Track / Stage 4 / 325 of 340 #585 of 1964

Problem 585

AMC 12 late, AIME early

Geometry Difficulty 5.0 Find the answer

5. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ and two planes $\alpha$ and $\beta$ . Plane $\alpha$ is perpendicular to the line $A_{1} C_{1}$ , and plane $\beta$ is parallel to the line $C D_{1}$ . Determine the smallest possible angle between the planes $\alpha$ and $\beta$ .

( 7 points)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Answer: $\frac{\pi}{6}$ .

Solution: First, let's prove one auxiliary statement. Let planes $\alpha$ and $\beta$ intersect along a line $l$ and form a dihedral angle $\varphi$ . OPEN REGIONAL INTERUNIVERSITY OLYMPIAD 2018-2019 MATHEMATICS (11th GRADE) INCLUSIVE STAGE VARIANT 2 (ANSWERS)

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.