Navvye/Olympiad-Problems-CSV huggingface dataset:
1956 Moscow Mathematical Olympiad
*  Let $A, B, C$ be three nodes of a graph paper. Prove that if $\vartriangle ABC$  is an acute one, then there is at least one more node either inside $\vartriangle ABC$ or on one of its sides.
1956 Moscow Mathematical Olympiad
* Prove that if the trihedral angles at each of the vertices of a triangular pyramid are formed by the identical planar angles, then all faces of this pyramid are equal.
1955 Moscow Mathematical Olympiad
a) The numbers $1, 2, . . . , 49$ are arranged in a square table as follows:



Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $36$ numbers, etc., $7$ times. Find the sum of the numbers selected.



b) The numbers $1, 2, . . . , k^2$ are arranged in a square table as follows:



Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $(k - 1)^2$ numbers, etc., $k$ times. Find the sum of the numbers selected.
1955 Moscow Mathematical Olympiad
We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.
1955 Moscow Mathematical Olympiad
Consider an equilateral triangle $\vartriangle ABC$ and points $D$ and $E$ on the sides $AB$ and $BC$csuch that $AE = CD$. Find the locus of intersection points of $AE$ with $CD$ as points $D$ and $E$ vary.
1955 Moscow Mathematical Olympiad
Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?
1955 Moscow Mathematical Olympiad
Find all rectangles that can be cut into $13$ equal squares.
1955 Moscow Mathematical Olympiad
Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$.

Prove that if $n > 3$, then $6$ divides $ab$.
1955 Moscow Mathematical Olympiad
Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.
1955 Moscow Mathematical Olympiad
a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present.

b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.
1955 Moscow Mathematical Olympiad
Which convex domains on a plane can contain an entire straight line?
1955 Moscow Mathematical Olympiad
There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.
1955 Moscow Mathematical Olympiad
Given two distinct nonintersecting circles none of which is inside the other.

Find the locus of the midpoints of all segments whose endpoints lie on the circles.
1955 Moscow Mathematical Olympiad
Find all real solutions of the system $\begin{cases} x^3 + y^3 = 1 \\
x^4 + y^4 = 1 \end{cases}$
1955 Moscow Mathematical Olympiad
Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$.

Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.
1955 Moscow Mathematical Olympiad
Inside $\vartriangle ABC$, there is fixed a point $D$ such that $AC - DA > 1$ and $BC - BD > 1$.

Prove that $EC - ED > 1$ for any point $E$ on segment $AB$.
1955 Moscow Mathematical Olympiad
Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute?
1955 Moscow Mathematical Olympiad
Find integer solutions of the equation $x^3 - 2y^3 - 4z^3 = 0$.
1955 Moscow Mathematical Olympiad
The quadratic expression $ax^2+bx+c$ is the $4$-th power (of an integer) for any integer $x$. Prove that $a = b = 0$.
1955 Moscow Mathematical Olympiad
The centers $O_1, O_2$ and $O_3$ of circles exscribed about $\vartriangle ABC$ are connected.

Prove that $O_1O_2O_3$ is an acute-angled one.
1955 Moscow Mathematical Olympiad
$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?
1955 Moscow Mathematical Olympiad
Cut a rectangle into $18$ rectangles so that no two adjacent ones form a rectangle.
1955 Moscow Mathematical Olympiad
* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$.

Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.
1955 Moscow Mathematical Olympiad
* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.
1955 Moscow Mathematical Olympiad
A point $O$ inside a convex $n$-gon $A_1A_2 . . .A_n$ is connected with segments to its vertices. The sides of this $n$-gon are numbered $1$ to $n$ (distinct sides have distinct numbers). The segments $OA_1,OA_2, . . . ,OA_n$ are similarly numbered.

a) For $n = 9$ find a numeration such that the sum of the sides’ numbers is the same for all triangles $A_1OA_2, A_2OA_3, . . . , A_nOA_1$.

b) Prove that for $n = 10$ there is no such numeration.
1955 Moscow Mathematical Olympiad
Let the inequality $$Aa(Bb + Cc) + Bb(Aa + Cc) + Cc(Aa + Bb) > \frac{ABc^2 + BCa^2 + CAb^2}{2}$$with given $a > 0, b > 0, c > 0$ hold for all $A > 0, B > 0, C > 0$.

Is it possible to construct a triangle with sides of lengths $a, b, c$?
1955 Moscow Mathematical Olympiad
Find all numbers $a$ such that



(1)  all numbers $[a], [2a], . . . , [Na]$ are distinct and



(2) all numbers $\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]$  are distinct.
1955 Moscow Mathematical Olympiad
Given $\vartriangle ABC$, points $C_1, A_1, B_1$ on sides $AB, BC, CA$, respectively, such that $\frac{AC_1}{C_1B}= \frac{BA_1}{A_1C}= \frac{CB_1}{B_1A}=\frac{1}{n}$ and points $C_2, A_2, B_2$ on sides $A_1B_1, B_1C_1, C_1A_1$ of $\vartriangle A_1B_1C_1$, respectively, such that $\frac{A_1C_2}{C_2B_1}= \frac{B_1A_2}{A_2C_1}= \frac{C_1B_2}{B_2A_1}= n$. Prove that $A_2C_2 //AC, C_2B_2 // CB, B_2A_2 // BA$.
1955 Moscow Mathematical Olympiad
On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any difference and any first term has a common point with a segment of the system.
1955 Moscow Mathematical Olympiad
Prove that the equation $x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0$, where $a_1 \ge 0, a_2 \ge  0, . . . , a_n \ge  0$, cannot have two positive roots.
1955 Moscow Mathematical Olympiad
Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.
1955 Moscow Mathematical Olympiad
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.
1955 Moscow Mathematical Olympiad
A right circular cone stands on plane $P$. The radius of the cone’s base is $r$, its height is $h$. A source of light is placed at distance $H$ from the plane, and distance $1$ from the axis of the cone. What is the illuminated part of the disc of radius $R$, that belongs to $P$ and is concentric with the disc forming the base of the cone?
1955 Moscow Mathematical Olympiad
What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?
1955 Moscow Mathematical Olympiad
Consider $\vartriangle A_0B_0C_0$ and points $C_1, A_1, B_1$ on its sides $A_0B_0, B_0C_0, C_0A_0$, points $C_2, A_2,B_2$ on the sides $A_1B_1, B_1C_1, C_1A_1$ of $\vartriangle A_1B_1C_1$, respectively, etc., so that

$\frac{A_0B_1}{B_1C_0}= \frac{B_0C_1}{C_1A_0}= \frac{C_0A_1}{A_1B_0}= k$, $\frac{A_1B_2}{B_2C_1}= \frac{B_1C_2}{C_2A_1}= \frac{C_1A_2}{A_2B_1}= \frac{1}{k^2}$

and, in general,

$\frac{A_nB_{n+1}}{B_{n+1}C_n}= \frac{B_nC_{n+1}}{C_{n+1}A_n}= \frac{C_nA_{n+1}}{A_{n+1}B_n}

=$ $k^{2n}$ for $n$ even , $\frac{1}{k^{2n}}$ for $n$ odd.

Prove that $\vartriangle ABC$ formed by lines $A_0A_1, B_0B_1, C_0C_1$ is contained in $\vartriangle A_nB_nC_n$ for any $n$.
1954 Moscow Mathematical Olympiad
A regular star-shaped hexagon is split into $4$ parts. Construct from them a convex polygon.
1954 Moscow Mathematical Olympiad
Given two convex polygons, $A_1A_2...A_n$ and $B_1B_2...B_n$ such that $A_1A_2 = B_1B_2$, $A_2A_3 =

B_2B_3$,$ ...$, $A_nA_1 = B_nB_1$ and $n - 3$ angles of one polygon are equal to the respective angles of the other. Find whether these polygons are equal.
1954 Moscow Mathematical Olympiad
Find a four-digit number whose division by two given distinct one-digit numbers goes along the following lines:
1954 Moscow Mathematical Olympiad
Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?
1954 Moscow Mathematical Olympiad
Define the maximal value of the ratio of a three-digit number to the sum of its digits.
1954 Moscow Mathematical Olympiad
* Cut out of a $3 \times 3$ square an unfolding of the cube with edge $1$.
1954 Moscow Mathematical Olympiad
From an arbitrary point $O$ inside a convex n-gon, perpendiculars are dropped to the (extensions of the) sides of the $n$-gon. Along each perpendicular a vector is constructed, starting from $O$, directed towards the side onto which the perpendicular is dropped, and of length equal to half the length of the corresponding side. Find the sum of these vectors.
1954 Moscow Mathematical Olympiad
Find all solutions of the system consisting of $3$ equations:

$x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0$ for $n = 1, 2, 3$.
1954 Moscow Mathematical Olympiad
Prove that if $x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0$

and $4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0$,

then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of  $(x - x_0)^2$.
1954 Moscow Mathematical Olympiad
Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.
1954 Moscow Mathematical Olympiad
a) Given $100$ numbers $a_1, ..., a_{100}$ such that $\begin{cases}

a_1 - 3a_2 + 2a_3 \ge 0, \\
a_2 - 3a_3 + 2a_4 \ge 0, \\
a_3 - 3a_4 + 2a_5 \ge 0, \\
 ...  \\
a_{99} - 3a_{100} + 2a_1 \ge 0, \\
a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}$

prove that the numbers are equal.



b) Given numbers $a_1=1, ..., a_{100}$ such that  $\begin{cases}

a_1 - 4a_2 + 3a_3 \ge 0, \\
a_2 - 4a_3 + 3a_4 \ge 0, \\
a_3 - 4a_4 + 3a_5 \ge 0, \\
 ...  \\
a_{99} - 4a_{100} + 3a_1 \ge 0, \\
a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}$

Find $a_2, a_3, ... , a_{100}.$
1954 Moscow Mathematical Olympiad
Consider $\vartriangle ABC$ and a point $S$ inside it. Let $A_1, B_1, C_1$ be the intersection points of $AS, BS, CS$ with $BC, AC, AB$, respectively. Prove that at least in one of the resulting quadrilaterals $AB_1SC_1, C_1SA_1B, A_1SB_1C$ both angles at either $C_1$ and $B_1$, or $C_1$ and $A_1$, or $A_1$ and $B_1$ are not acute.
1954 Moscow Mathematical Olympiad
Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?
1954 Moscow Mathematical Olympiad
Find all real solutions of the equation $x^2 + 2x \sin (xy) + 1 = 0$.
1954 Moscow Mathematical Olympiad
Given a piece of graph paper with a letter assigned to each vertex of every square such that on every segment connecting two vertices that have the same letter and are on the same line of the mesh, there is at least one vertex with another letter. What is the least number of distinct letters needed to plot such a picture?
1954 Moscow Mathematical Olympiad
Solve the system $\begin{cases}

10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\
11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\
15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\
2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\
6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\
3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\
4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}$
1954 Moscow Mathematical Olympiad
How many axes of symmetry can a heptagon have?
1954 Moscow Mathematical Olympiad
a) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot  3 \cdot  5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot  23 \cdot  29 \cdot 31$ (the product of primes $2$ to $31$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers.

Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$.

b) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot  3 \cdot  5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot  23 \cdot  29 \cdot 31 \cdot 37$ (the product of primes $2$ to $37$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers.

Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$.
1954 Moscow Mathematical Olympiad
The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections; $6$ streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points $A$ and $B$ situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through $120^o$ to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.)
1954 Moscow Mathematical Olympiad
A $17 \times 17$ square is cut out of a sheet of graph paper. Each cell of this square has one of thenumbers from $1$ to $70$. Prove that there are $4$ distinct squares whose centers $A, B, C, D$ are the vertices of a parallelogramsuch that $AB // CD$, moreover, the sum of the numbers in the squares with centers $A$ and $C$ is equal to that in the squares with centers $B$ and $D$.
1954 Moscow Mathematical Olympiad
Given four straight lines, $m_1, m_2, m_3, m_4$, intersecting at $O$ and numbered clockwise with $O$ as the center of the clock, we draw a line through an arbitrary point $A_1$ on $m_1$ parallel to $m_4$ until the line meets $m_2$ at $A_2$. We draw a line through $A_2$ parallel to $m_1$ until it meets $m_3$ at $A_3$. We also draw a line through $A_3$ parallel to $m_2$ until it meets $m_4$ at $A_4$. Now, we draw a line through$ A_4$ parallel to $m_3$ until it meets $m_1$ at $B$. Prove that

a) $OB< \frac{OA_1}{2}$ .



b) $OB \le \frac{OA_1}{4}$ .
1954 Moscow Mathematical Olympiad
Rays $l_1$ and $l_2$ pass through a point $O$. Segments $OA_1$ and $OB_1$ on $l_1$, and $OA_2$ and $OB_2$ on $l_2$, are drawn so that $\frac{OA_1}{OA_2} \ne \frac{OB_1}{OB_2}$ . Find the set of all intersection points of lines $A_1A_2$ and $B_1B_2$ as $l_2$ rotates around $O$ while $l_1$ is fixed.
1954 Moscow Mathematical Olympiad
*. Positive numbers $x_1, x_2, ..., x_{100}$ satisfy the system $\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\ 

x_1 + x_2 + ...+ x_{100} < 300 \end{cases}$

Prove that among these numbers there are three whose sum is greater than $100$.
1954 Moscow Mathematical Olympiad
Given a sequence of numbers $a_1, a_2, ..., a_{15}$, one can always construct a new sequence $b_1,b_2, ..., b_{15}$, where $b_i$ is equal to the number of terms in the sequence $\{a_k\}^{15}_{k=1}$  less than $a_i$ ($i = 1, 2,..., 15$).

Is there a sequence $\{a_k\}^{15}_{k=1}$ for which the sequence $\{b_k\}^{15}_{k=1}$ is $1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13$?
1954 Moscow Mathematical Olympiad
Consider five segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can exit either $5$ segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to $1001$? (A free endpoint is an endpoint from which no segment begins.)
1954 Moscow Mathematical Olympiad
How many planes of symmetry can a triangular pyramid have?
1954 Moscow Mathematical Olympiad
The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.
1954 Moscow Mathematical Olympiad
Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.
1953 Moscow Mathematical Olympiad
Prove that the sum of angles at the longer base of a trapezoid is less than the sum of angles at the shorter base.
1953 Moscow Mathematical Olympiad
Find the smallest number of the form $1...1$ in its decimal expression which is divisible by   $\underbrace{\hbox{3...3}}_{\hbox{100}}$,.
1953 Moscow Mathematical Olympiad
Divide a segment in halves using a right triangle.

(With a right triangle one can draw straight lines and erect perpendiculars but cannot drop perpendiculars.)
1953 Moscow Mathematical Olympiad
Prove that $n^2 + 8n + 15$ is not divisible by $n + 4$ for any positive integer $n$.
1953 Moscow Mathematical Olympiad
Three circles are pair-wise tangent to each other.

Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.
1953 Moscow Mathematical Olympiad
Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then

$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$
1953 Moscow Mathematical Olympiad
On the plane find the locus of points whose coordinates satisfy $sin(x + y) = 0$.
1953 Moscow Mathematical Olympiad
Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints.

Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.
1953 Moscow Mathematical Olympiad
Prove that the polynomial $x^{200}  y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.
1953 Moscow Mathematical Olympiad
Let $A$ be a vertex of a regular star-shaped pentagon, the angle at $A$ being less than $180^o$ and the broken line $AA_1BB_1CC_1DD_1EE_1$ being its contour. Lines $AB$ and $DE$ meet at $F$. Prove that polygon $ABB_1CC_1DED_1$ has the same area as the quadrilateral $AD_1EF$.
1953 Moscow Mathematical Olympiad
Given a right circular cone and a point $A$. Find the set of vertices of cones equal to the given one, with axes parallel to that of the given one, and with $A$ inside them.
1953 Moscow Mathematical Olympiad
Prove that $gcd (a + b, lcm(a, b)) = gcd (a, b)$ for any $a, b$.
1953 Moscow Mathematical Olympiad
A quadrilateral is circumscribed around a circle. Its diagonals intersect at the center of the circle. Prove that the quadrilateral is a rhombus.
1953 Moscow Mathematical Olympiad
a) On a plane, $11$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?

b) On a plane, $n$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
1953 Moscow Mathematical Olympiad
Inside a convex $1000$-gon, $500$ points are selected so that no three of the $1500$ points — the ones selected and the vertices of the polygon — lie on the same straight line. This $1000$-gon is then divided into triangles so that all $1500$ points are vertices of the triangles, and so that these triangles have no other vertices.

How many triangles will there be?
1953 Moscow Mathematical Olympiad
a) Solve the system $\begin{cases}

x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 = 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 = 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 = 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 = 4 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 = 5 \end{cases}$



b) Solve the system  $\begin{cases}

x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 +...+ 2x_{100}= 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 +...+ 4x_{100}= 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 +...+ 6x_{100}= 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 +...+ 8x_{100}= 4 \\
... \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 +...+ 199x_{100}= 100 \end{cases}$
1953 Moscow Mathematical Olympiad
Let $a, b, c, d$ be the lengths of consecutive sides of a quadrilateral, and $S$ its area.

Prove that $S \le \frac{ (a + b)(c + d)}{4}$
1953 Moscow Mathematical Olympiad
Somebody wrote $1953$ digits on a circle. The $1953$-digit number obtained by reading these figures clockwise, beginning at a certain point, is divisible by $27$. Prove that if one begins reading the figures at any other place, one gets another $1953$-digit number also divisible by $27$.
1953 Moscow Mathematical Olympiad
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?
1953 Moscow Mathematical Olympiad
Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1,  \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.
1953 Moscow Mathematical Olympiad
Given the equations

(1) $ax^2 + bx + c = 0$

(2)$ -ax^2 + bx + c = 0$

prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively,

then there is a root $x_3$ of the equation $\frac{a}{2}x^2 + bx + c = 0$

such that either $x_1 \le x_3 \le  x_2$ or $x_1 \ge x_3 \ge x_2$.
1953 Moscow Mathematical Olympiad
Given a $101\times 200$ sheet of graph paper, we start moving from a corner square in the direction of the square’s diagonal (not the sheet’s diagonal) to the border of the sheet, then change direction obeying the laws of light’s reflection. Will we ever reach a corner square?
1953 Moscow Mathematical Olympiad
Divide a cube into three equal pyramids.
1953 Moscow Mathematical Olympiad
Find roots of the equation

$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$.
1953 Moscow Mathematical Olympiad
Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2  < 10^{-9}$.
1953 Moscow Mathematical Olympiad
A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.