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Stage 4 · Geometry

10 problems · AMC 12 late, AIME early · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Of a triangle with vertices $A, B, C$ we know that $AB = 5$, $BC = 4$ and $AC = AM$, where $M$ is the midpoint of side $BC$. What is the length of side $AC$?

   1. A4
   2. B$\sqrt{17}$
   3. C$3\sqrt{2}$
   4. D$2\sqrt{5}$
   5. E$\sqrt{21}$

   Italian Mathematical Olympiad - February Round

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2. A polygon is regular if all its sides and angles are the same. Find the measure of each angle in a regular dodecagon (12-sided polygon).

   Berkeley Math Circle: Monthly Contest 5

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3. The area of a trapezoid is three times that of an equilateral triangle. If the heights of the trapezoid and the triangle are both equal to $8 \sqrt{3}$, what is the length of the median of the trapezoid?

   Tenth Philippine Mathematical Olympiad

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4. Compute $\arctan \left(\tan 65^{\circ}-2 \tan 40^{\circ}\right)$. (Express your answer in degrees as an angle between $0^{\circ}$ and $180^{\circ}$.)

   Annual Harvard-MIT Mathematics Tournament

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5. A parallelogram has 2 sides of length 20 and 15. Given that its area is a positive integer, find the minimum possible area of the parallelogram.

   HMMT November

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6. A regular decagon $A_{0} A_{1} A_{2} \cdots A_{9}$ is given in the plane. Compute $\angle A_{0} A_{3} A_{7}$ in degrees.

   HMMT February

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7. Let $L O V E R$ be a convex pentagon such that $L O V E$ is a rectangle. Given that $O V=20$ and $L O=V E=R E=R L=23$, compute the radius of the circle passing through $R, O$, and $V$.

   HMMT November

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8. Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length $\frac{1}{2}$ and edges parallel to those of the unit cube that contains all three points?

   HMMT February 2019

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9. Let $A B C D$ be an isosceles trapezoid with $A D=B C=255$ and $A B=128$. Let $M$ be the midpoint of $C D$ and let $N$ be the foot of the perpendicular from $A$ to $C D$. If $\angle M B C=90^{\circ}$, compute $\tan \angle N B M$.

   HMMT November

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10. Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\sqrt{19}$. Find the distance from the ball's point of first contact with a wall to the nearest vertex.

    HMMT November

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Answer key — Stage 4 · Geometry

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

1. $B$ open
2. $150°$ open
3. $24$ open
4. $25°$ open
5. $1$ open
6. $54^{\circ}$ open
7. $23$ open
8. $1/8$ open
9. $\frac{120}{353}$ open
10. $2$ open

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.

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