Two strings each consist of 100 letters, all $A$ or $B$. In one step one may insert any number of identical letters at a chosen position, or delete any number of consecutive identical letters. Prove that either string can be transformed into the other in at most100 steps.
Triangle $ABC$ has perimeter1. Its $A$-excircle touches $BC$ and the extensions of $AB,AC$ along rays $AB,AC$ at $P,Q$, respectively. The line through the midpoints of $AB,AC$ meets the circumcircle of $APQ$ at $X,Y$. Find $XY$.
Let $h>1$ be an integer. A positive fraction, not necessarily reduced, is good if its positive numerator and denominator sum to $h$. Call $h$ remarkable if every positive fraction whose denominator is less than $h$ can be expressed using good fractions, not necessarily distinct, and only addition and subtraction. Prove that $h$ is remarkable if and only if it is prime.
Chameleons on an island have five possible colors. When one bites another, only the bitten chameleon changes color, according to a fixed rule depending solely on their two current colors. Suppose2023 initially red chameleons can, by a sequence of bites among themselves, all become blue. Find the smallest$k$ for which one can always guarantee that$k$ initially red chameleons can all become blue under the same rule.
For example, a rule might say that a red chameleon biting a green one turns the victim blue; a green one biting a red one leaves the victim red; a red one biting a red one turns the victim yellow. Other rules are possible.
