MathDB

14

Part of 2021 AMC 10 Fall

Problems(2)

System of Equations with Absolute Values

Source: 2021 Fall AMC 10A 14

11/11/2021
How many ordered pairs (x,y)(x,y) of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9\\ (|x|+|y|-4)^2&=1\\ \end{align*} <spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 7<span class='latex-bold'>(A)</span>\: 1\qquad<span class='latex-bold'>(B)</span> \: 2\qquad<span class='latex-bold'>(C)</span> \: 3\qquad<span class='latex-bold'>(D)</span> \: 5\qquad<span class='latex-bold'>(E)</span> \: 7
AMCAMC 10AMC 10 Aalgebrasystem of equationsabsolute value
Coin toss bijection

Source: AMC 10B 2021 #14

11/17/2021
Una rolls 66 standard 66-sided dice simultaneously and calculates the product of the 66{ } numbers obtained. What is the probability that the product is divisible by 4?4?
<spanclass=′latex−bold′>(A)</span> 34<spanclass=′latex−bold′>(B)</span> 5764<spanclass=′latex−bold′>(C)</span> 5964<spanclass=′latex−bold′>(D)</span> 187192<spanclass=′latex−bold′>(E)</span> 6364<span class='latex-bold'>(A)</span>\: \frac34\qquad<span class='latex-bold'>(B)</span> \: \frac{57}{64}\qquad<span class='latex-bold'>(C)</span> \: \frac{59}{64}\qquad<span class='latex-bold'>(D)</span> \: \frac{187}{192}\qquad<span class='latex-bold'>(E)</span> \: \frac{63}{64}
probabilityAMCAMC 10AMC 10 BAMC 12AMC 12 B