Navvye/Olympiad-Problems-CSV huggingface dataset:
1952 Moscow Mathematical Olympiad
The circle is inscribed in $\vartriangle ABC$. Let $L, M, N$ be the tangent points of the circle with sides $AB, AC, BC$, respectively. Prove that $\angle MLN$ is always an acute angle.
1952 Moscow Mathematical Olympiad
Prove the identity:

a) $(ax + by + cz)^2 + (bx + cy + az)^2 + (cx + ay + bz)^2 =(cx + by + az)^2 + (bx + ay + cz)^2 + (ax + cy + bz)^2$

b) $(ax + by + cz + du)^2+(bx + cy + dz + au)^2 +(cx + dy + az + bu)^2 + (dx + ay + bz + cu)^2 =$

$(dx + cy + bz + au)^2+(cx + by + az + du)^2 +(bx + ay + dz + cu)^2 + (ax + dy + cz + bu)^2$.
1952 Moscow Mathematical Olympiad
Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.
1952 Moscow Mathematical Olympiad
Two men, $A$ and $B$, set out from town $M$ to town $N$, which is $15$ km away. Their walking speed is $6$ km/hr. They also have a bicycle which they can ride at $15$ km/hr. Both $A$ and $B$ start simultaneously, $A$ walking and $B$ riding a bicycle until $B$ meets a pedestrian girl, $C$, going from $N$ to $M$. Then $B$ lends his bicycle to $C$ and proceeds on foot; $C$ rides the bicycle until she meets $A$ and gives $A$ the bicycle which $A$ rides until he reaches $N$. The speed of $C$ is the same as that of $A$ and $B$. The time spent by $A$ and $B$ on their trip is measured from the moment they started from $M$ until the arrival of the last of them at $N$.

a) When should the girl $C$ leave $N$ for $A$ and $B$ to arrive simultaneously in $N$?

b) When should $C$ leave $N$ to minimize this time?
1952 Moscow Mathematical Olympiad
Prove that if the orthocenter divides all heights of a triangle in the same proportion, the triangle is equilateral.
1952 Moscow Mathematical Olympiad
Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.
1952 Moscow Mathematical Olympiad
Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.
1952 Moscow Mathematical Olympiad
$\vartriangle ABC$ is divided by a straight line $BD$ into two triangles. Prove that the sum of the radii of circles inscribed in triangles $ABD$ and $DBC$ is greater than the radius of the circle inscribed in $\vartriangle ABC$.
1952 Moscow Mathematical Olympiad
A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$
1952 Moscow Mathematical Olympiad
Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.
1952 Moscow Mathematical Olympiad
How $arc \sin(\cos(arc \sin x))$ and $arc \cos(\sin(arc \cos x))$ are related with each other?
1952 Moscow Mathematical Olympiad
Prove that $(1 - x)^n + (1 + x)^n < 2^n$ for an integer $n \ge 2$ and $|x| < 1$.
1952 Moscow Mathematical Olympiad
A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.
1952 Moscow Mathematical Olympiad
Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.
1952 Moscow Mathematical Olympiad
a) Solve the system of equations $\begin{cases}

1 - x_1x_2 = 0 \\
1 - x_2x_3 = 0 \\
...\\
1 - x_{14}x_{15} = 0 \\
1 - x_{15}x_1 = 0 \end{cases}$



b) Solve the system of equations $\begin{cases}

1 - x_1x_2 = 0 \\
1 - x_2x_3 = 0 \\
...\\
1 - x_{n-1}x_{n} = 0 \\
1 - x_{n}x_1 = 0 \end{cases}$



How does the solution vary for distinct values of $n$?
1952 Moscow Mathematical Olympiad
In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.
1952 Moscow Mathematical Olympiad
You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.
1952 Moscow Mathematical Olympiad
a) Prove that if the square of a number begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$, then the number itself begins with  $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$,.

b) Calculate $\sqrt{0.9...9}$ ($60$ nines) to $60$ decimal places
1952 Moscow Mathematical Olympiad
From a point $C$, tangents $CA$ and $CB$ are drawn to a circle $O$. From an arbitrary point $N$ on the circle, perpendiculars $ND, NE, NF$ are dropped to $AB, CA$ and $CB$, respectively. Prove that the length of $ND$ is the mean proportional of the lengths of $NE$ and $NF$.
1952 Moscow Mathematical Olympiad
Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.
1952 Moscow Mathematical Olympiad
$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.
1952 Moscow Mathematical Olympiad
How to arrange three right circular cylinders of diameter $a/2$ and height $a$ into an empty cube with side $a$ so that the cylinders could not change position inside the cube? Each cylinder can, however, rotate about its axis of symmetry.
1952 Moscow Mathematical Olympiad
In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$  and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .
1952 Moscow Mathematical Olympiad
$200$ soldiers occupy in a rectangle (military call it a square and educated military a carree): $20$ men (per row) times $10$ men (per column).

In each row, we consider the tallest man (if some are of equal height, choose any of them) and of the $10$ men considered we select the shortest (if some are of equal height, choose any of them). Call him $A$.

Next the soldiers assume their initial positions and in each column the shortest soldier is selected, of these $20$, the tallest is chosen. Call him $B$.

Two colonels bet on which of the two soldiers chosen by these two distinct procedures is taller: $A$ or $B$.

Which colonel wins the bet?
1952 Moscow Mathematical Olympiad
Prove that for arbitrary fixed $a_1, a_2,.. , a_{31}$ the sum $\cos 32x + a_{31} \cos 31x +... + a_2 cos 2x + a_1 \cos x$ can take both positive and negative values as $x$ varies.
1952 Moscow Mathematical Olympiad
Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.
1951 Moscow Mathematical Olympiad
Prove that $x^{12} - x^9 + x^4 - x + 1 > 0$ for all $x$.
1951 Moscow Mathematical Olympiad
Let $ABCD$ and $A'B'C'D'$ be two convex quadrilaterals whose corresponding sides are equal, i.e., $AB = A'B', BC = B'C'$, etc. Prove that if $\angle A > \angle A'$, then $\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'$.
1951 Moscow Mathematical Olympiad
Which number is greater:

$\frac{2.00 000 000 004}{(1.00 000 000 004)^2 + 2.00 000 000 004}$ or $\frac{2.00 000 000 002}{(1.00 000 000 002)^2 + 2.00 000 000 002}$ ?
1951 Moscow Mathematical Olympiad
Given an isosceles trapezoid $ABCD$ and a point $P$.

Prove that a quadrilateral can be constructed from segments $PA, PB, PC, PD$.
1951 Moscow Mathematical Olympiad
a) Given a chain of $60$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $59$ g, $60$ g? A broken link also weighs $1$ g.

b) Given a chain of $150$ links each weighing $1$ g. Find the smallest number of links that need to be broken if we want to be able to get from the obtained parts all weights $1$ g, $2$ g, . . . , $149$ g, $150$ g? A broken link also weighs $1$ g.
1951 Moscow Mathematical Olympiad
Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.
1951 Moscow Mathematical Olympiad
One side of a convex polygon is equal to $a$, the sum of exterior angles at the vertices not adjacent to this side are equal to $120^o$. Among such polygons, find the polygon of the largest area.
1951 Moscow Mathematical Olympiad
We have two concentric circles. A polygon is circumscribed around the smaller circle and is contained entirely inside the greater circle. Perpendiculars from the common center of the circles to the sides of the polygon are extended till they intersect the greater circle. Each of the points obtained is connected with the endpoints of the corresponding side of the polygon . When is the resulting star-shaped polygon the unfolding of a pyramid?
1951 Moscow Mathematical Olympiad
Given three equidistant parallel lines. Express by points of the corresponding lines the values of the resistance, voltage and current in a conductor so as to obtain the voltage $V = I \cdot  R$ by connecting with a ruler the points denoting the resistance $R$ and the current $I$. (Each point of each scale denotes only one number).



similar wordingThree parallel straight lines are given at equal distances from each other. How to depict by points of the corresponding straight lines the values of resistance, voltage and the current in the conductor, so that, applying a ruler to to points depicting the values of resistance R and values of current I, obtain on the voltage scale a point depicting the value of voltage V = I R (point each scale represents one and only one number).
1951 Moscow Mathematical Olympiad
Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.
1951 Moscow Mathematical Olympiad
*  On a plane, given points $A, B, C$ and angles $\angle D, \angle E, \angle F$ each less than $180^o$ and the sum equal to $360^o$, construct with the help of ruler and protractor a point $O$ such that $\angle AOB = \angle D, \angle BOC = \angle E$ and $\angle COA = \angle F.$
1951 Moscow Mathematical Olympiad
Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.
1951 Moscow Mathematical Olympiad
What figure can the central projection of a triangle be?

(The center of the projection does not lie on the plane of the triangle.)
1951 Moscow Mathematical Olympiad
To prepare for an Olympiad $20$ students went to a coach. The coach gave them $20$ problems and it turned out that

(a) each of the students solved two problems and

(b) each problem was solved by twostudents.

Prove that it is possible to organize the coaching so that each student would discuss one of the problems that (s)he had solved, and so that all problems would be discussed.
1951 Moscow Mathematical Olympiad
Dividing $x^{1951} - 1$ by $P(x) = x^4 + x^3 + 2x^2 + x + 1$ one gets a quotient and a remainder.

Find the coefficient of $x^{14}$ in the quotient.
1951 Moscow Mathematical Olympiad
A sphere is inscribed in an $n$-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then

(1) all tangent points of these faces to the sphere would coincide with one point, $H$, and

(2) the vertices of the faces would lie on a circle centered at $H$.
1951 Moscow Mathematical Olympiad
*  Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.
1951 Moscow Mathematical Olympiad
Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.
1951 Moscow Mathematical Olympiad
Consider a curve with the following property:

inside the curve one can move an inscribed equilateral triangle so that each vertex of the triangle moves along the curve and draws the whole curve.

Clearly, every circle possesses the property.

Find a closed planar curve without self-intersections, that has the property but is not a circle.
1951 Moscow Mathematical Olympiad
*  A bus route has $14$ stops (counting the first and the last). A bus cannot carry more than $25$ passengers. We assume that a passenger takes a bus from $A$ to $B$ if (s)he enters the bus at $A$ and gets off at $B$.

Prove that for any bus route:

a) there are $8$ distinct stops $A_1, B_1, A_2, B_2, A_3, B_3, A_4, B_4$ such that no passenger rides from $A_k$ to $B_k$ for all $k = 1, 2, 3, 4$ (#)

b) there might not exist $10$ distinct stops $A_1, B_1, . . . , A_5, B_5$ with property (#).
1950 Moscow Mathematical Olympiad
On a chess board, the boundaries of the squares are assumed to be black. Draw a circle of the greatest possible radius lying entirely on the black squares.
1950 Moscow Mathematical Olympiad
a) Given $555$ weights: of $1$ g, $2$ g, $3$ g, . . . , $555$ g, divide them into three piles of equal mass.

b) Arrange $81$ weights of $1^2, 2^2, . . . , 81^2$ (all in grams) into three piles of equal mass.
1950 Moscow Mathematical Olympiad
a) We are given $n$ circles $O_1, O_2, . . . , O_n$, passing through one point $O$. Let $A_1, . . . , A_n$ denote the second intersection points of $O_1$ with $O_2, O_2$ with $O_3$, etc., $O_n$ with $O_1$, respectively. We choose an arbitrary point $B_1$ on $O_1$ and draw a line segment through $A_1$ and $B_1$ to the second intersection with $O_2$ at $B_2$, then draw a line segment through $A_2$ and $B_2$ to the second intersection with $O_3$ at $B_3$, etc., until we get a point $B_n$ on $O_n$. We draw the line segment through $B_n$ and $A_n$ to the second intersection with $O_1$ at $B_{n+1}$. If $B_k$ and $A_k$ coincide for some $k$, we draw the tangent to $O_k$ through $A_k$ until this tangent intersects $O_{k+1}$ at $B_{k+1}$. Prove that $B_{n+1}$ coincides with $B_1$.

b) for $n=3$ the same problem
1950 Moscow Mathematical Olympiad
Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles.

Prove that $Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$ .
1950 Moscow Mathematical Olympiad
In a country, one can get from some point $A$ to any other point either by walking, or by calling a cab, waiting for it, and then being driven. Every citizen always chooses the method of transportation that requires the least time. It turns out that the distances and the traveling times are as follows: $1$ km takes $10$ min, $2$ km takes $15$ min, $3$ km takes $17.5 $ min. We assume that the speeds of the pedestrian and the cab, and the time spent waiting for cabs, are all constants. How long does it take to reach a point which is $6$ km from $A$?
1950 Moscow Mathematical Olympiad
Let $A$ be an arbitrary angle,let $B$ and $C$ be acute angles.

Is there an angle $x$ such that $\sin x =\frac{\sin B \cdot \sin C}{1 - \cos B \cdot \cos C  \cdot \cos A}$ ?
1950 Moscow Mathematical Olympiad
Two triangular pyramids have common base. One pyramid contains the other. Can the sum of the lengths of the edges of the inner pyramid be longer than that of the outer one?
1950 Moscow Mathematical Olympiad
Solve the equation $\sqrt {x + 3 - 4 \sqrt{x -1}} +\sqrt{x + 8 - 6 \sqrt{x - 1}}= 1$.
1950 Moscow Mathematical Olympiad
a) In a convex $13$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?

b) In a convex $1950$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?
1950 Moscow Mathematical Olympiad
Prove that $\frac{1}{2}  \frac{3}{4} \frac{5}{6} \frac{7}{8}   ... \frac{99}{100 } <\frac{1}{10}$.
1950 Moscow Mathematical Olympiad
A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.
1950 Moscow Mathematical Olympiad
* On a circle, $20$ points are chosen. Ten non-intersecting chords without mutual endpoints connect some of the points chosen. How many distinct such arrangements are there?
1950 Moscow Mathematical Olympiad
The numbers $1, 2, 3, . . . , 101$ are written in a row in some order. Prove that it is always possible to erase $90 $ of the numbers so that the remaining $11$ numbers remain arranged in either increasing or decreasing order.
1950 Moscow Mathematical Olympiad
A spatial quadrilateral is circumscribed around a sphere. Prove that all the tangent points lie in one plane.
1950 Moscow Mathematical Olympiad
Is it possible to draw $10$ bus routes with stops such that for any $8$ routes there is a stop that does not belong to any of the routes, but any $9$ routes pass through all the stops?
1949 Moscow Mathematical Olympiad
Prove that $27 195^8 - 10 887^8 + 10 152^8$ is divisible by $26 460$.
1949 Moscow Mathematical Olympiad
a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.

b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.
1949 Moscow Mathematical Olympiad
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.

b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.
1949 Moscow Mathematical Olympiad
Consider a closed broken line of perimeter $1$ on a plane. Prove that a disc of radius $\frac14$ can cover this line.
1949 Moscow Mathematical Olympiad
Prove that for any triangle the circumscribed circle divides the line segment connecting the center of its inscribed circle with the center of one of the exscribed circles in halves.
1949 Moscow Mathematical Olympiad
Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ ,  $\left(0 < a < \frac14 \right)$ .
1949 Moscow Mathematical Olympiad
Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.
1949 Moscow Mathematical Olympiad
Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.
1949 Moscow Mathematical Olympiad
There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?
1949 Moscow Mathematical Olympiad
Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$.  Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.
1949 Moscow Mathematical Olympiad
Consider $13$ weights of integer mass (in grams). It is known that any $6$ of them may be placed onto two pans of a balance achieving equilibrium. Prove that all the weights are of equal mass.
1949 Moscow Mathematical Olympiad
The midpoints of alternative sides of a hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide.
1949 Moscow Mathematical Olympiad
Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.
1949 Moscow Mathematical Olympiad
Construct a convex polyhedron of equal “bricks” shown in Figure.
1949 Moscow Mathematical Olympiad
What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?
1949 Moscow Mathematical Olympiad
*  Prove that a number of the form $2^n$ for a positive integer $n$ may begin with any given combination of digits.
1949 Moscow Mathematical Olympiad
Two squares are said to be juxtaposed  if their intersection is a point or a segment. Prove that it is impossible to juxtapose  to a square more than eight non-overlapping squares of the same size.
1948 Moscow Mathematical Olympiad
The sum of the reciprocals of three positive integers is equal to $1$. What are all the possible such triples?
1948 Moscow Mathematical Olympiad
Find all possible arrangements of $4$ points on a plane, so that the distance between each pair of points is equal to either $a$ or $b$.

For what ratios of $a : b$ are such arrangements possible?
1948 Moscow Mathematical Olympiad
On a plane, $n$ straight lines are drawn. Two domains are called adjacent  if they border by a line segment. Prove that the domains into which the plane is divided by these lines can be painted two colors so that no two adjacent  domains are of the same color.
1948 Moscow Mathematical Olympiad
Prove that if $\frac{2^n- 2}{n} $ is an integer, then so is $\frac{2^{2^n-1}-2}{2^n - 1}$ .
1948 Moscow Mathematical Olympiad
Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$
1948 Moscow Mathematical Olympiad
Consider two triangular pyramids $ABCD$ and $A'BCD$, with a common base $BCD$, and such that $A'$ is inside $ABCD$. Prove that the sum of planar angles at vertex $A'$ of pyramid $A'BCD$ is greater than the sum of planar angles at vertex $A$ of pyramid $ABCD$.
1948 Moscow Mathematical Olympiad
Consider a circle and a point $A$ outside it. We start moving from $A$ along a closed broken line consisting of segments of tangents to the circle (the segment itself should not necessarily be tangent to the circle) and terminate back at $A$. (On the links of the broken line are solid.) We label parts of the segments with a plus sign if we approach the circle and with a minus sign otherwise. Prove that the sum of the lengths of the segments of our path, with the signs given, is zero.
1948 Moscow Mathematical Olympiad
a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$).

b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).
1948 Moscow Mathematical Olympiad
Let $R$ and $r$ be the radii of the circles circumscribed and inscribed, respectively, in a triangle.

Prove that $R \ge 2r$, and that $R = 2r$ only for an equilateral triangle.
1948 Moscow Mathematical Olympiad
Can a figure have a greater than $1$ and finite number of centers of symmetry?
1948 Moscow Mathematical Olympiad
The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.
1948 Moscow Mathematical Olympiad
a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections.

b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.
1948 Moscow Mathematical Olympiad
* What is the radius of the largest possible circle inscribed into a cube with side $a$?
1948 Moscow Mathematical Olympiad
How many different integer solutions to the inequality $|x| + |y| < 100$ are there?
1948 Moscow Mathematical Olympiad
What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?
1947 Moscow Mathematical Olympiad
Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.
1947 Moscow Mathematical Olympiad
a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected.

b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.
1947 Moscow Mathematical Olympiad
Find the coefficients of $x^{17}$ and $x^{18}$ after expansion and collecting the terms of $(1+x^5+x^7)^{20}$.