Navvye/Olympiad-Problems-CSV huggingface dataset:
1989 All Soviet Union Mathematical Olympiad
$N$ is the set of positive integers. Does there exist a function $f: N \to N$ such that $f(n+1) = f( f(n) ) + f( f(n+2) )$ for all $n$?
1989 All Soviet Union Mathematical Olympiad
A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length $< 1$. Show that the area of the polygon is $< \pi /4$.
1988 All Soviet Union Mathematical Olympiad
A book contains $30$ stories. Each story has a different number of pages under $31$. The first story starts on page $1$ and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
1988 All Soviet Union Mathematical Olympiad
$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.
1988 All Soviet Union Mathematical Olympiad
Show that there are infinitely many triples of distinct positive integers $a, b, c$ such that each divides the product of the other two and $a + b = c + 1$.
1988 All Soviet Union Mathematical Olympiad
Given a sequence of $19$ positive integers not exceeding $88$ and another sequence of $88$ positive integers not exceeding $19$. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum.
1988 All Soviet Union Mathematical Olympiad
The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.
1988 All Soviet Union Mathematical Olympiad
The numbers $1$ and $2$ are written on an empty blackboard.

Whenever the numbers $m$ and $n$ appear on the blackboard the number $m + n + mn$ may be written.

Can we obtain :

(1) $13121$,

(2) $12131$?
1988 All Soviet Union Mathematical Olympiad
If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.
1988 All Soviet Union Mathematical Olympiad
There are $21$ towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?
1988 All Soviet Union Mathematical Olympiad
Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.
1988 All Soviet Union Mathematical Olympiad
$A, B, C$ are the angles of a triangle. Show that $2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C$
1988 All Soviet Union Mathematical Olympiad
Form $10A$ has $29$ students who are listed in order on its duty roster. Form $10B$ has $32$ students who are listed in order on its duty roster. Every day two students are on duty, one from form $10A$ and one from form $10B$. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from $10A$ and one from $10B$) shared duty exactly once?
1988 All Soviet Union Mathematical Olympiad
In the triangle $ABC$, the angle $C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.
1988 All Soviet Union Mathematical Olympiad
Show that there are infinitely many odd composite numbers in the sequence $1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...$ .
1988 All Soviet Union Mathematical Olympiad
$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$
1988 All Soviet Union Mathematical Olympiad
What is the minimal value of $\frac{b}{c + d} + \frac{c}{a + b}$ for positive real numbers $b$ and $c$ and non-negative real numbers $a$ and $d$ such that $b + c\ge  a + d$?
1988 All Soviet Union Mathematical Olympiad
$n^2$ real numbers are written in a square $n \times n$ table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are $a_{ij}$ and we select row $i$, column $h$ and column $k$, then column h becomes $a_{1h} + a_{i1}, a_{2h} + a_{i2}, ... , a_{nh} + a_{in}$, column $k$ becomes $a_{1k} - a_{i1}, a_{2k} - a_{i2}, ... , a_{nk} - a_{in}$, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.
1988 All Soviet Union Mathematical Olympiad
In the acute-angled triangle $ABC$, the altitudes $BD$ and $CE$ are drawn. Let $F$ and $G$ be the points of the line $ED$ such that $BF$ and $CG$ are perpendicular to $ED$. Prove that $EF = DG$.
1988 All Soviet Union Mathematical Olympiad
Find the minimum value of $\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}$ for positive reals $x, y, z$ with $x^2 + y^2 + z^2 = 1$.
1988 All Soviet Union Mathematical Olympiad
A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?
1988 All Soviet Union Mathematical Olympiad
Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.
1988 All Soviet Union Mathematical Olympiad
A polygonal line with a finite number of segments has all its vertices on a parabola. Any two adjacent segments make equal angles with the tangent to the parabola at their point of intersection. One end of the polygonal line is also on the axis of the parabola. Show that the other vertices of the polygonal line are all on the same side of the axis.
1988 All Soviet Union Mathematical Olympiad
What is the smallest $n$ for which there is a solution to $\sin x_1 + \sin x_2 + ... + \sin x_n = 0, \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100$ ?
1988 All Soviet Union Mathematical Olympiad
The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.
1988 All Soviet Union Mathematical Olympiad
Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.
1987 All Soviet Union Mathematical Olympiad
Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws).

Let $xi$ be the number of $i$-th player victories, $yi$ -- losses.

Prove that $x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$
1987 All Soviet Union Mathematical Olympiad
It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs.
1987 All Soviet Union Mathematical Olympiad
Given right heptagon ($7$-angle) $A_1...A_7$. Prove that $\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$.
1987 All Soviet Union Mathematical Olympiad
The "Sea battle" game.

a)  You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely?

b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.
1987 All Soviet Union Mathematical Olympiad
Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.
1987 All Soviet Union Mathematical Olympiad
An L is an arrangement of $3$ adjacent unit squares formed by deleting one unit square from a $2 \times 2$ square.

a) How many Ls can be placed on an $8 \times  8$ board (with no interior points overlapping)?

b) Show that if any one square is deleted from a $1987 \times 1987$ board, then the remaining squares can be covered with Ls (with no interior points overlapping).
1987 All Soviet Union Mathematical Olympiad
Three lines are drawn parallel to the sides of the triangles in the opposite to the vertex, not belonging to the side, part of the plane. The distance from each side to the corresponding line equals the length of the side.  Prove that six intersection points of those lines with the continuations of the sides are situated on one circumference.
1987 All Soviet Union Mathematical Olympiad
Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point.  Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.
1987 All Soviet Union Mathematical Olympiad
Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.
1987 All Soviet Union Mathematical Olympiad
Given a convex pentagon. The angles $ABC$ and $ADE$ are equal. The angles $AEC$ and $ADB$ are equal too.  Prove that the angles $BAC$ and $DAE$ are equal also.
1987 All Soviet Union Mathematical Olympiad
Prove such $a$, that all the numbers $\cos a, \cos 2a, \cos 4a, ... , \cos (2^na)$ are negative.
1987 All Soviet Union Mathematical Olympiad
The positive numbers $a,b,c,A,B,C$ satisfy a condition $a + A = b + B = c + C = k$.

Prove that $aB + bC + cA \le  k^2$.
1987 All Soviet Union Mathematical Olympiad
Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$.  Prove that the sum of all the numbers is not greater than $1987$.
1987 All Soviet Union Mathematical Olympiad
The $B$ vertex of the $ABC$ angle lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. The $K$ point belongs to the intersection of the $[BA)$ beam and the circumference. The $KP$ chord is orthogonal to the angle $ABC$ bisector. The $(KP)$ line intersects the $BC$ beam in the M point.  Prove that the $[PM]$ segment is twice as long as the distance from the circle centre to the angle $ABC$ bisector.
1987 All Soviet Union Mathematical Olympiad
Two players are writting in turn natural numbers not exceeding $p$.

The rules forbid to write the divisors of the numbers already having been written.

Those who cannot make his move looses.

a) Who, and how, can win if $p=10$?

b) Who wins if $p=1000$?
1987 All Soviet Union Mathematical Olympiad
Every evening uncle Chernomor (see Pushkin's tales) appoints either $9$ or $10$ of his 33 "knights"  in the "night guard".  When it can happen, for the first time, that every knight has been on duty the same number of times?
1987 All Soviet Union Mathematical Olympiad
Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.
1987 All Soviet Union Mathematical Olympiad
The convex $n$-angle ($n\ge 5$) is cut along all its diagonals.

Prove that there are at least a pair of parts with the different areas.
1987 All Soviet Union Mathematical Olympiad
The $T_0$ set consists of all the numbers, representable as $(2k)!, k = 0, 1, 2, ... , n, ...$.

The $T_m$ set is obtained from $T_{m-1}$ by adding all the finite sums of different numbers, that belong to $T_{m-1}$.

Prove that there is a natural number, that doesn't belong to $T_{1987}$.
1987 All Soviet Union Mathematical Olympiad
The plot of the $y=f(x)$ function, being rotated by the (right) angle around the $(0,0)$ point is not changed.

a) Prove that the equation $f(x)=x$ has the unique solution.

b) Give an example of such a function.
1987 All Soviet Union Mathematical Olympiad
All the faces of a convex polyhedron are the triangles.

Prove that it is possible to paint all its edges in red and blue colour in such a way, that it is possible to move from the arbitrary vertex to every vertex along the blue edges only and along the red edges only.
1987 All Soviet Union Mathematical Olympiad
Prove that for every natural $n$ the following inequality is held: $(2n + 1)^n \ge (2n)^n + (2n - 1)^n$.
1986 All Soviet Union Mathematical Olympiad
The square polynomial $x^2+ax+b+1$ has natural roots.

Prove that $(a^2+b^2)$ is a composite number.
1986 All Soviet Union Mathematical Olympiad
Two equal squares, one with red sides, another with blue ones, give an octagon in intersection.

Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.
1986 All Soviet Union Mathematical Olympiad
The point $M$ belongs to the $[AC]$ side of the acute-angle triangle $ABC$. Two circles are outscribed around $ABM$ and $BCM$ triangles. What $M$ position corresponds to the minimal area of those circles intersection?
1986 All Soviet Union Mathematical Olympiad
Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres.  Is it possible for

a) $n=6$

b) $n=1986$ ?
1986 All Soviet Union Mathematical Olympiad
Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.
1986 All Soviet Union Mathematical Olympiad
Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.
1986 All Soviet Union Mathematical Olympiad
Two circumferences, with the distance $d$ between centres, intersect in $P$ and $Q$ points. Two lines are drawn through the $A$ point on the first circumference ($Q\ne A\ne P$) and $P$ and $Q$ points. They intersect the second circumference in the $B$ and $C$ points.

a) Prove that the radius of the circle, circumscribed around the $ABC$ triangle, equals $d$.

b) Describe the set of the new circle's centres, if the $A$ point moves along all the first circumference.
1986 All Soviet Union Mathematical Olympiad
Given right hexagon. Each side is divided onto $1000$ equal segments. All the points of division are connected with the segments, parallel to sides. Let us paint in turn the triples of unpainted nodes of obtained net, if they are vertices of the unilateral triangle, doesn't matter of what size an orientation. Suppose, we have managed to paint all the vertices except one.  Prove that the unpainted node is not a hexagon vertex.
1986 All Soviet Union Mathematical Olympiad
Find all the natural numbers equal to the square of its divisors number.
1986 All Soviet Union Mathematical Olympiad
Prove that the following inequality holds for all positive $\{a_i\}$:

$$\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)$$
1986 All Soviet Union Mathematical Olympiad
A line is drawn through the $A$ vertex of $ABC$ triangle with $|AB|\ne|AC|$.

Prove that the line can not contain more than one point $M$ such, that $M$ is not a triangle vertex, and the angles $ABM$ and $ACM$ are equal.  What lines do not contain such a point $M$ at all?
1986 All Soviet Union Mathematical Olympiad
A cube with edge of length $n$ ($n\ge 3$) consists of $n^3$ unit cubes.

Prove that it is possible to write different $n^3$ integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge.
1986 All Soviet Union Mathematical Olympiad
The decimal notation of three natural numbers consists of equal digits:  $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$.

For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$
1986 All Soviet Union Mathematical Olympiad
Given two points inside a convex dodecagon (twelve sides) situated 1$0$ cm far from each other.

Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.
1986 All Soviet Union Mathematical Olympiad
Given $30$ equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups.  Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?
1986 All Soviet Union Mathematical Olympiad
Find the relation of the black part length and the white part length for the main diagonal of the

a) $100\times 99$ chess-board;

b) $101\times 99$ chess-board.
1986 All Soviet Union Mathematical Olympiad
Given right $n$-angle $A_1A_2...A_n$. Prove that if

a) $n$ is even number, than for the arbitrary point $M$ in the plane, it is possible to choose signs in an expression

$\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}$to make it equal to the zero vector .

b) $n$ is odd, than the abovementioned expression equals to the zero vector for the finite set of $M$ points only.
1986 All Soviet Union Mathematical Olympiad
All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules:

At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column.

a) What is the minimal

b) What is the maximal possible number of $+1$ in the obtained table?
1986 All Soviet Union Mathematical Olympiad
Prove that for every natural $n$ the following inequality is valid $|\sin 1| + |\sin 2| + |\sin (3n-1)| + |\sin 3n| > \frac{8n}{5}$
1986 All Soviet Union Mathematical Olympiad
Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.
1986 All Soviet Union Mathematical Olympiad
A triangle and a square are outscribed around the unit circle.

Prove that the intersection area is more than $3.4$.

Is it possible to assert that it is more than $3.5$?
1986 All Soviet Union Mathematical Olympiad
Let us call a polynomial admissible if all it's coefficients are $0, 1, 2$ or $3$.

For given $n$ find the number of all the admissible  polynomials $P$ such, that $P(2) = n$.
1986 All Soviet Union Mathematical Olympiad
Consider all the tetrahedrons $AXBY$, outscribed around the sphere. Let $A$ and $B$ points be fixed.

Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on $X$ and $Y$ points.
1985 All Soviet Union Mathematical Olympiad
Two perpendiculars are drawn from the middles of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.
1985 All Soviet Union Mathematical Olympiad
Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?
1985 All Soviet Union Mathematical Olympiad
What maximal number of the dames  can be put on the chess-board $8\times 8$ so, that every dame can be taken by at least one other dame?
1985 All Soviet Union Mathematical Olympiad
You should paint all the sides and diagonals of the right $n$-angle so, that every pair of segments, having the common point, would be painted with different colours. How many colours will you require?
1985 All Soviet Union Mathematical Olympiad
Given a straight line $\ell$ and the point $O$ out of the line.

Prove that it is possible to move an arbitrary point $A$ in the same plane to the $O$ point, using only rotations around $O$ and symmetry with respect to the $\ell$.
1985 All Soviet Union Mathematical Olympiad
The senior coefficient a in the square polynomial $P(x) = ax^2 + bx + c$ is more than $100$.

What is the maximal number of integer values of $x$, such that $|P(x)|<50$.
1985 All Soviet Union Mathematical Olympiad
In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$?

j

h  i

e  f  g

a  b  c  d
1985 All Soviet Union Mathematical Olympiad
Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that

a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:

$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... +  \frac{a_k}{a_{k-1} }< k - 1$

b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:

$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... +  \frac{a_k}{a_{k-1} }< k - 1985$
1985 All Soviet Union Mathematical Olympiad
Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le  0$.
1985 All Soviet Union Mathematical Olympiad
The convex pentagon $ABCDE$ was drawn in the plane.

$A_1$ was symmetric to $A$ with respect to $B$.

$B_1$ was symmetric to $B$ with respect to $C$.

$C_1$ was symmetric to $C$ with respect to $D$.

$D_1$ was symmetric to $D$ with respect to $E$.

$E_1$ was symmetric to $E$ with respect to $A$.

How is it possible to restore the initial pentagon with the compasses and ruler, knowing $A_1,B_1,C_1,D_1,E_1$ points?
1985 All Soviet Union Mathematical Olympiad
The sequence $a_1, a_2, ... , a_k, ...$ is built according to the rules: $a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$.

Prove that it is non-periodical sequence.
1985 All Soviet Union Mathematical Olympiad
$n$ straight lines are drawn in a plane.

They divide the plane onto several parts. Some of the parts are painted.

Not a pair of painted parts has non-zero length common bound.

Prove that the number of painted parts is not more than $\frac{n^2 + n}{3}$.
1985 All Soviet Union Mathematical Olympiad
Given a cube, a cubic box, that exactly suits for the cube, and six colours.

First man paints each side of the cube with its (side's) unique colour. Another man does the same with the box.

Prove that the third man can put the cube in the box in such a way, that every cube side will touch the box side of different colour.
1985 All Soviet Union Mathematical Olympiad
The $[A_0A_5]$ diameter divides a circumference with the $O$ centre onto two hemicircumferences. One of them is divided onto five equal arcs $A_0A_1, A_1A_2, A_2A_3, A_3A_4, A_4A_5$. The $(A_1A_4)$ line crosses $(OA_2)$ and $(OA_3)$ lines in $M$ and $N$ points.

Prove that $(|A_2A_3| + |MN|)$ equals to the circumference radius.
1985 All Soviet Union Mathematical Olympiad
If there are four numbers $(a,b,c,d)$ in four registers of the calculating machine, they turn into $(a-b,b-c,c-d,d-a)$ numbers whenever you press the button. Prove that if not all the initial numbers are equal, machine will obtain at least one number more than $1985$ after some number of the operations.