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. Beside a straight edge a pentagon leaves 180 − 108 = 72°, so pentagons in a row along a line leave 72° gaps.

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.

Subtract one equation from the other

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.