Mathematics Olympiad Preparation

Each minute the distance between the robots decreases by the sum of their speeds. Divide the track length by this sum to find the time, then multiply by the first robot’s speed.

Find the difference and divide it by 2. Each moved apple reduces the first amount by one and increases the second by one, so the difference falls by two.

For every coloured cell, mark its reflection across the vertical axis, the horizontal axis and both axes together. Count only the newly required cells.

A week repeats every 7 days. Multiply the interval by the number of intervals, find the remainder after division by 7, and move forward that many weekdays.

Start with the largest packs. Reduce their number and check whether the remainder can be made from the other two sizes. Among exact combinations, choose the one using the fewest packs.

The first balance shows that one symbol weighs as much as two copies of the lightest. The second shows that the third symbol equals the sum of the first two masses, so it is the heaviest.

Compare the way the cells connect rather than the current orientation. Count how far the shape extends from its corners. Rotation or reflection does not change which cells share edges.

Test boxes A, B, C and D in turn. For each possibility, count the true statements. The answer is the only possibility making exactly one statement true.

One 90° turn moves the top to the right, right to bottom, bottom to left and left to top. Reduce the number of turns modulo 4 because four turns return the mosaic to its starting position.

Mark the start with 0. Mark all reachable neighbouring cells with 1, newly reached cells with 2, and continue layer by layer. The first number reaching the green station is the shortest distance.