Q(x))=Q(P(x))$ (i.e. we obtain the same polynomial, having collected the similar terms).

a) For every a find all $Q$ such that the $Q$ degree is not greater than three, and $Q$ commutes with $(x^2 - a)$.

b) Let $P$ be a square polynomial, and $k$ is a natural number. Prove that there is not more than one commuting with $P$ $k$-degree polynomial.

c) Find the $4$-degree and $8$-degree polynomials commuting with the given square polynomial $P$.

d) $R$ and $Q$ commute with the same square polynomial $P$. Prove that $Q$ and $R$ commute.

e) Prove that there exists a sequence $P_2, P_3, ... , P_n, ...$ ($P_k$ is $k$-degree polynomial), such that $P_2(x) = x^2 - 2$, and all the polynomials in this infinite sequence pairwise commute.
1976 All Soviet Union Mathematical Olympiad
There are $50$ exact watches lying on a table.

Prove that there exist a certain moment, when the sum of the distances from the centre of the table to the ends of the minute hands is more than the sum of the distances from the centre of the table to the centres of the watches.
1976 All Soviet Union Mathematical Olympiad
A row of $1000$ numbers is written on the blackboard. We write a new row, below the first according to the rule:

We write under every number $a$ the natural number, indicating how many times the number $a$ is encountered in the first line. Then we write down the third line: under every number $b$ -- the natural number, indicating how many times the number $b$ is encountered in the second line, and so on.

a) Prove that there is a line that coincides with the preceding one.

b) Prove that the eleventh line coincides with the twelfth.

c) Give an example of the initial line such, that the tenth row differs from the eleventh.
1976 All Soviet Union Mathematical Olympiad
Given three circumferences of the same radius in a plane.

a) All three are crossing in one point $K$. Consider three arcs $AK,CK,EK$ : the $A,C,E$ are the points of the circumferences intersection and the arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the arcs is $180$ degrees.

b) Consider the case, when the three circles give a curvilinear triangle $BDF$ as their intersection (instead of one point $K$). The arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the $AB, CD$ and $EF$ arcs is $180$ degrees.
1976 All Soviet Union Mathematical Olympiad
The natural numbers $x_1$ and $x_2$ are less than $1000$. We build a sequence:

$x_3 = |x_1 - x_2|,$

$x_4 = min \{ |x_1 - x_2|, |x_1 - x_3|, |x_2 - x_3|\},$

$...$

$x_k = min \{ |x_i - x_j|, 0 <i < j < k\},$

$...$

Prove that $x_{21} = 0$.
1976 All Soviet Union Mathematical Olympiad
Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?
1976 All Soviet Union Mathematical Olympiad
Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$.

Prove the following inequality: $|a|+|b|+|c|+|d| \ge  |a+d|+|b+d|+|c+d|$
1976 All Soviet Union Mathematical Olympiad
Given right $1976$-angle. The middles of all the sides and diagonals are marked.

What is the greatest number of the marked points lying on one circumference?
1976 All Soviet Union Mathematical Olympiad
There are $n$ rectangles drawn on the rectangular sheet of paper with the sides of the rectangles parallel to the sheet sides. The rectangles do not have pairwise common interior points.

Prove that after cutting out the rectangles the sheet will split into not more than $n+1$ part.
1976 All Soviet Union Mathematical Olympiad
There are three straight roads. Three pedestrians are moving along those roads, and they are NOT on one line in the initial moment. Prove that they will be one line not more than twice
1976 All Soviet Union Mathematical Olympiad
Given a chess-board $99\times 99$ with a set $F$ of fields marked on it (the set is different in three tasks). There is a beetle sitting on every field of the set $F$. Suddenly all the beetles have raised into the air and flied to another fields of the same set. The beetles from the neighbouring fields have landed either on the same field or on the neighbouring ones (may be far from their starting point). (We consider the fields to be neighbouring if they have at least one common vertex.) Consider a statement:

"There is a beetle, that either stayed on the same field or moved to the neighbouring one".

Is it always valid if the figure $F$ is:

a) A central cross, i.e. the union of the $50$-th row and the $50$-th column?

b) A window frame, i.e. the union of the $1$-st, $50$-th and $99$-th rows and the $1$-st, $50$-th and $99$-th columns?

c) All the chess-board?
1976 All Soviet Union Mathematical Olympiad
Let us call "big" a triangle with all sides longer than $1$. Given a equilateral triangle with all the sides equal to $5$.

Prove that:

a) You can cut $100$ big  triangles out of given one.

b) You can divide the given triangle onto $100$ big  nonintersecting ones fully covering the initial one.

c) The same as b), but the triangles either do not have common points, or have one common side, or one common vertex.

d) The same as c), but the initial triangle has the side $3$.
1976 All Soviet Union Mathematical Olympiad
Given natural $n$. We shall call "universal" such a sequence of natural number $a_1, a_2, ... , a_k, k\ge n$, if we can obtain every transposition of the first $n$ natural numbers (i.e such a sequence of $n$ numbers, that every one is encountered only once) by deleting some its members. (Examples: ($1,2,3,1,2,1,3$) is universal for $n=3$, and ($1,2,3,2,1,3,1$) -- not, because you can't obtain ($3,1,2$) from it.) The goal is to estimate the length of the shortest universal sequence for given $n$.

a) Give an example of the universal sequence of $n2$ members.

b) Give an example of the universal sequence of $(n^2 - n + 1)$ members.

c) Prove that every universal sequence contains not less than $n(n + 1)/2$ members

d) Prove that the shortest universal sequence for $n=4$ contains 12 members

e) Find as short universal sequence, as you can. The Organising Committee knows the method for $(n^2 - 2n +4) $ members.
1976 All Soviet Union Mathematical Olympiad
$n$ numbers are written down along the circumference. Their sum equals to zero, and one of them equals $1$.

a) Prove that there are two neighbours with their difference not less than $n/4$.

b) Prove that there is a number that differs from the arithmetic mean of its two neighbours not less than on $8/(n^2)$.

c) Try to improve the previous estimation, i.e what number can be used instead of $8$?

d) Prove that for $n=30$ there is a number that differs from the arithmetic mean of its two neighbours not less than on $2/113$, give an example of such $30$ numbers along the circumference, that not a single number differs from the arithmetic mean of its two neighbours more than on $2/113$.
1976 All Soviet Union Mathematical Olympiad
Given right $n$-angle wit the point $O$ -- its centre. All the vertices are marked either with $+1$ or $-1$. We may change all the signs in the vertices of right $k$-angle ($2 \le k \le n$) with the same centre $O$. (By $2$-angle we understand a segment, being halved by $O$.)

Prove that in a), b) and c) cases there exists such a set of $(+1)$s and $(-1)$s, that we can never obtain a set of $(+1)$s only.

a) $n = 15$,

b) $n = 30$,

c) $n > 2$,

d) Let us denote $K(n)$ the maximal number of $(+1)$ and $(-1)$ sets such, that it is impossible to obtain one set from another. Prove, for example, that $K(200) = 2^{80}$
1976 All Soviet Union Mathematical Olympiad
Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called "equator". We shall use the words "pole", "parallel","meridian" as self-explanatory.

a) Let $g(x)$, where $x$ is a point on the sphere, be the distance from this point to the equator plane.

Prove that $g(x)$ has the property if $x_1, x_2, x_3$ are the ends of the pairwise orthogonal radiuses, then

$g(x_1)^2 + g(x_2)2 + g(x_3)^2 = 1$. (*)



Let function $f(x)$ be an arbitrary nonnegative function on a sphere that satisfies (*) property.

b) Let $x_1$ and $x_2$ points be on the same meridian between the north pole and equator, and $x_1$ is closer to the pole than $x_2$. Prove that $f(x_1) > f(x_2)$.

c) Let $y_1$ be closer to the pole than $y_2$. Prove that $f(y_1) > f(y_2)$.

d) Let $z_1$ and $z_2$ be on the same parallel. Prove that $f(z_1) = f(z_2)$.

e) Prove that for all $x , f(x) = g(x)$.
1975 All Soviet Union Mathematical Olympiad
a) The triangle $ABC$ was turned around the centre of the outscribed circle by the angle less than $180$ degrees and thus was obtained the triangle $A_1B_1C_1$. The corresponding segments $[AB]$ and $[A_1B_1]$ intersect in the point $C_2, [BC]$ and $[B_1C_1]$ -- $A_2, [AC]$ and $[A_1C_1]$ -- $B_2$. Prove that the triangle $A_2B_2C_2$ is similar to the triangle $ABC$.

b) The quadrangle $ABCD$ was turned around the centre of the outscribed circle by the angle less than $180$ degrees and thus was obtained the quadrangle $A_1B1C_1D_1$. Prove that the points of intersection of the corresponding lines ( $(AB$) and $(A_1B_1), (BC)$ and $(B_1C_1), (CD)$ and $(C_1D_1), (DA)$ and $(D_1A_1)$ ) are the vertices of the parallelogram.
1975 All Soviet Union Mathematical Olympiad
Given a triangle $ABC$ with the unit area. The first player chooses a point $X$ on the side $[AB]$, than the second -- $Y$ on $[BC]$ side, and, finally, the first chooses a point $Z$ on $[AC]$ side. The first tries to obtain the greatest possible area of the $XYZ$ triangle, the second -- the smallest. What area can obtain the first for sure and how?
1975 All Soviet Union Mathematical Olympiad
What is the smallest perimeter of the convex $32$-angle, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?
1975 All Soviet Union Mathematical Olympiad
a) Given a big square consisting of $7\times 7$ squares. You should mark the centres of $k$ points in such a way, that no quadruple of the marked points will be the vertices of a rectangle with the sides parallel to the sides of the given squares. What is the greatest $k$ such that the problem has solution?

b) The same problem for $13\times 13$ square.
1975 All Soviet Union Mathematical Olympiad
Denote the middles of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2, A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.
1975 All Soviet Union Mathematical Olympiad
Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.
1975 All Soviet Union Mathematical Olympiad
Given a finite set of polygons in the plane. Every two of them have a common point.

Prove that there exists a straight line, that crosses all the polygons.
1975 All Soviet Union Mathematical Olympiad
Prove that for all the positive numbers $a,b,c$ the following inequality is valid:

$a^3+b^3+c^3+3abc>ab(a+b)+bc(b+c)+ac(a+c)$
1975 All Soviet Union Mathematical Olympiad
Three flies are crawling along the perimeter of the $ABC$ triangle in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the $ABC$ triangle. (The centre of masses for the triangle is the point of medians intersection.
1975 All Soviet Union Mathematical Olympiad
Several zeros, ones and twos are written on the blackboard. An anonymous clean in turn pairs of different numbers, writing, instead of cleaned, the number not equal to each. ($0$ instead of pair $\{1,2\}, 1$ instead of $\{0,2\}, 2$ instead of $\{0,1\}$). Prove that if there remains one number only, it does not depend on the processing order.
1975 All Soviet Union Mathematical Olympiad
Given a horizontal strip on the plane (its sides are parallel lines) and $n$ lines intersecting the strip. Every two of them intersect inside the strip, and not a triple has a common point. Consider all the paths along the segments of those lines, starting on the lower side of the strip and ending on the upper side with the properties: moving along such a path we are constantly rising up, and, having reached the intersection, we are obliged to turn to another line. Prove that:

a) there are not less than $n/2$ such a paths without common points;

b) there is a path consisting of not less than of $n$ segments;

c) there is a path that goes along not more than along $n/2+1$ lines;

d) there is a path that goes along all the $n$ lines.
1975 All Soviet Union Mathematical Olympiad
For what $k$ is it possible to construct a cube $kxkxk$ of the black and white cubes $1x1x1$ in such a way that every small cube has the same colour, that have exactly two his neighbours. (Two cubes are neighbours, if they have the common face.)
1975 All Soviet Union Mathematical Olympiad
Given a polynomial $P(x)$ with

a) natural coefficients;

b) integer coefficients;

Let us denote with $a_n$ the sum of the digits of $P(n)$ value.

Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.
1975 All Soviet Union Mathematical Olympiad
The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams.

What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if:

a) it is hockey (draws allowed)?

b) it is volleyball (no draws)?
1975 All Soviet Union Mathematical Olympiad
a) Given real numbers $a_1,a_2,b_1,b_2$ and positive $p_1,p_2,q_1,q_2$. Prove that in the table $2\times 2$

$(a_1 + b_1)/(p_1 + q_1)$ ,$ (a_1 + b_2)/(p_1 + q_2) $

$(a_2 + b_1)/(p_2 + q_1)$ ,$ (a_2 + b_2)/(p_2 + q_2)$



there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point).



b) Given real numbers $a_1, a_2, ... , a_n, b_1, b_2, ... , b_n$ and positive $p_1, p_2, ... , p_n, q_1, q_2, ... , q_n$. We build the table $n\times n$, with the numbers ($0 < i,j \le n$)



$(a_i + b_j)/(p_i + q_j)$



in the intersection of the $i$-th row and $j$-th column.

Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point).
1974 All Soviet Union Mathematical Olympiad
Given some cards with either "$-1$" or "$+1$" written on the opposite side. You are allowed to choose a triple of cards and ask about the product of the three numbers on the cards.

What is the minimal number of questions allowing to determine all the numbers on the cards ...

a) for $30$ cards,

b) for $31$ cards,

c) for $32$ cards.

(You should prove, that you cannot manage with less questions.)

d) Fifty above mentioned cards are lying along the circumference. You are allowed to ask about the product of three consecutive numbers only. You need to determine the product af all the $50$ numbers.

What is the minimal number of questions allowing to determine it?
1974 All Soviet Union Mathematical Olympiad
Among all the numbers representable as $36^k - 5^l$ ($k$ and $l$ are natural numbers) find the smallest.

Prove that it is really the smallest.
1974 All Soviet Union Mathematical Olympiad
a) Each of the side of the convex hexagon ($6$-angle) is longer than $1$. Does it necessary have a diagonal longer than $2$?

b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?
1974 All Soviet Union Mathematical Olympiad
Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.
1974 All Soviet Union Mathematical Olympiad
Given $n$ vectors of unit length in the plane. The length of their total sum is less than one.

Prove that you can rearrange them to provide the property:

for every $k, k\le  n$, the length of the sum of the first $k$ vectors is less than $2$.
1974 All Soviet Union Mathematical Olympiad
Find all the real $a,b,c$ such that the equality $|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$ is valid for all the real $x,y,z$.
1974 All Soviet Union Mathematical Olympiad
Given a square $ABCD$. Points $P$ and $Q$ are in the sides $[AB]$ and $[BC]$ respectively. $|BP|=|BQ|$. Let $H$ be the base of the perpendicular from the point $B$ to the segment $[PC]$. Prove that the angle $DHQ$ is a right one.
1974 All Soviet Union Mathematical Olympiad
Given some red and blue points. Some of them are connected by the segments. Let us call "exclusive" the point, if its colour differs from the colour of more than half of the connected points. Every move one arbitrary "exclusive" point is repainted to the other colour. Prove that after the finite number of moves there will remain no "exclusive" points.
1974 All Soviet Union Mathematical Olympiad
Find all the natural $n$ and $k$ such that $n^n$ has $k$ digits and $k^k$ has $n$ digits.
1974 All Soviet Union Mathematical Olympiad
Given points $D$ and $E$ on the legs $[CA]$ and $[CB]$, respectively, of the isosceles right triangle. $|CD| = |CE|$. The extensions of the perpendiculars from $D$ and $C$ to the line $AE$ cross the hypotenuse $AB$ in the points $K$ and $L$. Prove that $|KL| = |LB|$
1974 All Soviet Union Mathematical Olympiad
Two are playing the game "cats and rats" on the chess-board $8\times 8$. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions.

a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat?

b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape.
1974 All Soviet Union Mathematical Olympiad
a) Prove that you can rearrange the numbers $1, 2, ... , 32$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean.

b) Can you rearrange the numbers $1, 2, ... , 100$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean?
1974 All Soviet Union Mathematical Olympiad
Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.
1974 All Soviet Union Mathematical Olympiad
Given a convex polygon. You can put no triangle with area $1$ inside it.

Prove that you can put the polygon inside a triangle with the area $4$.
1974 All Soviet Union Mathematical Olympiad
Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$.

For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $f(x_1+x_2)\ge f(x_1)+f(x_2)$.

a) Prove that for all $x$ holds $f(x) \le  2x$.

b) Is the inequality $f(x) \le  1.9x$ valid?
1974 All Soviet Union Mathematical Olympiad
Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the middles of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?