IMC 2015

360°

提示Walk once round the shape: by the time you face the way you started, how far have you turned?

为什么One full turn, whatever the number of sides. A regular polygon with n sides therefore has an exterior angle of 360 ÷ n. The interior sum grows with n; the exterior sum never does.

108°

提示Exterior angle first: divide a full turn by five, then take that off a straight line.

为什么A regular pentagon has exterior angles of 72° and interior angles of 108° — a pair worth knowing cold. Stand one on a straight line with a side on the line and the next side rises at 180 − 108 = 72°, which is where most "pentagon on a line" angle chases start. Two pentagons cannot meet at such a vertex, though: 108° + 108° = 216° is more than the 180° a straight line has to give. So a row of pentagons on a line is spaced out, and the gap between two of them closes at 180 − 72 − 72 = 36° — the same 36° that three pentagons round a point leave over (360 − 324), which is why regular pentagons will not tile.

6

提示Which whole numbers, when squared, land between ten and ninety-nine?

为什么They are 16, 25, 36, 49, 64 and 81 — the squares of 4 to 9. A question about two-digit squares is a question about six numbers, so just list them.

two

提示Think about what 1 is missing that 2 has.

为什么The number 1 has only one factor, so by definition it is not prime — a convention chosen so that every number has a single prime factorisation. Competition questions lean on this: an expression that comes out as 1 does not count as prime, and a sequence that must begin with a prime cannot begin with 1.

four; twelve; five

提示Three solids, three different counts — picture each one in turn.

为什么Faces, edges and vertices are the three counts every solids question turns on: the tetrahedron has 4 faces, 4 vertices and 6 edges; the cube 6, 8 and 12; the square-based pyramid 5, 5 and 8. For every one of them, faces + vertices − edges = 2.

Their total: mean × count

提示Undo the averaging: instead of dividing, do the opposite.

为什么Mean questions are almost always total questions in disguise. Once the total is fixed, "what is the largest one of them could be?" becomes "make the others as small as the rules allow" — and if they must be different positive whole numbers, the smallest others are 1, 2, 3 and so on.

Exactly half

提示Drop a vertical line from the top vertex and look at the two pieces separately.

为什么Wherever the top vertex sits along the opposite side, the triangle is half the rectangle: each piece of the rectangle is cut in two by a diagonal. So a row of shaded triangles across a rectangle shades exactly half of it.

Multiply the choices together

提示Independent decisions do not add up; think of a grid with one row per first choice.

为什么The product rule: p options for the starter and q for the main give p × q meals. It extends to any number of positions — and a position locked in by earlier choices contributes a factor of 1, not another full set of options.

120

提示Every one of them could come first — and that is only the start.

为什么5 × 4 × 3 × 2 × 1 = 120, written 5!. n! grows fast: 6! = 720, 7! = 5040 — so a question about arranging a handful of things is usually about hundreds or thousands of cases, never a few.

Add or subtract the equations, scaling as needed

提示You cannot pin down each unknown, but combining the two lines often produces exactly the mix wanted.

为什么With fewer equations than unknowns you cannot solve for everything — and you are not asked to. Subtracting (or adding) the equations, then scaling, builds the combination the question wants. Check the coefficients you are aiming for and work towards them rather than towards single values.

half

提示Average the two parallel sides first.

为什么Area = ½(a + b)h. An L-shape is a rectangle minus a rectangle; a trapezium is the average of its parallel sides times the distance between them. Shaded regions inside a square are usually quickest by subtraction: the whole square minus the pieces you can name.

Areas in the ratio of their bases

提示Half base times height — one of those factors is shared.

为什么If two triangles have the same height, dividing one base by the other divides one area by the other. A point one third of the way along a side therefore cuts off a triangle worth one third of the whole, without any lengths or heights being calculated.

The decimal points

提示Pad the shorter figure with trailing zeros so every column matches.

为什么Once the points line up, every column holds digits of the same place value, so you can add or subtract straight down exactly as with whole numbers. A quick chain of additions and subtractions across shrinking place values (tenths, then hundredths, then thousandths) can often be tracked digit by digit without writing out the full column sum.

Round both to nearby 'nice' numbers

提示Swap the messy figures for tidy multiples that divide cleanly into each other.

为什么Rounding both the top and bottom of a division to numbers that share factors — for example treating 2977 ÷ 29 as roughly 3000 ÷ 30 — usually gives an answer close to the exact one, because the two roundings partly cancel. It turns an awkward division into one you can do in your head, which is exactly what an approximate 'how many per minute/per unit' question is really testing.

denominator

提示Find a shared multiple that every bottom number divides into exactly.

为什么A common denominator (often the lowest common multiple of the individual denominators) lets every fraction be rewritten with matching bottoms, so the numerators can simply be added. Skipping this step and combining fractions some other way — say, adding numerators and denominators separately — gives a wrong, and usually much smaller, result.

a to the power mn

提示Don't add the two indices — multiply them instead.

为什么This is why a number can be a hidden square even when it is not written as a familiar one: any number of the form (a to the power m) to the power 2, for any whole number m, is automatically a square, because doubling the outer index just multiplies the whole exponent by 2. Spotting a number written this way is often faster than trying to recognise it from its ordinary value.

Equilateral triangles (same side length as the hexagon)

提示Join the centre to every vertex and look at the angle each wedge makes.

为什么The angle at the centre of each wedge is a full turn shared six ways, 360° ÷ 6 = 60°. The two sides running from the centre to the hexagon's corners are equal (a hexagon happens to have that radius-equals-side property), so the third angle must also be 60° — giving an equilateral triangle. That is the quickest route into comparing a regular hexagon's area with any other shape built from equilateral triangles of the same side length.

4

提示Write the odd number as 2k + 1, multiply through by 4, then split off a full multiple of 8.

为什么4 × (2k + 1) = 8k + 4, which is a multiple of 8 plus exactly 4 more — so the remainder is always 4, whatever odd number is chosen. This is the standard move for remainder questions built from a product: pull out every factor of 2 you can find, then look at what is left over.

No — check each shared edge individually

提示Think about where the board has a jagged or non-rectangular outline.

为什么On a full rectangular grid, colour strictly alternates with every step, so after an even number of moves you are guaranteed back on the starting colour. Cut the grid into an irregular shape — an L, a cross, or anything with a 'dent' in its outline — and two squares of the same colour can end up sharing an edge at the join, breaking that guarantee. A problem that looks like a simple parity puzzle can hide exactly this kind of irregular boundary, so sketch the actual shape rather than trusting the general rule.

They are all equal, to the mean of the first two terms

提示Adding a new value that already matches the running average never changes that average.

为什么Call the first two terms a and b. The third term works out to be (a + b) ÷ 2, and once that value is in the list, adding another term equal to the current mean can never shift the mean — so the fourth, fifth and every later term is stuck at that same (a + b) ÷ 2, no matter how far down the sequence you look. So being told one distant term of such a sequence actually tells you the mean of the very first two.

Creates a smaller triangle similar to the original one

提示The angles stay identical; only the overall scale changes.

为什么Because the two triangles are similar, every length inside the smaller one scales by the same factor as its base compares with the original base — so an unknown height, base or side in the smaller triangle can be found from one ratio, without measuring the whole figure directly. This is the standard route into "strip" problems, where a shape is sliced into equal-width bands and each slice needs its own dimensions worked out.

Twice (once from each adjoining face)

提示Picture two neighbouring panels meeting along a seam — that seam belongs to both of them.

为什么So to count the actual distinct edges (or joins, seams, boundaries — anything shared between two neighbours), add up each piece’s own edge-count and divide by 2. It is the same idea behind counting handshakes at a party: if everyone reports how many hands they shook, the total is double the real number of handshakes, because each handshake got reported by both people involved in it.

overlap

提示Adding them separately counts the shared part twice — take one copy of it back off.

为什么This is the addition rule for overlapping regions, the same idea behind a two-circle Venn diagram: total covered = first area + second area − overlap. Miss the subtraction and the shared part gets double-counted, which is exactly the trap in any "total shaded area" question built from two overlapping shapes.

√2 times that step

提示Think of it as the diagonal cutting across a square, then relate the diagonal to one side using Pythagoras.

为什么This follows straight from Pythagoras: if both the horizontal and vertical legs equal d, the hypotenuse is the square root of (d² + d²), which simplifies to d√2. It is the fastest way to find the length of a diagonal segment built from equal horizontal and vertical steps, including diagonals threaded across a grid of identical small squares, without measuring anything directly.

Check which value fits every constraint from the diagram

提示One of the two answers could make a stated measurement negative or impossibly small — rule it out.

为什么Quadratics often produce two mathematically valid roots, but geometry adds real-world constraints: a length can't be negative, a point has to lie inside the shape, or one value must exceed some other given measurement. Always test both roots against the diagram's conditions before choosing an answer, rather than assuming the 'nicer' number is correct.