2024 IMC keystage3-IWYMIC team

Let $C,M,S$ be positive integers satisfying $C^2+M^2+2^{2024}=25\times2^S$. Given $C\le M$, what is the solution $(C,M,S)$?

Let $f(x)=ax^2+bx+c$, where $a,b,c$ are integers and $a\gt0$. Its two roots satisfy $0\lt x_1\lt x_2\lt1$. What is the minimum possible value of $a$?

Partition $\{1,2,\ldots,2024\}$ into $E,O$ according as the digit sum is even or odd. If $e,o$ are the sums of their elements, find $e-o$.

In triangle $ABC$, $I$ is the intersection of the internal bisectors of $\angle ABC,\angle ACB$. If $BI\cdot AC=CI\cdot AB$, prove that $ABC$ is isosceles.

In the diagram, $PQRS$ is a square inside a regular 9-gon, with $P$ a vertex of the 9-gon and $R$ the intersection of the two drawn diagonals. If a side of the 9-gon is 48 cm, find the area of $PQRS$ in $\mathrm{cm}^2$.

Let $\overline{ABCD}$ be a four-digit number. If the last two digits of $\overline{ABCD}^{\,2025}$ are 57, what is the largest possible two-digit number $\overline{CD}$?

Triangle $ABC$ is right-angled at $B$. Points $P\in BC,Q\in AB$ satisfy $\angle PAB=\angle CAB/3$ and $\angle QCB=\angle ACB/3$. Point $R$ is inside the triangle with $\angle RAC=\angle BAP$ and $\angle RCA=\angle BCQ$. If $\angle QRC$ is an integer number of degrees, find its minimum possible value.

A subsquare is a square of any size bounded by grid lines. An $m\times n$ grid has exactly 2024 subsquares. Find the smallest possible value of $mn$.

For example, a $2\times2$ grid has five subsquares: four of side 1 and one of side 2.

Donkeys numbered $1,\ldots,n$ around a circle say D,O,N,K,E,Y repeatedly, starting at 1. A donkey saying Y leaves; counting continues at the next donkey with D, until one remains. When $n=186$, the winner is 10. Find the smallest $n\gt186$ for which the winner is 100.

For four donkeys, the first one to say Y is number 2; after further counting, number 1 leaves, then number 4. Number 3 wins.

Captain Crook and his crew divide 10 gold bricks. Each visible chest contains at most one brick, and chains connecting the chests form a tree. Captain and crew alternately take a chest, Captain first. Whenever possible, the chosen chest must be adjacent to a previously taken chest; otherwise any chest may be taken. Captain may therefore start anywhere. The source diagrams illustrate this rule.

Can the crew arrange the tree so that Captain obtains at most one brick? If not, how can they minimize his possible gain? Explain a complete strategy.

In the first illustrated chain, three gold chests alternate with two empty ones; starting at the left gold chest gives Captain all three bricks. In the second illustrated tree, Captain can guarantee two bricks by starting at the leftmost or middle gold chest. In the third illustrated tree, Captain can get only one brick. The original illustrations below show these three arrangements.