Some figures stand in certain cells of a chess board. It is known that a figure stands on each row, and that different rows have a different number of figures. Prove that it is possible to mark $8$ figures so that on each row and column stands exactly one marked figure.
1997 Moscow Mathematical Olympiad
Grade 8
To get to the Stromboli Volcano from the observatory, one has to take a road and a passway, each taking $4$ hours. There are two craters on the top. The first crater erupts for $1$ hour, stays silent for $17$ hours, then repeats the cycle. The second crater erupts for $1$ hour, stays silent for $9$ hours, erupts for $1$ hour, stays silent for $17$ hours, and then repeats the cycle. During the eruption of the first crater, it is dangerous to take both the passway and the road, but the second crater is smaller, so it is still safe to take the road. At noon, scout Vanya saw both craters erupting simultaneously. Will it ever be possible for him to mount the top of the volcano without risking his life?
Inside acute $\angle{XOY},$ points $M$ and $N$ are taken so that $\angle{XON}=\angle{YOM}$. Point $Q$ is taken on segment $OX$ such that $\angle{NQO}=\angle{MQX}.$ Point $P$ is taken such that $\angle{NPO}=\angle{MPY}.$ Prove the lengths of the broken lines $MPN$ and $MQN$ are equal.
Prove that there exists a positive non-prime integer such that if any three of its neighboring digits are replaced with any given triple of the digits, the number remains non-prime. Does there exist a $1997$-digit such number?
In the rhombus $ABCD,$ the measure of $\angle{B}=40^{\circ}, E$ is the midpoint of $BC,$ and $F$ is the base of the perpendicular dropped from $A$ on $DE.$ Find the measure of $\angle{DFC}.$
A banker learned that among similarly looking golden coins, exactly one is counterfeit and has less weight. The banker asked an expert to determine the coin by means of a balance, and demanded each coin should participate in no more than two weightings in order to not wear out the coin, thereby losing market value. What is the largest number of coins the banker could have had, given that the expert successfully completed his task?
Grade 9
In a triangle one side is $3$ times shorter than the sum of the other two. Prove that the angle opposite said side is the smallest of the triangle’s angles.
$9$ different pieces of cheese are placed on a plate. Is it always possible to cut one of them into two parts so that the $10$ pieces obtained were divisible into two portions of equal mass of $5$ pieces each?
Convex octagon $AC_1BA_1CB_1$ satisfies: $AB_1=AC_1$, $BC_1=BA_1$, $CA_1=CB_1$ and $\angle{A}+\angle{B}+\angle{C}=\angle{A_1}+\angle{B_1}+\angle{C_1}$. Prove that the area of $\triangle{ABC}$ is equal to half the area of the octagon.
Along a circular railroad, $n$ trains circulate in the same direction at equal distances between them. Stations $A, B$ and $C$ on this railroad (denoted as the trains pass them) form an equilateral triangle. Ira enters station $A$ at the same time as Alex enters station $B$ in order to take the nearest train. It is knows that if they enter the stations at the same time as the driver Roma passes a forest, then Ira takes her train earlier than Alex; otherwise Alex takes the train earlier than or simultaneously with Ira. What part of the railroad goes through the forest (between which stations)?
Let $1+x+x^2+...+x^{n-1}=F(x)G(x)$, where $n>1$ and where $F$ and $G$ are polynomials whose coefficients are zeroes and units. Prove that one of the polynomials $F$ and $G$ can be represented in the form $(1+x+x^2+...x^{k-1})T(x),$ where $k>1$ and $T$ is a polynomial whose coefficients are zeroes and units.
Grade 10
Is there a convex body distinct from ball whose three orthogonal projections on three pairwise perpendicular planes are discs?
Prove that among the quadrilaterals with given lengths of the diagonals and the angle between them, the parallelogram has the least perimeter.
A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square. Generalization: Prove that if the same process is applied to any $n$-gon and the result is a regular $n$-gon, then the intial $n$-gon must also be regular.
Given real numbers $a_1\leq{a_2}\leq{a_3}$ and $b_1\leq{b_2}\leq{b_3}$ such that $$a_1+a_2+a_3=b_1+b_2+b_3,$$$$a_1a_2+a_2a_3+a_1a_3=b_1b_2+b_2b_3+b_1b_3.$$Prove that if $a_1\leq{b_1},$ then $a_3\leq{b_3}$
Consider the sequence formed by the first digits of the powers of $5$:$$1,5,2,1,6,...$$Prove any segment in this sequence, when written in reversed order, will be encountered in the sequence of the first digits of the powers of $2:$ $$1,2,4,8,1,3,6,1...$$