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Stage 5 · Geometry

10 problems · AIME late · mathsolympiadprep.com

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

1. Inside square $A B C D$, a point $E$ is chosen so that triangle $D E C$ is equilateral. Find the measure of $\angle A E B$.

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2. In $\triangle A B C$, it is known that $A B=A C, D$ is the midpoint of side $B C$, $B E \perp A C$ at point $E, B E$ intersects $A D$ at point $P$. If $B P=3, P E=1$, then $A E=(\quad)$.

   1. A$\frac{\sqrt{6}}{2}$
   2. B$\sqrt{2}$
   3. C$\sqrt{3}$
   4. D$\sqrt{6}$

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3. In the tetrahedron $A B C D$, it is known that
   $$A B=A C=A D=D B=5, B C=3, C D=4 \text {. }$$

   Then the volume of the tetrahedron is $\qquad$ .

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4. Given that $F_{1}, F_{2}$ are the common foci of an ellipse and a hyperbola, and $\mathrm{P}$ is one of their common points, with $\angle F_{1} P F_{2}=60^{\circ}$, then the minimum value of the product of the eccentricities of the ellipse and the hyperbola is

   1. A$\frac{\sqrt{3}}{3}$
   2. B$\frac{\sqrt{3}}{2}$
   3. C1
   4. D$\sqrt{3}$

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5. We have two perpendicular lines - like the axes of an ellipse - and a tangent $(t)$ with its point of tangency $(T)$. Construct the endpoints of the ellipse's axes $(A, B, C, D)$! How many solutions are there?

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6. The lateral faces of a triangular pyramid are equal in area and form angles $\alpha, \beta$ and $\gamma$ with the base. Find the ratio of the radius of the sphere inscribed in this pyramid to the radius of the sphere that touches the base of the pyramid and the extensions of the lateral faces.

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7. On a straight line, points $A, B$, and $C$ are given. It is known that $A B=5$, and segment $A C$ is one and a half times longer than $B C$. Find the segments $A C$ and $B C$.

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8. In triangle $ABC$, the bisector $BD$ is drawn, and in triangles $ABD$ and $CBD$ - the bisectors $DE$ and $DF$ respectively. It turned out that $EF \parallel AC$. Find the angle $DEF$. (I. Rubanov)

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9. Let $A B C D E F$ be a regular hexagon with area 1. Consider all triangles whose vertices belong to the set $\{A, B, C, D, E, F\}$: what is the sum of their areas?

   1. A3
   2. B4
   3. C5
   4. D6
   5. E7

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10. A cylinder's total surface area is in the ratio of $7: 4$ to the area of its base. What is the surface area and volume of the cylinder if the slant height of a cone inscribed in the cylinder is $30 \mathrm{~cm}$?

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Answer key — Stage 5 · Geometry

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

1. $150$ open
2. $\sqrt{2}$ open
3. $5 \sqrt{3}$ open
4. $\frac{\sqrt{3}}{2}$ open
5. $1$ open
6. $\frac{3-\cos\alpha-\cos\beta-\cos\gamma}{3+\cos\alpha+\cos\beta+\cos\gamma}$ open
7. $BC=10,AC=15orBC=2,AC=3$ open
8. $45$ open
9. $6$ open
10. $F=1512\pi\mathrm{}^{2},K=7776\pi\mathrm{}^{3}$ 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.