Navvye/Olympiad-Problems-CSV huggingface dataset:
1947 Moscow Mathematical Olympiad
Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass.
1947 Moscow Mathematical Olympiad
Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$.

Prove that the three circles which pass:

a) through $O, A, B$,

b) through $O, B, C$, and

c) through $O, C, A$, are equal
1947 Moscow Mathematical Olympiad
Find the coefficient of $x^2$ after expansion and collecting the terms of the following expression (there are $k$ pairs of parentheses): $((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$.
1947 Moscow Mathematical Olympiad
How many squares different in size or location can be drawn on an $8 \times  8$ chess board?

Each square drawn must consist of whole chess board’s squares.
1947 Moscow Mathematical Olympiad
Which of the polynomials,  $(1+x^2 -x^3)^{1000}$ or $(1-x^2 +x^3)^{1000}$, has the greater coefficient of $x^{20}$ after expansion and collecting the terms?
1947 Moscow Mathematical Olympiad
Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product $\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^2}\right)...\left(1-\frac{1}{10^{99}}\right)$
1947 Moscow Mathematical Olympiad
Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.
1947 Moscow Mathematical Olympiad
Twenty cubes of the same size and appearance are made of either aluminum or of heavier duralumin. How can one find the number of duralumin cubes using not more than $11$ weighings on a balance without weights? (We assume that all cubes can be made of aluminum, but not all of duralumin.)
1947 Moscow Mathematical Olympiad
How many digits are there in the decimal expression of $2^{100}$ ?
1947 Moscow Mathematical Olympiad
Position the $4$ points on plane so that when measuring of all pairwise distances between them, it turned out only two different numbers. Find all such locations.
1947 Moscow Mathematical Olympiad
a) Given $5$ points on a plane, no three of which lie on one line.

Prove that four of these points can be taken as vertices of a convex quadrilateral.

b) Inside a square, consider a convex quadrilateral and inside the quadrilateral, take a point $A$.

It so happens that no three of the $9$ points — the vertices of the square, of the quadrilateral and $A$ — lie on one line.

Prove that $5$ of these points are vertices of a convex pentagon.
1947 Moscow Mathematical Olympiad
Prove that no convex $13$-gon can be cut into parallelograms.
1947 Moscow Mathematical Olympiad
a) $101$ numbers are selected from the set $1, 2, . . . , 200$.

Prove that among the numbers selected there is a pair in which one number is divisible by the other.

b) One number less than $16$, and $99$ other numbers are selected from the set $1, 2, . . . , 200$.

Prove that among the selected numbers there are two such that one divides the other.
1947 Moscow Mathematical Olympiad
In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.
1947 Moscow Mathematical Olympiad
In the numerical triangle

$................1..............$

$...........1 ...1 ...1.........$

$......1... 2... 3 ... 2 ... 1....$

$.1...3...6...7...6...3...1$

$...............................$

each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.
1947 Moscow Mathematical Olympiad
Prove that if the four faces of a tetrahedron are of the same area they are equal.
1946 Moscow Mathematical Olympiad
What is the largest number of acute angles that a convex polygon can have?
1946 Moscow Mathematical Olympiad
Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and  $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)
1946 Moscow Mathematical Olympiad
Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.
1946 Moscow Mathematical Olympiad
Solve the system of equations: $\begin{cases} 

x_1 + x_2 + x_3 = 6 \\
x_2 + x_3 + x_4 = 9 \\
x_3 + x_4 + x_5 = 3 \\
x_4 + x_5 + x_6 = -3 \\
x_5 + x_6 + x_7 = -9 \\
x_6 + x_7 + x_8 = -6 \\
x_7 + x_8 + x_1 = -2 \\
x_8 + x_1 + x_2 = 2  \end{cases}$
1946 Moscow Mathematical Olympiad
Prove that after completing the multiplication and collecting the terms

$(1 - x + x^2 - x^3 +... - x^{99} + x^{100})(1 + x + x^2 + ...+ x^{99} + x^{100})$

has no monomials of odd degree.
1946 Moscow Mathematical Olympiad
Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and  passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.
1946 Moscow Mathematical Olympiad
Through a point $M$ inside an angle $a$ line is drawn. It cuts off this angle a triangle of the least possible area. Prove that $M$ is the midpoint of the segment on this line that the angle intercepts.
1946 Moscow Mathematical Olympiad
Prove that $n^2 + 3n + 5$ is not divisible by $121$ for any positive integer $n$.
1946 Moscow Mathematical Olympiad
Prove that for any positive integer $n$ the following identity holds $\frac{(2n)!}{n!}= 2^n \cdot (2n - 1)!!$
1946 Moscow Mathematical Olympiad
Prove that if $\alpha$ and $\beta$ are acute angles and $\alpha$ < $\beta$ , then $\frac{tan \alpha}{\alpha} < \frac{tan \beta}{\beta} $
1946 Moscow Mathematical Olympiad
a) Two seventh graders and several eightth graders take part in a chess tournament. The two seventh graders together scored eight points. The scores of eightth graders are equal. How many eightth graders took part in the tournament?

b) Ninth and tenth graders participated in a chess tournament. There were ten times as many tenth graders as ninth graders. The total score of tenth graders was $4.5$ times that of the ninth graders. What was the ninth graders score?
1946 Moscow Mathematical Olympiad
Prove that for any integers $x$ and $y$ we have $x^5 + 3x^4y - 5x^3y^2 - 15x^2y^3 + 4xy^4 + 12y^5 \ne 33$.
1946 Moscow Mathematical Olympiad
On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.
1946 Moscow Mathematical Olympiad
Towns $A_1, A_2, . . . , A_{30}$ lie on line $MN$. The distances between the consecutive towns are equal. Each of the towns is the point of origin of a straight highway. The highways are on the same side of $MN$ and form the following angles with it:





Thirty cars start simultaneously from these towns along the highway at the same constant speed. Each intersection has a gate. As soon as the first (in time, not in number) car passes the intersection the gate closes and blocks the way for all other cars approaching this intersection. Which cars will pass all intersections and which will be stopped?
1946 Moscow Mathematical Olympiad
a) A bus network is organized so that:

1) one can reach any stop from any other stop without changing buses;

2) every pair of routes has a single stop at which one can change buses;

3) each route has exactly three stops?

How many bus routes are there?



b) A town has $57$ bus routes. How many stops does each route have if it is known that

1) one can reach any stop from any other stop without changing buses;

2) for every pair of routes there is a single stop where one can change buses;

3) each route has three or more stops?
1946 Moscow Mathematical Olympiad
Given the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, ... ,$ ascertain whether among its first $(10^8+1)$ terms there is a number that ends with four zeros.
1946 Moscow Mathematical Olympiad
On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn.

Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB, \vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB, \vartriangle S_0CD, \vartriangle S0EF$.

Consider separately the case $\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}$ .
1945 Moscow Mathematical Olympiad
a) Divide $a^{128} - b^{128}$ by $(a + b)(a^2 + b^2)(a^4 + b^4)(a^8 + b^8)(a^{16} + b^{16})(a^{32} + b^{32})(a^{64} + b^{64}) $.

b) Divide $a^{2^k} - b^{2^k}$ by $(a + b)(a^2 + b^2)(a^4 + b^4) ... (a^{2^{k-1}} + b^{2^{k-1}})$
1945 Moscow Mathematical Olympiad
Prove that for any positive integer $n\ge 2$ the following inequality holds:

$$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}>\frac{1}{2}$$
1945 Moscow Mathematical Olympiad
Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ is a perfect square.
1945 Moscow Mathematical Olympiad
Prove that it is impossible to divide a scalene triangle into two equal triangles.
1945 Moscow Mathematical Olympiad
Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.
1945 Moscow Mathematical Olympiad
Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.
1945 Moscow Mathematical Olympiad
The system $\begin{cases}  x^2 - y^2 = 0 \\
(x - a)^2 + y^2 = 1 \end{cases}$

generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?
1945 Moscow Mathematical Olympiad
A right triangle $ABC$ moves along the plane so that the vertices $B$ and $C$ of the triangle’s acute angles slide along the sides of a given right angle. Prove that point $A$ fills in a line segment and find its length.
1945 Moscow Mathematical Olympiad
Given the $6$ digits: $0, 1, 2, 3, 4, 5$.

Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).
1945 Moscow Mathematical Olympiad
Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^o 30'$. It turned out that the polygons coincided (aligned again). What is the minimal possible number of sides of the polygons?
1945 Moscow Mathematical Olympiad
The side $AD$ of a parallelogram $ABCD$ is divided into $n$ equal segments. The nearest to $A$ division point $P$ is connected with $B$. Prove that line $BP$ intersects the diagonal $AC$ at point $Q$ such that $AQ = \frac{AC}{n + 1}$
1945 Moscow Mathematical Olympiad
Segments connect vertices $A, B, C$ of $\vartriangle ABC$ with respective points $A_1, B_1, C_1$ on the opposite sides of the triangle. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ do not belong to one straight line.
1945 Moscow Mathematical Olympiad
Solve in integers the equation $xy + 3x - 5y = - 3$.
1945 Moscow Mathematical Olympiad
The numbers $a_1, a_2, ..., a_n$ are equal to $1$ or $-1$. Prove that

$$2 \sin \left(a_1+\frac{a_1a_2}{2}+\frac{a_1a_2a_3}{4}+...+\frac{a_1a_2...a_n}{2^{n-1}}\right)\frac{\pi}{4}=a_1\sqrt{2+a_2\sqrt{2+a_3\sqrt{2+...+a_n\sqrt2}}}$$In particular, for $a_1 = a_2 = ... = a_n = 1$ we have

$$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$
1945 Moscow Mathematical Olympiad
A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.
1941 Moscow Mathematical Olympiad
Construct a triangle given its height and median — both from the same vertex — and the radius of the circumscribed circle.
1941 Moscow Mathematical Olympiad
Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.
1941 Moscow Mathematical Olympiad
Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.
1941 Moscow Mathematical Olympiad
A point $P$ lies outside a circle. Consider all possible lines drawn through $P$ so that they intersect the circle. Find the locus of the midpoints of the chords — segments the circle intercepts on these lines.
1941 Moscow Mathematical Olympiad
Prove that $1$ plus the product of any four consecutive integers is a perfect square.
1941 Moscow Mathematical Olympiad
On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.
1941 Moscow Mathematical Olympiad
A polynomial $P(x)$ with integer coefficients takes odd values at $x = 0$ and $x = 1$. Prove that $P(x)$ has no integer roots.
1941 Moscow Mathematical Olympiad
Given points $M$ and $N$, the bases of heights $AM$ and $BN$ of $\vartriangle ABC$ and the line to which the side $AB$ belongs. Construct $\vartriangle ABC$.
1941 Moscow Mathematical Olympiad
Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.
1941 Moscow Mathematical Olympiad
How many roots does equation $\sin x = \frac{x}{100}$ have?
1941 Moscow Mathematical Olympiad
a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes.

b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.
1941 Moscow Mathematical Olympiad
* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.
1941 Moscow Mathematical Olympiad
Consider $\vartriangle ABC$ and a point $M$ inside it. We move $M$ parallel to $BC$ until $M$ meets $CA$, then parallel to $AB$ until it meets $BC$, then parallel to $CA$, and so on. Prove that $M$ traverses a self-intersecting closed broken line and find the number of its straight segments.
1941 Moscow Mathematical Olympiad
a) Find an integer $a$ for which $(x - a)(x - 10) + 1$ factors in the product $(x + b)(x + c)$ with integers $b$ and $c$.

b) Find nonzero and nonequal integers $a, b, c$ so that $x(x - a)(x - b)(x - c) + 1$ factors into the product of two polynomials with integer coefficients.
1941 Moscow Mathematical Olympiad
Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.
1941 Moscow Mathematical Olympiad
Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.
1941 Moscow Mathematical Olympiad
On a plane, several points are chosen so that a disc of radius $1$ can cover every $3$ of them. Prove that a disc of radius $1$ can cover all the points.
1941 Moscow Mathematical Olympiad
Solve in integers the equation $x + y = x^2 - xy + y^2$.
1941 Moscow Mathematical Olympiad
Given two skew perpendicular lines in space, find the set of the midpoints of all segments of given length with the endpoints on these lines.
1941 Moscow Mathematical Olympiad
Construct a right triangle, given two medians drawn to its legs.
1940 Moscow Mathematical Olympiad
Factor $(b - c)^3 + (c - a)^3 + (a - b)^3$.
1940 Moscow Mathematical Olympiad
It takes a steamer $5$ days to go from Gorky to Astrakhan downstream the Volga river and $7$ days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan?
1940 Moscow Mathematical Olympiad
How many zeros does $100!$ have at its end in the usual decimal representation?
1940 Moscow Mathematical Olympiad
Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?
1940 Moscow Mathematical Olympiad
Solve the system  $\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\
 x + y = b \end{cases}$
1940 Moscow Mathematical Olympiad
Consider all positive integers written in a row: $123456789101112131415...$

Find the $206788$-th digit from the left.
1940 Moscow Mathematical Olympiad
Construct a circle equidistant from four points on a plane. How many solutions are there?
1940 Moscow Mathematical Olympiad
Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.
1940 Moscow Mathematical Olympiad
Find all $3$-digit numbers  $\overline {abc}$ such that $\overline {abc} = a! + b!  + c! $.
1940 Moscow Mathematical Olympiad
Find a four-digit number that is perfect square and such that the first two digits are the same and the last two as well.
1940 Moscow Mathematical Olympiad
Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.
1940 Moscow Mathematical Olympiad
How does one tile a plane, without gaps or overlappings, with the tiles equal to a given irregular quadrilateral?
1940 Moscow Mathematical Olympiad
How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^2 + y^2$ is divisible by $7$?
1940 Moscow Mathematical Olympiad
* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.
1940 Moscow Mathematical Olympiad
Which is greater: $300!$ or $100^{300}$?
1940 Moscow Mathematical Olympiad
The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.
1940 Moscow Mathematical Olympiad
Let $a_1, ...,, a_n$ be positive numbers. Prove the inequality:

$$\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+ ... +\frac{a_{n-1}}{a_n}+ \frac{a_n}{a_1} \ge n$$
1940 Moscow Mathematical Olympiad
How many positive integers $x$ less than $10 000$ are there such that $2^x - x^2$ is divisible by $7$ ?
1939 Moscow Mathematical Olympiad
Solve the system  $\begin{cases}  3xyz -x^3 - y^3-z^3 = b^3 \\
 x + y+ z = 2b \\
 x^2 + y^2-z^2 = b^2

\end{cases}$
1939 Moscow Mathematical Olympiad
Prove that $cos \frac{2\pi}{5}  +cos \frac{4\pi}{5} = -\frac{1}{2}$.
1939 Moscow Mathematical Olympiad
Consider points $A, B, C$. Draw a line through $A$ so that the sum of distances from $B$ and $C$ to this line is equal to the length of a given segment.
1939 Moscow Mathematical Olympiad
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.
1939 Moscow Mathematical Olympiad
Prove that for any triangle the bisector lies between the median and the height drawn from the same vertex.
1939 Moscow Mathematical Olympiad
Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients
1939 Moscow Mathematical Olympiad
Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.
1939 Moscow Mathematical Olympiad
Given two points $A$ and $B$ and a circle, find a point $X$ on the circle so that points $C$ and $D$ at which lines $AX$ and $BX$ intersect the circle are the endpoints of the chord $CD$ parallel to a given line $MN$.
1939 Moscow Mathematical Olympiad
Find the remainder after division of $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ by $7$.
1939 Moscow Mathematical Olympiad
Consider a regular pyramid and a perpendicular to its base at an arbitrary point $P$. Prove that the sum of the lengths of the segments connecting $P$ to the intersection points of the perpendicular with the planes of the pyramid’s faces does not depend on the location of $P$.
1939 Moscow Mathematical Olympiad
What is the greatest number of parts that $5$ spheres can divide the space into?
1938 Moscow Mathematical Olympiad
In space $4$ points are given. How many planes equidistant from these points are there? Consider separately

(a) the generic case (the points given do not lie on a single plane) and

(b) the degenerate cases.
1938 Moscow Mathematical Olympiad
The following operation is performed over points $O_1, O_2, O_3$ and $A$ in space.

The point $A$ is reflected with respect to $O_1$, the resultant point $A_1$ is reflected through $O_2$, and the resultant point $A_2$ through $O_3$. We get some point $A_3$ that we will also consecutively reflect through $O_1, O_2, O_3$.

Prove that the point obtained last coincides with $A$..
1938 Moscow Mathematical Olympiad
What is the largest number of parts into which $n$ planes can divide space?
1938 Moscow Mathematical Olympiad
Given the base, height and the difference between the angles at the base of a triangle, construct the triangle.
1938 Moscow Mathematical Olympiad
How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?
1937 Moscow Mathematical Olympiad
Solve the system  $\begin{cases} x+ y +z = a \\
 x^2 + y^2 + z^2 = a^2 \\
 x^3 + y^3 +z^3 = a^3

\end{cases}$
1937 Moscow Mathematical Olympiad
* On a plane two points $A$ and $B$ are on the same side of a line. Find point $M$ on the line such that $MA +MB$ is equal to a given length.
1937 Moscow Mathematical Olympiad
Two segments slide along two skew lines. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments
1937 Moscow Mathematical Olympiad
Given three points that are not on the same straight line. Three circles pass through each pair of the points so that the tangents to the circles at their intersection points are perpendicular to each other. Construct the circles.
1937 Moscow Mathematical Olympiad
* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?
1937 Moscow Mathematical Olympiad
Into how many parts can an $n$-gon be divided by its diagonals if no three diagonals meet at one point?
1936 Moscow Mathematical Olympiad
Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.
1936 Moscow Mathematical Olympiad
All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle.
1936 Moscow Mathematical Olympiad
Represent an arbitrary positive integer as an expression involving only $3$ twos and any mathematical signs.

(P. Dirac)
1936 Moscow Mathematical Olympiad
Consider a circle and a point $P$ outside the circle. The angle of given measure with vertex at $P$ subtends a diameter of the circle. Construct the circle’s diameter with ruler and compass.
1936 Moscow Mathematical Olympiad
Find $4$ consecutive positive integers whose product is $1680$.
1936 Moscow Mathematical Olympiad
Solve the system  $\begin{cases} x+y=a \\
x^5 +y^5 = b^5

\end{cases}$
1936 Moscow Mathematical Olympiad
Given an angle less than $180^o$, and a point $M$ outside the angle. Draw a line through $M$ so that the triangle, whose vertices are the vertex of the angle and the intersection points of its legs with the line drawn, has a given perimeter.
1936 Moscow Mathematical Olympiad
The lengths of a rectangle’s sides and of its diagonal are integers. Prove that the area of the rectangle is an integer multiple of $12$.