ITOT 44 Autumn 2022 · THCS · O (Basic)

Can 36 distinct numbers be written in a $6\times6$ table so that the sum in every $1\times5$ rectangle, horizontal or vertical, is either 2022 or 2023?

Does there exist a positive integer which can be written as a product of two palindromes in more than 100 ways? A palindrome is a positive integer read identically from left to right and right to left.

A pentagon $ABCDE$ is circumscribed about a circle. Its angles at $A,C,E$ are all $100^\circ$. Find $\angle ACE$. Enter the angle in degrees.

Can all positive integers greater than 1 be colored with three colors, using all three colors and one color per integer, so that the product of any two integers of different colors has the third color?

Alice has eight coins. She knows that seven are genuine and have equal weight; one is counterfeit and is either heavier or lighter. Bob has a balance which shows which pan is heavier, not the weight difference. Before each weighing Alice pays Bob one of her remaining coins. If it is genuine, he reports the true outcome; if it is counterfeit, he reports an arbitrary outcome. Can Alice guarantee identifying five genuine coins without giving any of those five to Bob?