MathDB

10

Part of 2008 AIME Problems

Problems(2)

Isosceles Trapezoid

Source: AIME 2008I Problem 10

3/23/2008
Let ABCD ABCD be an isosceles trapezoid with AD‾∥BC‾ \overline{AD}\parallel{}\overline{BC} whose angle at the longer base AD‾ \overline{AD} is π3 \dfrac{\pi}{3}. The diagonals have length 1021 10\sqrt {21}, and point E E is at distances 107 10\sqrt {7} and 307 30\sqrt {7} from vertices A A and D D, respectively. Let F F be the foot of the altitude from C C to AD‾ \overline{AD}. The distance EF EF can be expressed in the form mn m\sqrt {n}, where m m and n n are positive integers and n n is not divisible by the square of any prime. Find m \plus{} n.
geometrytrapezoidinequalitiestrigonometryquadraticsAMCAIME
Growing Path

Source: AIME 2008II Problem 10

4/3/2008
The diagram below shows a 4×4 4\times4 rectangular array of points, each of which is 1 1 unit away from its nearest neighbors. [asy]unitsize(0.25inch); defaultpen(linewidth(0.7));
int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j));[/asy]Define a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let m m be the maximum possible number of points in a growing path, and let r r be the number of growing paths consisting of exactly m m points. Find mr mr.
analytic geometryAMC