Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 Moldova National Olympiad
The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that:



$ m_a+m_b+m_c+m_d\leq \dfrac{16}{3}R$
2011 Morocco National Olympiad
Solve the following equation in $\mathbb{R}^+$ :

$$\left\{\begin{matrix}

\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\ 

x+y+z=\frac{3}{670}

\end{matrix}\right.$$
2011 Morocco National Olympiad
Solve in $\mathbb{R}$ the equation :

$(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$ $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$ $ 0$.
2011 Morocco National Olympiad
Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.
2011 Morocco National Olympiad
Let $ABC$ be a triangle. $F$ and $L$ are two points on the side $[AC]$ such that $AF=LC< AC/2$.

Find the mesure of the angle $\angle FBL$ knowing that $AB^{2}+BC^{2}=AL^{2}+LC^{2}$.
2011 Morocco National Olympiad
Prove that

$$2010< \frac{2^{2}+1}{2^{2}-1}+\frac{3^{2}+1}{3^{2}-1}+...+\frac{2010^{2}+1}{2010^{2}-1}< 2010+\frac{1}{2}.$$
2011 Morocco National Olympiad
Compute the sum

$$S=1+2+3-4-5+6+7+8-9-10+\dots-2010$$

where every three consecutive $+$ are followed by two $-$.
2011 Morocco National Olympiad
Solve in $\mathbb{R}^{3}$ the following system

$$\left\{\begin{matrix}

\sqrt{x^{2}-y}=z-1\\ 

\sqrt{y^{2}-z}=x-1\\ 

\sqrt{z^{2}-x}=y-1

\end{matrix}\right.$$
2011 Morocco National Olympiad
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side  $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$.  The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
2011 Morocco National Olympiad
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq  C(x+y)$ for all reals $x,y$.
2011 Morocco National Olympiad
Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$

($p$ and $q$ are two real parameters).
2011 Morocco National Olympiad
Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation

$$(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.$$
2011 Morocco National Olympiad
Let $ABCD$ be a convex quadrilateral with angles $\angle ABC$ and $\angle BCD$ not less than $120^{\circ}$. Prove that

$$AC + BD> AB+BC+CD$$
2011 Morocco National Olympiad
Find all positive integers n such that :

$-2^{0}+2^{1}-2^{2}+2^{3}-2^{4}+...-(-2)^{n}=4^{0}+4^{1}+4^{2}+...+4^{2010}$
2011 Morocco National Olympiad
One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$.

What integer was removed ?
2011 Morocco National Olympiad
When dividing an integer $m$ by a positive integer $n$, $(0<  n\leq 100)$, a student finds $\frac{m}{n}= 0,167a_{1}a_{2}...$.

Prove that the student made a mistake while computing.
2011 Morocco National Olympiad
$ (C)$ and $(C')$ are two circles which intersect in $A$ and $B$. $(D)$ is a line that moves and passes through $A$, intersecting $(C)$ in P and $(C')$ in P'.

Prove that the bisector of $[PP']$ passes through a non-moving point.
2011 Morocco National Olympiad
Given positive reals $a,b,c;$ show that we have

$$\left(a+\frac 1b\right)\left(b+\frac 1c\right)\left(c+\frac 1a\right)\geq 8.$$
2011 Morocco National Olympiad
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle.

$(a)$ Prove that

$$\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).$$

$(b)$ When do we have equality?
2011 Morocco National Olympiad
Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.
2011 Morocco National Olympiad
The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that

$$S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},$$

Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.
2011 Morocco National Olympiad
Let $a$ and $b$ be two positive real numbers such that $a+b=ab$.

Prove that $\frac{a}{b^{2}+4}+\frac{b}{a^{2}+4}\geq \frac{1}{2}$.
2011 Morocco National Olympiad
Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system :

$\left\{\begin{matrix}

x+y+z+t=4\\ 

\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt}

\end{matrix}\right.$
2011 Morocco National Olympiad
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that

$\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.
2011 Morocco National Olympiad
Let $ABC$ be a triangle. The inside bisector of the angle $\angle BAC$ cuts $[BC]$ in $L$ and the circle $(C)$ circumsbribed to the triangle $ABC$ in $D$. The perpendicular to $(AC)$ going through $D$ cuts $[AC]$ in $M$ and the circle $(C)$ in $K$. Find the value of $\frac{AM}{MC}$ knowing that $\frac{BL}{LC}=\frac{1}{2}$.
2011 Morocco National Olympiad
Let $a,b,c$ be three postive real numbers such that $a+b+c=1$.

Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.
2011 Morocco National Olympiad
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y, \in \mathbb{R}$,

$$xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).$$
2011 Morocco National Olympiad
Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.
2011 Morocco National Olympiad
Problem 3 (MAR CP 1992) :

From the digits $1,2,...,9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$, $123456798$, ..., $987654321$. What is the $100000th$ number ?
2011 Morocco National Olympiad
Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A,  P \in  C_{1}$ and $B, D, F, A,  Q \in  C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.
2011 Morocco National Olympiad
Let $x$, $y$, and $z$ be three real positive numbers such that $x^{2}+y^{2}+z^{2}+2xyz=1$.

Prove that $2(x+y+z)\leq 3$.
2011 Morocco National Olympiad
Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.



(Poland) Andrzej Komisarski and Marcin Kuczma
2011 Morocco National Olympiad
Let $a, b, c, d, m, n$  be positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}=1989$, $n^{2}=max\left \{ a,b,c,d \right \}$ and $a+b+c+d=m^{2}$.

Find the values of $m$ and $n$.
2005 Morocco National Olympiad
In a square $ABCD$ let $F$ be the midpoint of $\left[ CD\right] $ and let $E$ be a point on $\left[ AB\right] $ such that $AE>EB$ . the parallel with $\left( DE\right) $ passing by $F$ meets the segment $\left[ BC\right] $ at $H$.

Prove that the line $\left( EH\right) $ is tangent to the circle circumscribed with $ABCD$
2005 Morocco National Olympiad
Find all the positive integers $x,y,z$ satisfiing : $x^{2}+y^{2}+z^{2}=2xyz$
2005 Morocco National Olympiad
Consider $n$ points $A_1, A_2, \ldots, A_n$ on a circle.  How many ways are there if we want to color these points by $p$ colors, so that each two neighbors points are colored with two different colors?
1957 Moscow Mathematical Olympiad
Find all isosceles trapezoids that are divided into $2$ isosceles triangles by a diagonal.
1957 Moscow Mathematical Olympiad
a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$.

Prove that $a, b, c$ and $d$ are all divisible by $5$.

b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$.

Prove that $a, b, c, d$ and $e$ are all divisible by $7$.
1957 Moscow Mathematical Olympiad
A snail crawls over a table at a constant speed. Every $15$ minutes it turns by $90^o$, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.
1957 Moscow Mathematical Olympiad
For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to $1$ or all the numbers are equal to $0$.
1957 Moscow Mathematical Olympiad
The distance between towns $A$ and $B$ is $999$ km.

At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows:

$\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$

How many signs are there, with both distances written with the help of only two distinct digits?
1957 Moscow Mathematical Olympiad
Given two concentric circles and a pair of parallel lines. Find the locus of the fourth vertices of all rectangles with three vertices on the concentric circles, two vertices on one circle and the third on the other and with sides parallel to the given lines.
1957 Moscow Mathematical Olympiad
Of all parallelograms of a given area find the one with the shortest possible longer diagonal.
1957 Moscow Mathematical Olympiad
Solve the equation $x^3 - [x] = 3$.
1957 Moscow Mathematical Olympiad
In a quadrilateral $ABCD$ points $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, respectively. The line through $M$ and $N$ meets $AB$ and $CD$ at $M'$ and $N'$, respectively. Prove that if $MM' = NN'$, then $AD  // BC$.
1957 Moscow Mathematical Olympiad
a) A student takes a subway to an Olympiad, pays one ruble and gets his change. Prove that if he takes a tram (street car) on his way home, he will have enough coins to pay the fare without change.

b) A student is going to a club. (S)he takes a tram, pays one ruble and gets the change. Prove that on the way back by a tram (s)he will be able to pay the fare without any need to change.



Note: In $1957$, the price of a subway ticket was $50$ kopeks, that of a tram ticket $30$ kopeks, the denominations of the coins were $1, 2, 3, 5, 10, 15$, and $20$ kopeks. ($1$ rouble = $100$ kopeks.)
1957 Moscow Mathematical Olympiad
A planar polygon $A_1A_2A_3 . . .A_{n-1}A_n$ ($n > 4$) is made of rigid rods that are connected by hinges.

Is it possible to bend the polygon (at hinges only!) into a triangle?
1957 Moscow Mathematical Olympiad
For which integer $n$ is $N = 20^n + 16^n - 3^n - 1$ divisible by $323$?
1957 Moscow Mathematical Olympiad
The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.
1957 Moscow Mathematical Olympiad
Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.
1957 Moscow Mathematical Olympiad
(a) A radio lamp has a $7$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $7$ holes. Is it possible to number the contacts and the holes so that for any insertion at least one contact would match the hole with the same number?



(b) A radio lamp has a $20$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $20$ holes. Let the contacts in the plug and the socket be already numbered. Is it always possible to insert the plug so that none of the contacts matches its socket?
1957 Moscow Mathematical Olympiad
The lengths, $a$ and $b$, of two sides of a triangle are known.



(a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value?

(b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?
1957 Moscow Mathematical Olympiad
(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles.



(b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.
1957 Moscow Mathematical Olympiad
Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.
1957 Moscow Mathematical Olympiad
(a) Prove that the number of all digits in the sequence $1, 2, 3,... , 10^8$ is equal to the number of all zeros in the sequence $1, 2, 3, ... , 10^9$.



(b) Prove that the number of all digits in the sequence $1, 2, 3, ... ,  10^k$ is equal to the number of all zeros in the sequence $1, 2, 3,  ... , 10^{k+1}$.
1957 Moscow Mathematical Olympiad
(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} +  \frac{B'O}{B'G} +  \frac{C'O}{C'G} = 3.$$

(b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} +  \frac{B'O}{B'G} +  \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$
1957 Moscow Mathematical Olympiad
Solve the system: $$\frac{2x_1^2}{1+x_1^2}=x_2, \frac{2x_2^2}{1+x_2^2}=x_3, \frac{2x_3^2}{1+x_3^2}=x_1$$
1957 Moscow Mathematical Olympiad
Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.
1957 Moscow Mathematical Olympiad
Find all real solutions of the system :



(a) $$1-x_1^2=x_2, 1-x_2^2=x_3, ..., 1-x_{98}^2=x_{99}, 1-x_{99}^2=x_1$$(b)*  $$1-x_1^2=x_2, 1-x_2^2=x_3, ..., 1-x_{98}^2=x_{n}, 1-x_{n}^2=x_1$$
1957 Moscow Mathematical Olympiad
Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3!  \cdot ... \cdot a_{12}!$ is minimal.
1957 Moscow Mathematical Olympiad
* Three equal circles are tangent to each other externally and to the fourth circle internally. Tangent lines are drawn to the circles from an arbitrary point on the fourth circle. Prove that the sum of the lengths of two tangent lines equals the length of the third tangent.
1957 Moscow Mathematical Olympiad
Given quadrilateral $ABCD$ and the directions of its sides. Inscribe a rectangle in $ABCD$.
1957 Moscow Mathematical Olympiad
Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ($i = 1, 2, 3,..., n - 1$) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.
1956 Moscow Mathematical Olympiad
Prove that there are no four points $A, B, C, D$ on a plane such that all triangles $\vartriangle ABC,\vartriangle BCD, \vartriangle CDA, \vartriangle DAB$ are acute ones.
1956 Moscow Mathematical Olympiad
Find all two-digit numbers $x$ the sum of whose digits is the same as that of $2x$, $3x$, ... , $9x$.
1956 Moscow Mathematical Olympiad
A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.
1956 Moscow Mathematical Olympiad
a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$.

b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.
1956 Moscow Mathematical Olympiad
a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)?

b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.
1956 Moscow Mathematical Olympiad
On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.
1956 Moscow Mathematical Olympiad
a) In the decimal expression of a positive number, $a$, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of $a$ with accuracy to $0.01$ with deficiency). The number obtained is divided by $a$ and the quotient is similarly approximated with the same accuracy by a number $b$. What numbers $b$ can be thus obtained? Write all their possible values.

b) same as (a) but with accuracy to $0.001$

c) same as (a) but with accuracy to $0.0001$
1956 Moscow Mathematical Olympiad
On an infinite sheet of graph paper a table is drawn so that in each square of the table stands a number equal to the arithmetic mean of the four adjacent numbers. Out of the table a piece is cut along the lines of the graph paper. Prove that the largest number on the piece always occurs at an edge, where $x = \frac14 (a + b + c + d)$.
1956 Moscow Mathematical Olympiad
In a convex quadrilateral $ABCD$, consider quadrilateral $KLMN$ formed by the centers of mass of triangles $ABC, BCD, DBA, CDA$. Prove that the straight lines connecting the midpoints of the opposite sides of quadrilateral $ABCD$ meet at the same point as the straight lines connecting the midpoints of the opposite sides of $KLMN$.
1956 Moscow Mathematical Olympiad
Consider positive numbers $h, s_1, s_2$, and a spatial triangle $\vartriangle ABC$. How many ways are there to select a point $D$ so that the height of tetrahedron $ABCD$ dropped from $D$ equals $h$, and the areas of faces $ACD$ and $BCD$ equal $s_1$ and $s_2$, respectively?
1956 Moscow Mathematical Olympiad
A square of side $a$ is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if $r$ is the radius of the circle inscribed in the triangle, then $r\sqrt2 < a < 2r$.
1956 Moscow Mathematical Olympiad
Given a closed broken line $A_1A_2A_3...A_n$ in space and a plane intersecting all its segments, $A_1A_2$ at $B_1, A_2A_3$ at $B_2,... , A_nA_1$ at $Bn$,  prove that

$$\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1$$.
1956 Moscow Mathematical Olympiad
Prove that the system of equations $\begin{cases} x_1 - x_2 = a \\
x_3 - x_4 = b \\
x_1 + x_2 + x_3 + x_4 = 1\end{cases}$

has at least one solution in positive numbers ($x_1 ,x_2 ,x_3 ,x_4>0$) if and only if $|a| + |b| < 1$.
1956 Moscow Mathematical Olympiad
Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all heights of $\vartriangle A_1B_1C_1$ pass through $O$, and all heights of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.
1956 Moscow Mathematical Olympiad
a) Points $A_1, A_2, A_3, A_4, A_5, A_6$ divide a circle of radius $1$ into six equal arcs. Ray $\ell_1$ from $A_1$ connects $A_1$ with $A_2$, ray $\ell_2$ from $A_2$ connects $A_2$ with $A_3$, and so on, ray $\ell_6$ from $A_6$ connects $A_6$ with $A_1$. From a point $B_1$ on $\ell_1$ the perpendicular is dropped to $\ell_6$, from the foot of this perpendicular another perpendicular is dropped to $\ell_5$, and so on. Let the foot of the $6$-th perpendicular coincide with $B_1$. Find the length of segment $A_1B_1$.



b) Find points $B_1, B_2,... , B_n$ on the extensions of sides $A_1A_2, A_2A_3,... , A_nA_1$ of a regular $n$-gon $A_1A_2...A_n$ such that $B_1B_2 \perp  A_1A_2$, $B_2B_3 \perp  A_2A_3$,$ . . . $, $B_nB_1 \perp A_nA_1$.
1956 Moscow Mathematical Olympiad
a) $100$ numbers (some positive, some negative) are written in a row. All of the following three types of numbers are underlined: 1) every positive number, 2) every number whose sum with the number following it is positive,  3) every number whose sum with the two numbers following it is positive.

Can the sum of all underlined numbers be

(i) negative?

(ii) equal to zero?



b) $n$ numbers (some positive and some negative) are written in a row. Each positive number and each number whose sum with several of the numbers following it is positive is underlined. Prove that the sum of all underlined numbers is positive.
1956 Moscow Mathematical Olympiad
$64$ non-negative numbers whose sum equals $1956$ are arranged in a square table, eight numbers in each row and each column. The sum of the numbers on the two longest diagonals is equal to $112$. The numbers situated symmetrically with respect to any of the longest diagonals are equal.

(a) Prove that the sum of numbers in any column is less than $1035$.

(b) Prove that the sum of numbers in any row is less than $518$.
1956 Moscow Mathematical Olympiad
* Assume that the number of a tree’s leaves is a multiple of $15$. Neglecting the shade of the trunk and branches prove that one can rip off the tree $7/15$ of its leaves so that not less than $8/15$ of its shade remains.
1956 Moscow Mathematical Olympiad
* A shipment of $13.5$ tons is packed in a number of weightless containers. Each loaded container weighs not more than $350$ kg. Prove that $11$ trucks each of which is capable of carrying · $1.5$ ton can carry this load.
1956 Moscow Mathematical Olympiad
Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.
1956 Moscow Mathematical Olympiad
a) *  In a rectangle of area $5$ sq. units, $9$ rectangles of area $1$ are arranged. Prove that the area of the overlap of some two of these rectangles is  $\ge 1/9$

b) In a rectangle of area $5$ sq. units, lie $9$ arbitrary polygons each of area $1$. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$
1956 Moscow Mathematical Olympiad
$1956$ points are chosen in a cube with edge $13$.

Is it possible to fit inside the cube a cube with edge $1$ that would not contain any of the selected points?
1956 Moscow Mathematical Olympiad
Given three numbers $x, y, z$ denote the absolute values of the differences of each pair by $x_1,y_1, z_1$. From $x_1, y_1, z_1$ form in the same fashion the numbers $x_2, y_2, z_2$, etc. It is known that $x_n = x,y_n = y, z_n = z$ for some $n$. Find $y$ and $z$ if $x = 1$.
1956 Moscow Mathematical Olympiad
A quadrilateral is circumscribed around a circle. Prove that the straight lines connecting neighboring tangent points either meet on the extension of a diagonal of the quadrilateral or are parallel to it.