Mathematics Olympiad Preparation

The holes move with the disk while the outer numbers remain fixed. Shift every hole by the stated number of sectors clockwise and add the revealed numbers.

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.

Try alternating two colours while moving around the cycle. If the final route and the first route receive different colours, two are enough. If they match, one route requires a third colour.

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.

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.

Each lamp only needs to be considered pressed or not pressed, because two presses cancel. Check the possibilities systematically and track how many times every lamp is toggled. Choose the shortest successful option.

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.

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.

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.

List numbers giving the first remainder and test them in increasing order against the second condition. The first number satisfying both conditions is the answer.

Trace the beam one cell at a time. A / mirror swaps right with up and left with down; a \ mirror swaps right with down and left with up. Stop when the beam crosses the outer frame.

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.

Divide the sum of the first two areas by their combined width to obtain the common height. Then divide the sum of the second and third areas by this height.

Test each person as the truth-teller. That person’s statement must be true and the other two statements must be false. Only one choice satisfies all three conditions.

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.

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.

Trace the white cells from an end cell. Record how many neighbours each cell has and where turns occur. Only one option keeps the same connection structure under rotation or reflection.