Maths Olympiad Prep

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Stage 3 · Geometry

10 problems · AMC 10/12, early questions · mathsolympiadprep.com

The answer key prints on its own page at the end.

  1. An isosceles triangle has one side of length 33 and another side of length 88. The perimeter of this triangle is:

    1. A1414
    2. B1919
    3. C1111
    4. D1414 or 1919

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  2. The area of the triangle formed by the line x5+y2=1\frac{x}{5} + \frac{y}{2} = 1 and the coordinate axes is

    1. A2
    2. B5
    3. C7
    4. D10

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  3. A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:

    1. ALeast when the point is the center of gravity of the triangle\text{Least when the point is the center of gravity of the triangle}
    2. BGreater than the altitude of the triangle\text{Greater than the altitude of the triangle}
    3. CEqual to the altitude of the triangle\text{Equal to the altitude of the triangle}
    4. DOne-half the sum of the sides of the triangle\text{One-half the sum of the sides of the triangle}
    5. EGreatest when the point is the center of gravity\text{Greatest when the point is the center of gravity}

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  4. Given the parabola x2=2yx^{2}=2y and its focus coincides with one of the foci of the ellipse y2m+x22=1\frac {y^{2}}{m}+ \frac {x^{2}}{2}=1, then m=m=

    1. A11
    2. B22
    3. C33
    4. D94\frac {9}{4}

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  5. Given that FF is a focus of the hyperbola CC: y2−mx2=3m(m>0)y^{2}-mx^{2}=3m (m > 0), the distance from point FF to one asymptote of CC is:

    1. A3\sqrt {3}
    2. B2\sqrt {2}
    3. C22\dfrac { \sqrt {2}}{2}
    4. D33\dfrac { \sqrt {3}}{3}

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  6. Rectangle ABCDABCD is inscribed in a semicircle with diameter FE‾,\overline{FE}, as shown in the figure. Let DA=16,DA=16, and let FD=AE=9.FD=AE=9. What is the area of ABCD?ABCD?

    1. A240
    2. B248
    3. C256
    4. D264
    5. E272

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  7. On square ABCDABCD, point EE lies on side ADAD and point FF lies on side BCBC, so that BE=EF=FD=30BE=EF=FD=30. Find the area of the square ABCDABCD.

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  8. A line passing through the focus of the parabola C:y2=4xC: y^{2}=4x intersects the parabola at points A(x1,y1)A(x_{1},y_{1}) and B(x2,y2)B(x_{2},y_{2}). If x1+x2=9x_{1}+x_{2}=9, then ∣AB∣=|AB|= _____

    1. A1111
    2. B1010
    3. C66
    4. D44

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  9. Among the following sets of numbers, which one cannot constitute the side lengths of a right-angled triangle?

    1. A3, 4, 5
    2. B6, 8, 11
    3. C8, 15, 17
    4. D7, 24, 25

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  10. If the arc length corresponding to a central angle of 2 radians is 4 cm, then the area of the sector enclosed by this central angle is      cm2.

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Answer key — Stage 3 · Geometry

Worked solutions for every problem are on the site, one page per problem.

  1. BB open
  2. 55 open
  3. PA′+PB′+PC′=PA'+PB'+PC'= open
  4. DD open
  5. AA open
  6. 240240 open
  7. 810810 open
  8. AA open
  9. B:6,8,11B: 6, 8, 11 open
  10. 44 open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.