A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours

The natural number $A$ has three digits added to its right. The resulting number turned out to be equal to the sum of all natural numbers from $1$ to $A$. Find $A$.

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

There are $40$ identical gas cylinders, gas pressure values in which we are unknown and may be evil. It is allowed to connect any cylinders with each other in an amount not exceeding a given natural number $k$, and then separate them; while the pressure gas in the connected cylinders is set equal to the arithmetic average of the pressures in them before the connection. At what minimum $k$ is there a way to equalize the pressures in all $40$ cylinders, regardless of initial pressure distribution in the cylinders?

Prove that the numbers from $1$ to $ 15$ cannot be divided into two groups: $A$ of $2$ numbers and $B$ of $13$ numbers such that the sum of the numbers in group $B$ is equal to product of numbers in group $A$.

Given triangle $ABC$. Point $A_1$ is symmetric to vertex $A$ wrt line $BC$, and point $C_1$ is symmetric to vertex $C$ wrt line $AB$. Prove that if points $A_1$, $B$ and $C_1$ lie on the same line and $C_1B = 2A_1B$, then angle $\angle CA_1B$ is right.

The box contains a complete set of dominoes. Two players take turns choosing one dice from the box and placing them on the table, applying them to the already laid out chain on either of the two sides according to the rules of domino. The one who cannot make his next move loses. Who will win if they both played correctly?

An open chain was made from $54$ identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a $3\times 3 \times3$ surface with this chain of squares?

All natural numbers from $1$ to $N$, $ N \ge 2$ are written out in a certain order in a circle. Moreover, for any pair of neighboring numbers there is at least one digit appearing in the decimal notation of each of them. Find the smallest possible value of $N$.

In triangle $ABC$, on side $AC$ there are points $D$ and $E$, that $AB = AD$ and $BE = EC$ ($E$ between $A$ and $D$). Point $F$ is midpoint of arc $BC$ of circumcircle of triangle $ABC$. Prove that the points $B, E, D, F$ lie on the same circle.

The product of positive numbers $x, y$ and $z$ is equal to $1$. Prove that if it holds that $$\frac1x +\frac1y + \frac1z \ge x + y + z,$$then for any natural $k$, holds the inequality $$\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.$$

The maze is an $8 \times 8 $square, each cell contains $1 \times 1$ which has one of four arrows drawn (up, down, right, left). The upper side of the upper right cell is the exit from the maze.In the lower left cell there is a chip that, with each move, moves one square in the direction indicated by the arrow. After each move, the shooter in the cell in which there was just a chip rotates $90^o$ clockwise. If a chip must move, taking it outside the $8 \times 8$ square, it remains in place, and the arrow also rotates $90^o$ clockwise. Prove that sooner or later, the chip will come out of the maze.

All cells of the checkered plane are painted in $5$ colors so that in any figure of the species all colors are different. Prove that in any figure of the species $ \begin{tabular}{ | l | c| c | c | r| } \hline & & & &\\ \hline \end{tabular}$, all colors are different..

Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)

In triangle $ABC$ ($AB > BC$), $K$ and $M$ are the midpoints of sides $AB$ and $AC$, $O$ is the point of intersection of the angle bisectors. Let $P$ be the intersection point of lines $KM$ and $CO$, and the point $Q$ is such that $QP \perp KM$ and $QM \parallel BO$. Prove that $QO \perp AC$.

Given a circle $\omega$, a point $A$ lying inside $\omega$, and point $B$ ($B \ne A$). All possible triangles $BXY$ are considered, such that the points $X$ and $Y$ lie on $\omega$ and the chord $XY$ passes through the point $A$. Prove that the centers of the circumcircles of the triangles $BXY$ lie on the same straight line.

There are $n$ points in general position in space (no three lie on the same straight line, no four lie in the same plane). A plane is drawn through every three of them. Prove that If you take any whatever $n-3$ points in space, there is a plane from those drawn that does not contain any of these $n - 3$ points.

Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?

Triangle $ABC$ has an inscribed circle tangent to sides $AB$, $AC$ and $BC$ at points $C_1$, $B_1$ and $A_1 $ respectively. Let $K$ be a point on the circle diametrically opposite to point $C_1$, $D$ be the intersection point of lines $B_1C_1$ and $A_1K$. Prove that $CD = CB_1$.

Each voter in an election puts $n$ names of candidates on the ballot. There are $n + 1$ at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the $(n + 1)$-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.

Some natural numbers are marked. It is known that on every a segment of the number line of length $1999$ has a marked number. Prove that there is a pair of marked numbers, one of which is divisible by the other.

The function $f(x)$, defined on the entire real line, is known but that for any $a > 1 $ the function $f(x)+f(ax)$ is continuous on the entire line. Prove that $f(x)$ is also continuous along the entire line.

In the class, every talker is friends with at least one silent person. At this chatterbox is silent if there is an odd number of his friends in the office —silent. Prove that the teacher can invite you to an elective class without less than half the class so that all talkers are silent. original wordingВ классе каждый болтун дружит хотя бы с одним молчуном. При этом болтун молчит, если в кабинете находится нечетное число его друзей - молчунов. Докажите, что учительмо жет пригласитьна факультатив не менее половины класса так, чтобы все болтуны молчали

A polyhedron is circumscribed around a sphere. Let's call its face large if the projection of the sphere onto the plane of the face falls entirely within the face. Prove that there are no more than 6 large faces.

Are there real numbers $a, b$ and $c$ such that for all real $x$ and $y$ the following inequality holds: $$|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?$$

The cells of a $50\times 50$ square are painted in four colors. Prove that there is a cell, on four sides of which (i.e. top, bottom, left and on the right) there are cells of the same color as it (not necessarily adjacent to this cell).

For some polynomial there is an infinite set its values, each of which takes at least at two integer points. Prove that there is at most one the integer value that a polynomial takes at exactly one integer point.