Mathematics Olympiad Preparation

Count only cell edges touching an empty cell or the outside of the grid. A shared edge between two orange cells lies inside the shape and is not part of its perimeter.

Find the length of one repeating block. Divide the position number by the block length and use the remainder to locate the footprint. A remainder of 0 means the last footprint in the block.

An even number ends in an even digit. A number is divisible by 3 when its digit sum is divisible by 3. Check the possible triples, then place the hundreds and tens digits in decreasing order.

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.

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.

Place one cube in every coloured top-view cell. Then, in each column, raise one selected stack to the height required by the front view. Other occupied cells in that column may remain one cube high.

Check which edges of the white space are reached by red paths. The correct tile must have a path at every one of these edges and no path at the remaining edges.

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.

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.

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.

Check every cell. Count it when it is coloured on A or B, but not on both. Cells coloured on both sheets and cells empty on both do not count.

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.

Add the three pair totals. Every parcel is counted twice, so divide by 2 to obtain the total mass of all three. Subtract the pair that does not contain the requested symbol.

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.

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.

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

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.

The top face is the same in both views. The four faces adjacent to it appear in pairs across the two diagrams. The only remaining symbol must lie on the opposite face.