Navvye/Olympiad-Problems-CSV huggingface dataset:
1985 All Soviet Union Mathematical Olympiad
Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order.

Prove that $|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$.
1985 All Soviet Union Mathematical Olympiad
The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?
1985 All Soviet Union Mathematical Olympiad
One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle outscribed around $AMD$ triangle has radius $R$.
1985 All Soviet Union Mathematical Olympiad
Given right hexagon.

The lines parallel to all the sides are drawn from all the vertices and middles of the sides (consider only the interior, with respect to the hexagon, parts of those lines). Thus the hexagon is divided onto $24$ triangles, and the figure has $19$ nodes. $19$ different numbers are written in those nodes.

Prove that at least $7$ of $24$ triangles have the property: the numbers in its vertices increase (from the least to the greatest) counterclockwise.
1985 All Soviet Union Mathematical Olympiad
Solve the equation ("$2$" encounters $1985$ times):

$$\frac{x}{2+ \frac{x}{2+\frac{x}{2+... \frac{x}{2+\sqrt {1+x}}}}}=1$$
1985 All Soviet Union Mathematical Olympiad
All the points situated more close than $1$ cm to ALL the vertices of the right pentagon with $1$ cm side, are deleted from that pentagon.  Find the area of the remained figure.
1985 All Soviet Union Mathematical Olympiad
Given big enough sheet of cross-lined paper with the side of the squares equal to $1$. We are allowed to cut it along the lines only.

Prove that for every $m>12$ we can cut out a rectangle of the greater than $m$ area such, that it is impossible to cut out a rectangle of $m$ area from it.
1985 All Soviet Union Mathematical Olympiad
The $ABCDA_1B_1C_1D_1$ cube has unit length edges.

Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through $A,C$ and $B_1$ points.
1984 All Soviet Union Mathematical Olympiad
a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$.

b) Let $n$ is divisible by $4$.

Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.
1984 All Soviet Union Mathematical Olympiad
Prove that every positive $a$ and $b$ satisfy inequality $\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge  a\sqrt b + b\sqrt a$
1984 All Soviet Union Mathematical Olympiad
Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.)

We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively.

Prove that the triangle $ABC$ is equilateral.
1984 All Soviet Union Mathematical Olympiad
Given four colours and enough square plates $1\times 1$. We have to paint four edges of every plate with four different colours and combine plates, putting them with the edges of the same colour together.

Describe all the pairs $m,n$, such that we can combine those plates in a $n\times m$ rectangle, that has every edge of one colour, and its four edges have different colours.
1984 All Soviet Union Mathematical Olympiad
Prove that every positive $x,y$ and real $a$ satisfy inequality $x^{\sin ^2a}  y^{\cos^2a} < x + y$.
1984 All Soviet Union Mathematical Olympiad
Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win?
1984 All Soviet Union Mathematical Olympiad
$n$ natural numbers ($n>3$) are written on the circumference. The relation of the two neighbours sum to the number itself is a whole number.  Prove that the sum of those relations is

a) not less than $2n$

b)  less than $3n$
1984 All Soviet Union Mathematical Olympiad
The circle with the centre $O$ is inscribed in the $ABC$ triangle. The circumference touches its sides $[BC], [CA], [AB]$ in $A_1, B_1, C_1$ points respectively. The $[AO], [BO], [CO]$ segments cross the circumference in $A_2, B_2, C_2$ points respectively. Prove that $(A_1A_2),(B_1B_2)$ and $(C_1C_2)$ lines intersect in one point.
1984 All Soviet Union Mathematical Olympiad
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.
1984 All Soviet Union Mathematical Olympiad
$n$ real numbers are written in increasing order in a line. The same numbers are written in the second line below in unknown order. The third line contains the sums of the pairs of numbers above from two previous lines. It comes out, that the third line is arranged in increasing order.

Prove that the second line coincides with the first one.
1984 All Soviet Union Mathematical Olympiad
Given $ABC$ triangle. From the $P$ point three lines $(PA),(PB),(PC)$ are drawn. They cross the outscribed circumference in $A_1, B_1,C_1$ points respectively. It comes out that the $A_1B_1C_1$ triangle equals to the initial one.

Prove that there are not more than eight such a points $P$ in a plane.
1984 All Soviet Union Mathematical Olympiad
Positive $x,y,z$ satisfy a system: $\begin{cases}  x^2 + xy + y^2/3= 25 \\
 y^2/ 3  + z^2 = 9 \\
z^2 + zx + x^2 = 16 \end{cases}$

Find the value of  expression $xy + 2yz + 3zx$.
1984 All Soviet Union Mathematical Olympiad
The teacher wrote on a blackboard: " $x^2 + 10x + 20$ "

Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: " $x^2 + 20x + 10$ ".

Is it true that some time during the process there was written the square polynomial with the integer roots?
1984 All Soviet Union Mathematical Olympiad
The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is outscribed around the circle with radius $R$ ($R>r$).  Find the coin trace area (a sort of polygon ring).
1984 All Soviet Union Mathematical Olympiad
There are scales and $(n+1)$ weights with the total weight $2n$. Each weight is an integer. We put all the weights in turn on the lighter side of the scales, starting from the heaviest one, and if the scales is in equilibrium -- on the left side.

Prove that when all the weights will be put on the scales, they will be in equilibrium.
1984 All Soviet Union Mathematical Olympiad
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again.  Prove that its notation cannot contain more than three different digits.
1984 All Soviet Union Mathematical Olympiad
The $x$ and $y$ figures satisfy a condition:

for every $n\ge1$ the number $xx...x6yy...y4$ ($n$ times $x$ and $n$ times $y$) is an exact square.

Find all possible $x$ and $y$.
1984 All Soviet Union Mathematical Olympiad
The $A,B,C$ and $D$ points (from left to right) belong to the straight line.

Prove that every point $E$, that doesn't belong to the line satisfy: $|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$.
1984 All Soviet Union Mathematical Olympiad
Given a sequence $\{x_n\},x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$.

Prove that the sequence has limit and find it.
1984 All Soviet Union Mathematical Olympiad
The white fields of $1983\times 1984 $1983x1984  are filled with either $+1$ or $-1$.

For every black field, the product of neighbouring numbers is $+1$.

Prove that all the numbers are $+1$.
1984 All Soviet Union Mathematical Olympiad
What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$?  (No differential calculus allowed).
1984 All Soviet Union Mathematical Olympiad
Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn.

Prove that the quadrangle, they make, is outscribed around some circle and find its radius.
1984 All Soviet Union Mathematical Olympiad
Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.
1984 All Soviet Union Mathematical Olympiad
The white fields of $3x3$ chess-board are filled with either $+1$ or $-1$.

For every field, let us calculate the product of neighbouring numbers.

Then let us change all the numbers by the respective products.

Prove that we shall obtain only $+1$'s, having repeated this operation finite number of times.
1983 All Soviet Union Mathematical Olympiad
Every cell of a $4\times 4$ square grid net,  has $1\times 1$ size.

Is it possible to represent this net as a union of the following sets:

a) Eight broken lines of length five each?

b) Five broken lines of length eight each?
1983 All Soviet Union Mathematical Olympiad
Three numbers were written with a chalk on the blackboard.

The following operation was repeated several times: One of the numbers was cleared and the sum of two other numbers, decreased by $1$, was written instead of it. The final set of numbers is $\{17, 1967, 1983\}$.

Is it possible to admit that the initial numbers were

a) $\{2, 2, 2\}$?

b) $\{3, 3, 3\}$?
1983 All Soviet Union Mathematical Olympiad
Three disks touch pairwise from outside in the points $X,Y,Z$. Then the radiuses of the disks were expanded by $2/\sqrt3$ times, and the centres were reserved.  Prove that the $XYZ$ triangle is completely covered by the expanded disks.
1983 All Soviet Union Mathematical Olympiad
Find all the solutions of the system

$y^2 = x^3 - 3x^2 + 2x$

$x^2 = y^3 - 3y^2 + 2y$
1983 All Soviet Union Mathematical Olympiad
Natural number $k$ has $n$ digits in its decimal notation.

It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times.

Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$.

(Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)
1983 All Soviet Union Mathematical Olympiad
The point $D$ is the middle of the $[AB]$ side of the $ABC$ triangle. The points $E$ and F belong to $[AC]$ and $[BC]$ respectively.  Prove that the $DEF$ triangle area does not exceed the sum of the $ADE$ and $BDF$ triangles areas.
1983 All Soviet Union Mathematical Olympiad
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?
1983 All Soviet Union Mathematical Olympiad
Two acute angles $a$ and $b$ satisfy condition $\sin^2a+\sin^2b = \sin(a+b)$

Prove that $a + b = \pi /2$.
1983 All Soviet Union Mathematical Olympiad
The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ lines.
1983 All Soviet Union Mathematical Olympiad
The pupil is training in the square equation solution.

Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$.  Prove that the process cannot be infinite.

What maximal number of the equations he will have to solve?
1983 All Soviet Union Mathematical Olympiad
Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$.

Prove that $m^k$ is divisible by $k^m$.
1983 All Soviet Union Mathematical Olympiad
The Abba tribe language alphabet contains two letters only. Not a word of this language is a beginning of another word. Can this tribe vocabulary contain $3$ four-letter, $10$ five-letter, $30$ six-letter and $5$ seven-letter words?
1983 All Soviet Union Mathematical Olympiad
Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be

a) $10$?

b) $1$?
1983 All Soviet Union Mathematical Olympiad
The points $A_1,B_1,C_1$ belong to $[BC],[CA],[AB]$ sides of the $ABC$ triangle respectively. The $[AA_1], [BB_1], [CC_1]$ segments split the $ABC$ onto $4$ smaller triangles and $3$ quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?
1983 All Soviet Union Mathematical Olympiad
The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.
1983 All Soviet Union Mathematical Olympiad
One side of the rectangle is $1$ cm. It is known that the rectangle can be divided by two orthogonal lines onto four rectangles, and each of the smaller rectangles has the area not less than $1$ square centimetre, and one of them is not less than $2$ square centimetres. What is the least possible length of another side of big rectangle?
1983 All Soviet Union Mathematical Olympiad
Given a point $O$ inside $ABC$ triangle.

Prove that $S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C *  \overrightarrow{OC} = \overrightarrow{0}$,

where $S_A, S_B, S_C$ denote $BOC, COA, AOB$ triangles areas respectively.
1983 All Soviet Union Mathematical Olympiad
Given $(2m+1)$ different integers, each absolute value is not greater than $(2m-1)$.

Prove that it is possible to choose three numbers among them, with their sum equal to zero.
1983 All Soviet Union Mathematical Olympiad
The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the $ABC$ triangle respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the $DEF, DEA, DBF, CEF$, triangles respectively. Prove that $d_0 \ge \frac{\sqrt3}{2}  min\{d_1, d_2, d_3\}$. When the equality takes place?
1983 All Soviet Union Mathematical Olympiad
The $M$ set consists of $k$ non-intersecting segments on the line. It is possible to put an arbitrary segment shorter than $1$ cm on the line in such a way, that his ends will belong to $M$. Prove that the total sum of the segment lengths is not less than $1/k$ cm.
1983 All Soviet Union Mathematical Olympiad
The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation.  Prove that if for some $n$, $u_n \le  (n+8)$, than $x$ is a rational number.
1982 All Soviet Union Mathematical Olympiad
Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$.

The circumference with radius $r_2$ and centre $O_2$ is inscribed in $MO_1K$ angle.

Find the $MO_1KO_2$ quadrangle area.
1982 All Soviet Union Mathematical Olympiad
Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)?
1982 All Soviet Union Mathematical Olympiad
a) Let $m$ and $n$ be natural numbers.

For some nonnegative integers $k_1, k_2, ... , k_n$ the number $2^{k_1}+2^{k_2}+...+2^{k_n}$ is divisible by $(2^m-1)$.

Prove that $n \ge  m$.

b) Can You find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?
1982 All Soviet Union Mathematical Olympiad
A nonnegative real number is written at every cube's vertex. The sum of those numbers equals to $1$. Two players choose in turn faces of the cube, but they cannot choose the face parallel to already chosen one (the first moves twice, the second -- once).  Prove that the first player can provide the number, at the common for three chosen faces vertex, to be not greater than $1/6$.
1982 All Soviet Union Mathematical Olympiad
Once upon a time, three boys visited a library for the first time.

The first decided to visit the library every second day.

The second decided to visit the library every third day.

The third decided to visit the library every fourth day.

The librarian noticed, that the library doesn't work on Wednesdays.

The boys decided to visit library on Thursdays, if they have to do it on Wednesdays, but to restart the day counting in these cases.

They strictly obeyed these rules.

Some Monday later I met them all in that library.

What day of week was when they visited a library for the first time?
1982 All Soviet Union Mathematical Olympiad
The parallelogram $ABCD$ isn't a diamond. The relation of the diagonal lengths $|AC|/|BD|$ equals to $k$. The $[AM)$ ray is symmetric to the $[AD)$ ray with respect to the $(AC)$ line. The $[BM)$ ray is symmetric to the $[BC)$ ray with respect to the $(BD)$ line. ($M$ point is those rays intersection.)  Find the $|AM|/|BM|$ relation.
1982 All Soviet Union Mathematical Olympiad
$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$.  Prove that some two of the marked points are the ends of one diameter.
1982 All Soviet Union Mathematical Olympiad
Given a point $M$ inside a right tetrahedron.

Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$.

(e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le  -1/3$)
1982 All Soviet Union Mathematical Olympiad
Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$.

Sort those numbers in increasing order.
1982 All Soviet Union Mathematical Olympiad
The closed broken line $M$ has odd number of vertices -- $A_1,A_2,..., A_{2n+1}$ in sequence.  Let us denote with $S(M)$ a new closed broken line with vertices $B_1,B_2,...,B_{2n+1}$ -- the middles of the first line links: $B_1$ is the middle of $[A_1A_2], ... , B_{2n+1}$ -- of $[A_{2n+1}A_1]$.  Prove that in a sequence $M_1=S(M), ... , M_k = S(M_{k-1}), ...$ there is a broken line, homothetic to the $M$.
1982 All Soviet Union Mathematical Olympiad
All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in computer's memory.

The program looks through all the sequent pairs (first and second, second and third,...) and exchanges numbers in the pair, if the number on the lower place is greater than another.

Then the program repeats the process, but moves from another end of the array.

The number, that stand initially on the $100$-th place reserved its place.

Find that number.
1982 All Soviet Union Mathematical Olympiad
Cucumber river in the Flower city  has parallel banks with the distance between them $1$ metre.

It has some islands with the total perimeter $8$ metres.

Mr. Know-All claims that it is possible to cross the river in a boat from the arbitrary point,

and the trajectory will not exceed $3$ metres. Is he right?
1982 All Soviet Union Mathematical Olympiad
There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted.

Can you restore them with the help of compass and ruler?
1982 All Soviet Union Mathematical Olympiad
The square table $n\times n$ is filled by integers.

If the fields have common side, the difference of numbers in them doesn't exceed $1$.

Prove that some number is encountered not less than

a) not less than $[n/2]$ times ($[ ...]$ mean the whole part),

b) not less than $n$ times.
1982 All Soviet Union Mathematical Olympiad
Prove that the following inequality is valid for the positive $x$:

$$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$
1982 All Soviet Union Mathematical Olympiad
What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property:

none of the remained numbers equals to the product of two other remained numbers?
1982 All Soviet Union Mathematical Olympiad
Every square on the infinite sheet of cross-lined paper contains some real number.

Prove that some square contains a number that does not exceed at least four of eight neighbouring numbers.
1982 All Soviet Union Mathematical Olympiad
Given a sequence of real numbers $a_1, a_2, ... , a_n$.

Prove that it is possible to choose some of the numbers providing $3$ conditions:

a) not a triple of successive members is chosen,

b) at least one of every triple of successive members is chosen,

c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.
1982 All Soviet Union Mathematical Olympiad
Given the square table $n\times n$ with $(n-1)$ marked fields.

Prove that it is possible to move all the marked fields below the diagonal by moving rows and columns.
1982 All Soviet Union Mathematical Olympiad
Prove that the following inequality holds for all real $a$ and natural $n$: $|a| \cdot |a-1|\cdot |a-2|\cdot ...\cdot |a-n| \ge  \frac{n!F(a)}{2n}$ .

$F(a)$ is the distance from $a$ to the closest integer.
1982 All Soviet Union Mathematical Olympiad
Can You find three polynomials $P,Q,R$ of three variables $x,y,z$, providing the condition:

a)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-2x+1)^3 = 1$,

b)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-x+1)^3 = 1$,

for all $x,y,z$?
1982 All Soviet Union Mathematical Olympiad
The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron.  Prove that $KLMN$ perimeter is less than $4/3$ $ABCD$ perimeter.
1981 All Soviet Union Mathematical Olympiad
Two equal chess-boards ($8\times 8$) have the same centre, but one is rotated by $45$ degrees with respect to another.  Find the total area of black fields intersection, if the fields have unit length sides.
1981 All Soviet Union Mathematical Olympiad
Given points $A,B,M,N$ on the circumference. Two chords $[MA_1]$ and $[MA_2]$ are orthogonal to $(NA)$ and $(NB)$ lines respectively.  Prove that $(AA_1)$ and $(BB_1)$ lines are parallel.
1981 All Soviet Union Mathematical Olympiad
Let us say, that a natural number has the property $P(k)$ if it can be represented as a product of $k$ succeeding natural numbers greater than $1$.

a) Find k such that there exists n which has properties $P(k)$ and $P(k+2)$ simultaneously.

b) Prove that there is no number having properties $P(2)$ and $P(4)$ simultaneously
1981 All Soviet Union Mathematical Olympiad
The rectangular table has four rows.

The first one contains arbitrary natural numbers (some of them may be equal). The consecutive lines are filled according to the rule: we look through the previous row from left to the certain number $n$ and write the number $k$ if $n$ was met $k$ times.

Prove that the second row coincides with the fourth one.
1981 All Soviet Union Mathematical Olympiad
Given real $a$.

Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities

$y \le -x^2 $ and $y \ge  x^2 - 2x + a$
1981 All Soviet Union Mathematical Olympiad
Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal.  Prove that the triangle $BHD$ is also equilateral