MathDB

18

Part of 2022 AMC 10

Problems(2)

Coordinate Plane

Source: 2022 AMC 10A P18 / 2022 AMC 12A P18

11/11/2022
Let TkT_k be the transformation of the coordinate plane that first rotates the plane kk degrees counterclockwise around the origin and then reflects the plane across the yy-axis. What is the least positive integer nn such that performing the sequence of transformations transformations T1,T2,T3,…,TnT_1, T_2, T_3, \dots, T_n returns the point (1,0)(1,0) back to itself?
<spanclass=′latex−bold′>(A)</span>359<spanclass=′latex−bold′>(B)</span>360<spanclass=′latex−bold′>(C)</span>719<spanclass=′latex−bold′>(D)</span>720<spanclass=′latex−bold′>(E)</span>721<span class='latex-bold'>(A) </span> 359 \qquad <span class='latex-bold'>(B) </span> 360\qquad <span class='latex-bold'>(C) </span> 719 \qquad <span class='latex-bold'>(D) </span> 720 \qquad <span class='latex-bold'>(E) </span> 721
AMCAMC 10AMC 122022 AMC 10a2022 AMC 12A2022 AMCtransformations
linear algebra time i guess

Source: 2022 AMC 10B Problem 18

11/17/2022
Consider systems of three linear equations with unknowns x,x, y,y, and z,z, \begin{align*} a_1 x + b_1 y + c_1 z = 0 \\ a_2 x + b_2 y + c_2 z = 0 \\ a_3 x + b_3 y + c_3 z = 0 \end{align*} where each of the coefficients is either 00 or 11 and the system has a solution other than x=y=z=0.x = y = z = 0. For example, one such system is {1x+1y+0z=0,0x+1y+1z=0,0x+0y+0z=0}\{1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0\} with a nonzero solution of {x,y,z}={1,−1,1}.\{x, y, z\} = \{1, -1, 1\}. How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)
<spanclass=′latex−bold′>(A)</span>302<spanclass=′latex−bold′>(B)</span>338<spanclass=′latex−bold′>(C)</span>340<spanclass=′latex−bold′>(D)</span>343<spanclass=′latex−bold′>(E)</span>344<span class='latex-bold'>(A) </span> 302 \qquad <span class='latex-bold'>(B) </span> 338 \qquad <span class='latex-bold'>(C) </span> 340 \qquad <span class='latex-bold'>(D) </span> 343 \qquad <span class='latex-bold'>(E) </span> 344
algebralinear equationAMCAMC 102022 AMC2022 AMC 10Bsystem of equations