Mathematics Olympiad Preparation

Both allowed changes share a common divisor. Therefore the starting number and every reachable result have the same remainder modulo that divisor. Find the option with a different remainder.

A number is divisible by 6 when it is divisible by both 2 and 3. Check that the digit sum is divisible by 3, reserve an even final digit, and arrange the remaining digits as large as possible.

Apply each clue to every remaining digit order. Reject a code if its numbers of correct-place and wrong-place matches differ from the clue. One code remains after all clues.

Start at the upper container and mark every node reachable through an open channel. Continue only along open channels leaving marked nodes. Count reached lower containers, not the number of different paths.

Count one top face for every non-empty stack. On each of the four sides, add only the part of a stack higher than its neighbour; outside the construction, the neighbouring height is zero.

Add the train speeds and divide the distance by this sum to find the meeting time. Multiply that time by the speed of the train leaving A.

A hole away from a fold is reflected across that fold. A hole on neither fold creates four images, a hole on one fold creates two, and a hole at the intersection creates one. Add the contributions of all punches.

Imagine folding the four faces in a row into a band around the cube. The first and third band faces become opposite, as do the second and fourth. The faces above and below the band are also opposite.

Turn every statement into an arrow from the earlier finisher to the later one. Join the arrows into one order of six runners and select the fourth.

At each base position, the stack height cannot exceed either the corresponding front height or side height. Its maximum is therefore the smaller of those two values. Find this for all 9 cells and add them.