VWO 2004

The author of this posting is : Peter VDD

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most of us didn't really expect to get this, but here it goes (flanders mathematical olympiad 2004, today)

triangle with sides 501m, 668m, 835m

how many lines can be draws so that the line halves both area and circumference?

Two bags contain some numbers, and the total number of numbers is prime.

When we tranfer the number 170 from 1 bag to bag 2, the average in both bags increases by one.

If the total sum of all numbers is 2004, find the number of numbers.

Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex $T$ is on the perpendicular line through the center $O$ of the base of the prism (see figure). Let $s$ denote the side of the base, $h$ the height of the cell and $\theta$ the angle between the line $TO$ and $TV$ .

(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi. (b) the total surface area of the cell is given by the formula $6sh - \dfrac{9s^2}{2\tan\theta} + \dfrac{s^2 3\sqrt{3}}{2\sin\theta}$