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Stage 3 · Geometry

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

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

1. What is the smallest possible value of $n$ if a solid cube is made of white plastic and has dimensions $n \times n \times n$, the six faces of the cube are completely covered with gold paint, the cube is then cut into $n^{3}$ cubes, each of which has dimensions $1 \times 1 \times 1$, and the number of $1 \times 1 \times 1$ cubes with 0 gold faces is strictly greater than the number of $1 \times 1 \times 1$ cubes with exactly 1 gold face?

   CEMC Pascal

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2. Aaron has 144 identical cubes, each with edge length 1 cm. He uses all of the cubes to construct a solid rectangular prism, which he places on a flat table. If the perimeter of the base of the prism is 20 cm, what is the sum of all possible heights of the prism?

   CEMC Fermat

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3. Suppose for a quadrilateral $ABCD$, $\angle DAB = 90^\circ$, $\angle ABC = \angle BCD = 60^\circ$. If $AB = 5$ and $CD = 4$, what is the value of $BC$? Here for a line segment $XY$, its length is also denoted by $XY$.

   

   Japan Junior Mathematical Olympiad

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4. In a rhombus $P Q R S$ with $P Q=Q R=R S=S P=S Q=6$ and $P T=R T=14$, what is the length of $S T$?

   CEMC Fermat

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5. A cube has edge length 4 m. One end of a rope of length 5 m is anchored to the centre of the top face of the cube. What is the integer formed by the rightmost two digits of the integer closest to 100 times the area of the surface of the cube that can be reached by the other end of the rope?

   CEMC Fermat

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6. A rectangular piece of paper $P Q R S$ has $P Q=20$ and $Q R=15$. The piece of paper is glued flat on the surface of a large cube so that $Q$ and $S$ are at vertices of the cube. What is the shortest distance from $P$ to $R$, as measured through the cube?

   CEMC Fermat

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7. A rectangle with dimensions 100 cm by 150 cm is tilted so that one corner is 20 cm above a horizontal line, as shown. To the nearest centimetre, the height of vertex $Z$ above the horizontal line is $(100+x) \mathrm{cm}$. What is the value of $x$?

   CEMC Fermat

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8. Quadrilateral $ABCD$ has $\angle BCD=\angle DAB=90^{\circ}$. The perimeter of $ABCD$ is 224 and its area is 2205. One side of $ABCD$ has length 7. The remaining three sides have integer lengths. What is the integer formed by the rightmost two digits of the sum of the squares of the side lengths of $ABCD$?

   CEMC Fermat

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9. A triangle has sides of length $9$, $40$, and $41$. What is its area?

   Annual Harvard-MIT November Tournament

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10. Three equal circles are inscribed into the equilateral triangle with the sides of length $1$ (see figure). What is the length of the radii of the circles?
    

    1. A$\frac{1}{3}$
    2. B$\frac{1}{4}$
    3. C$\frac{\sqrt{3}}{4}$
    4. D$\frac{\sqrt{3}+1}{4}$
    5. E$\frac{\sqrt{3}-1}{4}$

    Slovenia — National Math Olympiad

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Answer key — Stage 3 · Geometry

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

1. $9$ open
2. $31$ open
3. $6$ open
4. $10$ open
5. $81$ open
6. $18.4$ open
7. $67$ open
8. $60$ open
9. $180$ open
10. $E$ 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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