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2015 MBMT Geometry Round - Montgomery Blair Math Tournament

Source:

October 14, 2023
geometryMBMT

Problem Statement

[hide=F stands for Fermat, E stands for Euler]they had two problem sets under those two names
F1. A circle has area π\pi. Find the circumference of the circle.
F2. In triangle ABCABC, AB=5AB = 5, BC=12BC = 12, and ∠B=90o\angle B = 90^o. Compute ACAC.
F3 / E1. A square has area 20152015. Find the length of the square's diagonal.
F4. I have two cylindrical candles. The first candle has diameter 11 and height 11. The second candle has diameter 22 and height 22. Both candles are lit at 1:001:00 PM and both burn at the same constant rate (volume per time period). The first candle burns out at 1:501:50 PM. When does the second candle burn out? Specify AM or PM.
F5 / E2. In triangle ABCABC, BCBC has length 1212, the altitude from AA to BCBC has length 66, and the altitude from CC to ABAB has length 88. Compute the length of ABAB.
F6 / E3. Let ABCABC be an isosceles triangle with base ACAC. Suppose that there exists a point DD on side ABAB such that AC=CD=BDAC = CD = BD. Find the degree measure of ∠ABC\angle ABC.
F7 / E6. In concave quadrilateral ABCDABCD, ∠ABC=60o\angle ABC = 60^o and ∠ADC=240o\angle ADC = 240^o. If AD=CD=4AD = CD = 4, compute BDBD.
F8 / E7. A circle of radius 55 is inscribed in an isosceles trapezoid with legs of length 1414. Compute the area of the trapezoid.
E4. The Egyptian goddess Isil has a staff consisting of a pole with a circle on top. The length of the pole is 3232 inches, and the tangent segment from the bottom of the pole to the circle is 4040 inches. Find the radius of the circle, in inches. https://cdn.artofproblemsolving.com/attachments/5/b/01ea1819aa58c4bde105b9b885f658b3823494.png
E5. The two concentric circles shown below have radii 11 and 22. A chord of the larger circle that is tangent to the smaller circle is drawn. Find the area of the shaded region, bounded by the chord and the larger circle. https://cdn.artofproblemsolving.com/attachments/e/5/425735d2717548552fda8363c201dc8043da13.png
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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