2004 Flanders Math Olympiad

Part of Flanders Math Olympiad

Subcontests

(4)

1

Nightmare beehive.

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

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}

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1

strange geometry

The author of this posting is : Peter VDD ____________________________________________________________________ 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?

1

2 Bags...

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.

1

flanders MO -- good way of buying stuff

A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably) While the salesmen isn't watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?