A right triangle has an angle of $30^\circ$. The supplied Vietnamese text asks to prove that one angle bisector is shorter than half another angle bisector. The official original asks for one bisector to be exactly half as long as another. Preserve this distinction when interpreting the question.
One bacterium is in a cell of a $10\times10$ board. In each move, one existing bacterium moves to a cell sharing a side with its current cell and splits into two, both remaining in that new cell. Can all cells ever contain equal numbers of bacteria after some moves?
Call a positive integer good if its decimal digits are only 0 and 1. Suppose two good integers have a product which is also good. Must the digit sum of the product equal the product of the digit sums of the two factors?
Hexagon $AB^{\prime} CA^{\prime} BC^{\prime}$ has $ABC$ equilateral, its angles $A^{\prime} BC^{\prime},C^{\prime} AB^{\prime},B^{\prime} CA^{\prime}$ greater than $120^\circ$, and $AB^{\prime}=AC^{\prime}$, $BA^{\prime}=BC^{\prime}$, $CA^{\prime}=CB^{\prime}$. Prove that $AB^{\prime},BC^{\prime},CA^{\prime}$ are the side lengths of a triangle.
The integers 1 through 100 are colored: 50 red, 25 yellow, 25 green. The red and yellow numbers can be partitioned into 25 triples, each containing two red numbers and a yellow number strictly between them. The same is possible for the red and green numbers. Must all 100 numbers be partitionable into 25 quadruples, each with two red numbers, one yellow and one green, both nonred numbers strictly between the red numbers?
Let $X$ be a set of integers which can be partitioned into $N$ disjoint increasing arithmetic progressions, infinite in both directions, but not into fewer such progressions. Is a partition into $N$ progressions necessarily unique for every such $X$ when (a) $N=2$; (b) $N=3$? Original weights: 4 points each.
