IMC 2015

360°

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

The whyOne 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°

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

The whyA 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

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

The whyThey 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

HintThink about what 1 is missing that 2 has.

The whyThe 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

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

The whyFaces, 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

HintUndo the averaging: instead of dividing, do the opposite.

The whyMean 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

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

The whyWherever 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

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

The whyThe 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

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

The why5 × 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

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

The whyWith 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

HintAverage the two parallel sides first.

The whyArea = 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

HintHalf base times height — one of those factors is shared.

The whyIf 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.