MathDB

MBMT Geometry Rounds

Part of Montgomery Blair

Subcontests

(7)
2023
1

2023 MBMT Geometry Round - Montgomery Blair Math Tournament

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names
B1. If the values of two angles in a triangle are 6060 and 7575 degrees respectively, what is the measure of the third angle?
B2. Square ABCDABCD has side length 11. What is the area of triangle ABCABC?
B3 / G1. An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is 88, what is the square’s side length?
B4 / G2. What is the maximum possible number of sides and diagonals of equal length in a quadrilateral?
B5. A square of side length 44 is put within a circle such that all 44 corners lie on the circle. What is the diameter of the circle?
B6 / G3. Patrick is rafting directly across a river 2020 meters across at a speed of 55 m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of 1212 m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled?
B7 / G4. Quadrilateral ABCDABCD has side lengths AB=7AB = 7, BC=15BC = 15, CD=20CD = 20, and DA=24DA = 24. It has a diagonal length of BD=25BD = 25. Find the measure, in degrees, of the sum of angles ABCABC and ADCADC.
B8 / G5. What is the largest PP such that any rectangle inscribed in an equilateral triangle of side length 11 has a perimeter of at least PP?
G6. A circle is inscribed in an equilateral triangle with side length ss. Points AA,BB,CC,DD,EE,FF lie on the triangle such that line segments ABAB, CDCD, and EFEF are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon ABCDEF=932ABCDEF = \frac{9\sqrt3}{2} , find ss.
G7. Let △ABC\vartriangle ABC be such that ∠A=105o\angle A = 105^o, ∠B=45o\angle B = 45^o, ∠C=30o\angle C = 30^o. Let MM be the midpoint of ACAC. What is ∠MBC\angle MBC?
G8. Points AA, BB, and CC lie on a circle centered at OO with radius 1010. Let the circumcenter of △AOC\vartriangle AOC be PP. If AB=16AB = 16, find the minimum value of PBPB. The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.

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2022
1

2022 MBMT Geometry Round - Montgomery Blair Math Tournament

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
D1. A Giant Hopper is 200200 meters away from you. It can hop 5050 meters. How many hops would it take for it to reach you?
D2. A rope of length 66 is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges?
D3 / Z1. Point EE is on side ABAB of rectangle ABCDABCD. Find the area of triangle ECDECD divided by the area of rectangle ABCDABCD.
D4 / Z2. Garb and Grunt have two rectangular pastures of area 3030. Garb notices that his has a side length of 33, while Grunt’s has a side length of 55. What’s the positive difference between the perimeters of their pastures?
D5. Let points AA and BB be on a circle with radius 66 and center OO. If ∠AOB=90o\angle AOB = 90^o, find the area of triangle AOBAOB.
D6 / Z3. A scalene triangle (the 33 side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle?
D7. Square ABCDABCD has side length 66. If triangle ABEABE has area 99, find the sum of all possible values of the distance from EE to line CDCD.
D8 / Z4. Let point EE be on side AB‾\overline{AB} of square ABCDABCD with side length 22. Given DE=BC+BEDE = BC+BE, find BEBE.
Z5. The two diagonals of rectangle ABCDABCD meet at point EE. If ∠AEB=2∠BEC\angle AEB = 2\angle BEC, and BC=1BC = 1, find the area of rectangle ABCDABCD.
Z6. In △ABC\vartriangle ABC, let DD be the foot of the altitude from AA to BCBC. Additionally, let XX be the intersection of the angle bisector of ∠ACB\angle ACB and ADAD. If BD=AC=2AX=6BD = AC = 2AX = 6, find the area of ABCABC.
Z7. Let △ABC\vartriangle ABC have ∠ABC=40o\angle ABC = 40^o. Let DD and EE be on AB‾\overline{AB} and AC‾\overline{AC} respectively such that DE is parallel to BC‾\overline{BC}, and the circle passing through points DD, EE, and CC is tangent to AB‾\overline{AB}. If the center of the circle is OO, find ∠DOE\angle DOE.
Z8. Consider △ABC\vartriangle ABC with AB=3AB = 3, BC=4BC = 4, and AC=5AC = 5. Let DD be a point of ACAC other than AA for which BD=3BD = 3, and EE be a point on BCBC such that ∠BDE=90o\angle BDE = 90^o. Find ECEC.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2019
1

2019 MBMT Geometry Round - Montgomery Blair Math Tournament

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
D1. Triangle ABCABC has AB=3AB = 3, BC=4BC = 4, and ∠B=90o\angle B = 90^o. Find the area of triangle ABCABC.
D2 / L1. Let ABCDEFABCDEF be a regular hexagon. Given that AD=5AD = 5, find ABAB.
D3. Caroline glues two pentagonal pyramids to the top and bottom of a pentagonal prism so that the pentagonal faces coincide. How many edges does Caroline’s figure have?
D4 / L3. The hour hand of a clock is 66 inches long, and the minute hand is 1010 inches long. Find the area of the region swept out by the hands from 8:458:45 AM to 9:159:15 AM of a single day, in square inches.
D5 / L2. Circles AA, BB, and CC are all externally tangent, with radii 11, 1010, and 100100, respectively. What is the radius of the smallest circle entirely containing all three circles?
D6. Four parallel lines are drawn such that they are equally spaced and pass through the four vertices of a unit square. Find the distance between any two consecutive lines.
D7 / L4. In rectangle ABCDABCD, AB=2AB = 2 and AD>ABAD > AB. Two quarter circles are drawn inside of ABCDABCD with centers at AA and CC that pass through BB and DD, respectively. If these two quarter circles are tangent, find the area inside of ABCDABCD that is outside both of the quarter circles.
D8 / L6. Triangle ABCABC is equilateral. A circle passes through AA and is tangent to side BCBC. It intersects sides AB and ACAC again at EE and FF, respectively. If AE=10AE = 10 and AF=11AF = 11, find ABAB.
L5. Find the area of a triangle with side lengths 2\sqrt{2}, 58\sqrt{58}, and 2172\sqrt{17}.
L7. Triangle ABCABC has area 8080. Point DD is in the interior of △ABC\vartriangle ABC such that AD=6AD =6, BD=4BD = 4, CD=16CD = 16, and the area of △ADC=48\vartriangle ADC = 48. Determine the area of △ADB\vartriangle ADB.
L8. Given two points AA and BB in the plane with AB=1AB = 1, define f(C)f(C) to be the circumcenter of triangle ABCABC, if it exists. Find the number of points XX so that f2019(X)=Xf^{2019}(X) = X.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2018
1

2018 MBMT Geometry Round - Montgomery Blair Math Tournament

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
C1. A circle has circumference 6π6\pi. Find the area of this circle.
C2 / G2. Points AA, BB, and CC are on a line such that AB=6AB = 6 and BC=11BC = 11. Find all possible values of ACAC.
C3. A trapezoid has area 8484 and one base of length 55. If the height is 1212, what is the length of the other base?
C4 / G1. 2727 cubes of side length 1 are arranged to form a 3×3×33 \times 3 \times 3 cube. If the corner 1×1×11 \times 1 \times 1 cubes are removed, what fraction of the volume of the big cube is left?
C5. There is a 5050-foot tall wall and a 300300-foot tall guard tower 5050 feet from the wall. What is the minimum aa such that a flat “XX” drawn on the ground aa feet from the side of the wall opposite the guard tower is visible from the top of the guard tower?
C6. Steven’s pizzeria makes pizzas in the shape of equilateral triangles. If a pizza with side length 8 inches will feed 2 people, how many people will a pizza of side length of 16 inches feed?
C7 / G3. Consider rectangle ABCDABCD, with 1=AB<BC1 = AB < BC. The angle bisector of ∠DAB\angle DAB intersects BC‾\overline{BC} at EE and DC‾\overline{DC} at FF. If FE=FDFE = FD, find BCBC.
C8 / G6. △ABC\vartriangle ABC. is a right triangle with ∠A=90o\angle A = 90^o. Square ADEFADEF is drawn, with DD on AB‾\overline{AB}, FF on AC‾\overline{AC}, and EE inside △ABC\vartriangle ABC. Point GG is chosen on BC‾\overline{BC} such that EGEG is perpendicular to BCBC. Additionally, DE=EGDE = EG. Given that ∠C=20o\angle C = 20^o, find the measure of ∠BEG\angle BEG.
G4. Consider a lamp in the shape of a hollow cylinder with the circular faces removed with height 4848 cm and radius 77 cm. A point source of light is situated at the center of the lamp. The lamp is held so that the bottom of the lamp is at a height 4848 cm above an infinite flat sheet of paper. What is the area of the illuminated region on the flat sheet of paper, in cm2cm^2? https://cdn.artofproblemsolving.com/attachments/c/6/6e5497a67ae5ff5a7bff7007834a4271ce3ca7.png
G5. There exist two triangles ABCABC such that AB=13AB = 13, BC=122BC = 12\sqrt2, and ∠C=45o\angle C = 45^o. Find the positive difference between their areas.
G7. Let ABCABC be an equilateral triangle with side length 22. Let the circle with diameter ABAB be Γ\Gamma. Consider the two tangents from CC to Γ\Gamma, and let the tangency point closer to AA be DD. Find the area of ∠CAD\angle CAD.
G8. Let ABCABC be a triangle with ∠A=60o\angle A = 60^o, AB=37AB = 37, AC=41AC = 41. Let HH and OO be the orthocenter and circumcenter of ABCABC, respectively. Find OHOH. The orthocenter of a triangle is the intersection point of the three altitudes. The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2017
1

2017 MBMT Geometry Round - Montgomery Blair Math Tournament

[hide=R stands for Ramanujan, P stands for Pascal]they had two problem sets under those two names
R1. What is the distance between the points (6,0)(6, 0) and (−2,0)(-2, 0)?
R2 / P1. Angle XX has a degree measure of 3535 degrees. What is the supplement of the complement of angle XX? The complement of an angle is 9090 degrees minus the angle measure. The supplement of an angle is 180180 degrees minus the angle measure.
R3. A cube has a volume of 729729. What is the side length of the cube?
R4 / P2. A car that always travels in a straight line starts at the origin and goes towards the point (8,12)(8, 12). The car stops halfway on its path, turns around, and returns back towards the origin. The car again stops halfway on its return. What are the car’s final coordinates?
R5. A full, cylindrical soup can has a height of 1616 and a circular base of radius 33. All the soup in the can is used to fill a hemispherical bowl to its brim. What is the radius of the bowl?
R6. In square ABCDABCD, the numerical value of the length of the diagonal is three times the numerical value of the area of the square. What is the side length of the square?
R7. Consider triangle ABCABC with AB=3AB = 3, BC=4BC = 4, and AC=5AC = 5. The altitude from BB to ACAC intersects ACAC at HH. Compute BHBH.
R8. Mary shoots 55 darts at a square with side length 22. Let xx be equal to the shortest distance between any pair of her darts. What is the maximum possible value of xx?
P3. Let ABCABC be an isosceles triangle such that AB=BCAB = BC and all of its angles have integer degree measures. Two lines, ℓ1\ell_1 and ℓ2\ell_2, trisect ∠ABC\angle ABC. ℓ1\ell_1 and ℓ2\ell_2 intersect ACAC at points DD and EE respectively, such that DD is between AA and EE. What is the smallest possible integer degree measure of ∠BDC\angle BDC?
P4. In rectangle ABCDABCD, AB=9AB = 9 and BC=8BC = 8. WW, XX, YY , and ZZ are on sides ABAB, BCBC, CDCD, and DADA, respectively, such that AW=2WBAW = 2WB, CX=3BXCX = 3BX, CY=2DYCY = 2DY , and AZ=DZAZ = DZ. If WYWY and XZXZ intersect at OO, find the area of OWBXOWBX.
P5. Consider a regular nn-gon with vertices A1A2...AnA_1A_2...A_n. Find the smallest value of nn so that there exist positive integers i,j,k≤ni, j, k \le n with ∠AiAjAk=34o5\angle A_iA_jA_k = \frac{34^o}{5}.
P6. In right triangle ABCABC with ∠A=90o\angle A = 90^o and AB<ACAB < AC, DD is the foot of the altitude from AA to BCBC, and MM is the midpoint of BCBC. Given that AM=13AM = 13 and AD=5AD = 5, what is ABAC\frac{AB}{AC} ?
P7. An ant is on the circumference of the base of a cone with radius 22 and slant height 66. It crawls to the vertex of the cone XX in an infinite series of steps. In each step, if the ant is at a point PP, it crawls along the shortest path on the exterior of the cone to a point QQ on the opposite side of the cone such that 2QX=PX2QX = PX. What is the total distance that the ant travels along the exterior of the cone?
P8. There is an infinite checkerboard with each square having side length 22. If a circle with radius 11 is dropped randomly on the checkerboard, what is the probability that the circle lies inside of exactly 33 squares?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2016
1

2016 MBMT Geometry Round - Montgomery Blair Math Tournament

[hide=E stands for Euclid, L stands for Lobachevsky]they had two problem sets under those two names
E1. What is the perimeter of a rectangle if its area is 2424 and one side length is 66?
E2. John moves 3 miles south, then 22 miles west, then 77 miles north, and then 55 miles east. What is the length of the shortest path, in miles, from John's current position to his original position?
E3. An equilateral triangle ABCABC is drawn with side length 22. The midpoints of sides ABAB, BCBC, and CACA are constructed, and are connected to form a triangle. What is the perimeter of the newly formed triangle?
E4. Let triangle ABCABC have sides AB=74AB = 74 and AC=5AC = 5. What is the sum of all possible integral side lengths of BC?
E5. What is the area of quadrilateral ABCDABCD on the coordinate plane with A(1,0)A(1, 0), B(0,1)B(0, 1), C(1,3)C(1, 3), and D(5,2)D(5, 2)?
E6 / L1. Let ABCDABCD be a square with side length 3030. A circle centered at the center of ABCDABCD with diameter 3434 is drawn. Let EE and FF be the points at which the circle intersects side ABAB. What is EFEF?
E7 / L2. What is the area of the quadrilateral bounded by ∣2x∣+∣3y∣=6|2x| + |3y| = 6?
E8. A circle OO with radius 22 has a regular hexagon inscribed in it. Upon the sides of the hexagon, equilateral triangles of side length 22 are erected outwards. Find the area of the union of these triangles and circle OO.
L3. Right triangle ABCABC has hypotenuse ABAB. Altitude CDCD divides ABAB into segments ADAD and DBDB, with AD=20AD = 20 and DB=16DB = 16. What is the area of triangle ABCABC?
L4. Circle OO has chord ABAB. Extend ABAB past BB to a point CC. A ray from CC is drawn, and this ray intersects circle OO. Let point DD be the point of intersection of the ray and the circle that is closest to point CC. Given AB=20AB = 20, BC=16BC = 16, and OA=2016OA = \frac{201}{6} , find the longest possible length of CDCD.
L5. Consider a circular cone with vertex AA. The cone's height is 44 and the radius of its base is 33. Inscribe a sphere inside the cone. Find the ratio of the volume of the cone to the volume of the sphere.
L6. A disk of radius 12\frac12 is randomly placed on the coordinate plane. What is the probability that it contains a lattice point (point with integer coordinates)?
L7. Let ABCABC be an equilateral triangle of side length 22. Let DD be the midpoint of BCBC, and let PP be a variable point on ACAC. By moving PP along ACAC, what is the minimum perimeter of triangle BDPBDP?
L8. Let ABCDABCD be a rectangle with AB=8AB = 8 and BC=9BC = 9. Let DEFGDEFG be a rhombus, where GG is on line BCBC and AA is on line EFEF. If m∠EFG=30o,whatism\angle EFG = 30^o, what is DE$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2015
1

2015 MBMT Geometry Round - Montgomery Blair Math Tournament

[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.