2025 IMC keystage3-IWYMIC team

Let $a,b,c$ be positive integers such that $1/a^2+1/b^2=1/c^2$. Find the minimum possible value of $a$.

How many ordered integer triples $(a,b,c)$ are roots of $x^3+px^2+q=0$ for some real $p,q$, with exactly two of $|a|,|b|,|c|$ prime?

For $p(x)=x^2-6x+849$, find the sum of all integers $m$ for which $p(m)$ is a perfect square.

For real $x$, let $[x]=\lfloor x\rfloor$ and $\{x\}=x-[x]$. Positive integers $(n,p,q)$ satisfy $q\gt p\gt3$, $n\le2025$ and $\{n/2\}+\{n/3\}+\{n/p\}+\{n/q\}=3$. Among triples minimizing $p+q$, how many possible values of $n$ are there?

$ABCD$ is a trapezium of area $2025\ \mathrm{cm}^2$, with $AB\parallel CD$, and $E\in AB$. Points $M,R,T$ are the centroids of $ADE,CDE,BCE$, respectively. Find the area of $MRT$ in $\mathrm{cm}^2$.

What are the last 10 digits of the base-2 representation of $47^{2025}$?

Alan gets 1 point for the first win in a run, and one more point for each consecutive win. After a loss the next win is again worth 1. He plays 100 games and wins 80. Find his minimum total score.

A camp lasts 8 days. Every student has kitchen duty on exactly 3 days, and no pair of students serves together more than once. What is the largest possible number of students?

An isosceles trapezium has three sides of equal length. The radius of its circumcircle equals the difference between its parallel sides. Determine all possible acute angles in degrees.

A restaurant in Đà Nẵng offers four options: Bèo, Nậm, Bèo-and-Nậm Combo, and Xèo. An IMC participant visits on all six nights, 14–19 August, and orders exactly one option each night. He does not eat the same dish type on consecutive nights. Thus Bèo cannot be followed by Bèo or the Combo; a Combo on night 5 excludes Bèo, Nậm and Combo on nights 4 and 6. How many six-night order sequences are possible?