MathDB

12

Part of 2007 AIME Problems

Problems(2)

Rotating Isosceles Triangle

Source: AIME I 2007 #12

3/15/2007
In isosceles triangle ABCABC, AA is located at the origin and BB is located at (20,0)(20, 0). Point CC is in the first quadrant with AC=BCAC = BC and ∠BAC=75∘\angle BAC = 75^\circ. If △ABC\triangle ABC is rotated counterclockwise about point AA until the image of CC lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form p2+q3+r6+sp\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s where pp, qq, rr, ss are integers. Find (p−q+r−s)/2(p-q+r-s)/2.
rotationgeometryAMCAIME
Log of Geometric Sequence

Source: AIME II 2007 #12

3/29/2007
The increasing geometric sequence x0,x1,x2,…x_{0},x_{1},x_{2},\ldots consists entirely of integral powers of 3.3. Given that ∑n=07log⁡3(xn)=308and56≤log⁡3(∑n=07xn)≤57,\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57, find log⁡3(x14).\log_{3}(x_{14}).
calculusintegrationlogarithmsratioAMCUSA(J)MOUSAMO