One day, Rabbit was about to go for a meeting with Donkey, but Winnie the Pooh and Duck unexpectedly came to his home. Being well-bred, Rabbit offered the guests some refreshments. Pooh tied Duck’s mouth by a napkin and ate $10$ pots of honey and $22$ cups of condensed milk alone, whereby he needed two minutes for each pot of honey and $1$ minute for each cup of milk. Knowing that there was nothing sweet left in the house, Pooh released the Duck. Afflicted Rabbit observed that he wouldn’t have been late for the meeting with Donkey if Pooh had shared the refreshments with Duck. Knowing that Duck needs $5$ minutes for a pot of honey and $3$ minutes for a cup of milk, he computed the time the guests would have needed to devastate his supplies. What is that time?

Cities $A,B,C,D$ are positioned in such a way that $A$ is closer to $C$ than to $D$, and $B$ is closer to $C$ than to $D$. Prove that every point on the straight road from $A$ to $B$ is closer to $C$ than to $D$.

Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

On the world conference of parties of liars and truth-lovers there were $32$ participants which were sitting in four rows with $8$ chairs each. During a break each participant claimed that among his neighbors (by row or column) there are members of both parties. It is known that liars always lie, whereas truth-lovers always tell truth. What is the smallest number of liars at the conference for which this situation is possible?

The equation $ax^5 + bx^4 + c = 0$ has three distinct roots. Show that so does the equation $cx^5 +bx+a = 0$.

Point $ P$ is taken inside a right angle $ KLM$. A circle $ S_1$ with center $ O_1$ is tangent to the rays $ LK,LP$ of angle $ KLP$ at $ A,D$ respectively. A circle $ S_2$ with center $ O_2$ is tangent to the rays of angle $ MLP$, touching $ LP$ at $ B$. Suppose $ A,B,O_1$ are collinear. Let $ O_2D,KL$ meet at $ C$. Prove that $ BC$ bisects angle $ ABD$.

Find all prime numbers $p,q,r,s$ such that their sum is a prime number and $p^2+qs$ and $p^2 +qr$ are squares of integers.

There are $ 16$ pupils in a class. Every month, the teacher divides the pupils into two groups. Find the smallest number of months after which it will be possible that every two pupils were in two different groups during at least one month.

We have seven equal pails with water, filled to one half, one third, one quarter, one fifth, one eighth, one ninth, and one tenth, respectively. We are allowed to pour water from one pail into another until the first pail empties or the second one fills to the brim. Can we obtain a pail that is filled to (a) one twelfth, (b) one sixth after several such steps?

The equation $ x^2 + ax + b = 0$ has two distinct real roots. Prove that the equation $ x^4 + ax^3 + (b - 2)x^2 - ax + 1 = 0$ has four distinct real roots.

A circle with center O is inscribed in a quadrilateral ABCD and touches its non-parallel sides BC and AD at E and F respectively. The lines AO and DO meet the segment EF at K and N respectively, and the lines BK and CN meet at M. Prove that the points O,K,M and N lie on a circle.

A rectangle of size $ m \times n$ has been filled completely by trominoes (a tromino is an L-shape consisting of 3 unit squares). There are four ways to place a tromino 1st way: let the "corner" of the L be on top left 2nd way: let the "corner" of the L be on top right 3rd way: let the "corner" of the L be on bottom left 4th way: let the "corner" of the L be on bottom right Prove that the difference between the number of trominoes placed in the 1st and the 4th way is divisible by $ 3$.

Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).

Find the free coefficient of the polynomial $P(x)$ with integer coefficients, knowing that it is less than $1000$ in absolute value and that $P(19) = P(94) = 1994$.

In a convex pentagon $ ABCDE$ side $ AB$ is perpendicular to $ CD$ and side $ BC$ is perpendicular to $ DE$. Prove that if $ AB = AE = ED = 1$, then $ BC + CD < 1$.

In the Flower-city there are $ n$ squares and $ m$ streets, where $ m \geq n + 1$. Each street connects two squares and does not pass through other squares. According to a tradition in the city, each street is named either blue or red. Every year, a square is selected and the names of all streets emanating from that square are changed. Show that the streets can be initially named in such a way that, no matter how the names will be changed, the streets will never all have the same name.

Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.

It was noted that during one day in a town, each person made at most one phone call. Prove that the people in the town can be divided into three groups such that no two persons in the same group talked by phone that day.

A circle with center $O$ is tangent to the sides $AB$, $BC$, $AC$ of a triangle $ABC$ at points $E,F,D$ respectively. The lines $AO$ and $CO$ meet $EF$ at points $N$ and $M$. Prove that the circumcircle of triangle $OMN$ and points $O$ and $D$ lie on a line.

On the vertices of a convex $ n$-gon are put $ m$ stones, $ m > n$. In each move we can choose two stones standing at the same vertex and move them to the two distinct adjacent vertices. After $ N$ moves the number of stones at each vertex was the same as at the beginning. Prove that $ N$ is divisible by $ n$.

Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$for all $x \ne 1$.

Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.

Points $ A_1,A_2, ... ,A_n$ inside a circle and points $ B_1,B_2,...,B_n$ on its boundary are positioned so that the segments $ A_1B_1,A_2B_2, ... ,A_nB_n$ do not intersect. A bug can go from point $ A_i$ to $ A_j$ if the segment $ A_iA_j$ does not intersect any segment $ A_kB_k$, $ k \neq i, j$. Prove that the bug can go from any point $ A_p$ to any point $ A_q$ in a finite number of steps.