Problems

Liam saw an unusual clock in the museum: the clock had no digits, and it’s not clear how the clock should be rotated. That is, we know that $1$ is the next digit clockwise from $12$, $2$ is the next digit clockwise from $1$, and so on. Moreover all the arrows (hour, minute, and second) have the same length, so it’s not clear which is which. What time does the clock show?

Two circles are tangent to each other and the smaller circle with the center $A$ is located inside the larger circle with the center $C$. The radii $CD$ and $CE$ are tangent to the smaller circle and the angle $\mathrm{\angle }DCE={60}^{\circ }$. Find the ratio of the radii of the circles.

For positive real numbers $a,b,c$ prove the inequality: $$(a^{2}b+b^{2}c+c^{2}a)(a{b}^2+b{c}^2+c{a}^2)\ge 9a^2{b}^2{c}^2.$$

Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$ and the center $A$. Let $B$ and $D$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $BD$ past $D$ until it meets the circle $C_2$ in $C$. The tangent to $C_2$ at $C$ and the tangent to $C_1$ at $B$ meet in a point $E$. Draw from $E$ the second tangent to $C_2$ which meets $C_2$ at the point $F$. Show that $BE$ bisects angle $\mathrm{\angle }FBC$.

Due to a mistake in the bakery, a cake that was supposed to be shaped as two concentric pieces (like on the right diagram below) came out like the left diagram below. Find the smallest number of pieces the cake should be cut into in order to rearrange the pieces into the cake on the right side of the picture.

Note that the cake is $\mathit{\text{not}}$ tiered like a wedding cake, but is shaped like a cylinder with a flat top. Curved cuts are allowed.

Katie and Charlotte had $4$ sheets of paper. They cut some of the sheets into $4$ pieces. They then cut some of the newly obtained papersheets also into $4$ pieces. They did this several more times, cutting a piece of paper into $4$. In the end they counted the number of sheets. Could this number be $2024$?

There is a scout group where some of the members know each other. Amongst any four members there is at least one of them who knows the other three. Prove that there is at least one member who knows the entirety of the scout group.

The distance between two villages equals $999$ kilometres. When you go from one village to the other, every kilometre you see a sign on the road, saying $0\mid 999,1\mid 998,2\mid 997,...,999\mid 0$. The signs show the distances to the two villages. Find the number of signs that contain only two different digits. For example, the sign $0\mid 999$ contains only two digits, namely $0$ and $9$, whereas the sign $1\mid 998$ contains three digits, namely $1$, $8$ and $9$.

Two players are playing a game. The first player is thinking of a finite sequence of positive integers $a_1$, $a_2$, ..., $a_n$. The second player can try to find the first player’s sequence by naming their own sequence $b_1$, $b_2$, ..., $b_n$. After this, the first player will give the result $a_1{b}_1+a_2{b}_2+...+a_n{b}_n$. Then the second player can say another sequence $c_1$, $c_2$, ..., $c_n$ to get another answer $a_1{c}_1+a_2{c}_2+...+a_n{c}_n$ from the first player. Find the smallest number of sequences the second player has to name to find out the sequence $a_1$, $a_2$, ..., $a_n$.

The letters $A$, $R$, $S$ and $T$ represent different digits from $1$ to $9$. The same letters correspond to the same digits, while different letters correspond to different digits.
Find $ART$, given that $ARTS+STAR=10,T31$.