MathDB

20

Part of 2021 AMC 10 Fall

Problems(2)

Coefficient Switcharoo

Source: 2021 AMC 12A #17, AMC 10A #20

11/11/2021
How many ordered pairs of positive integers (b,c)(b,c) exist where both x2+bx+c=0x^2+bx+c=0 and x2+cx+b=0x^2+cx+b=0 do not have distinct, real solutions?
<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>6<spanclass=′latex−bold′>(C)</span>8<spanclass=′latex−bold′>(D)</span>10<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span> 4 \qquad <span class='latex-bold'>(B) </span> 6 \qquad <span class='latex-bold'>(C) </span> 8 \qquad <span class='latex-bold'>(D) </span> 10 \qquad <span class='latex-bold'>(E) </span> 12 \qquad
2021 amc 12aAMCAMC 12AMC 12 A
Nice Symmetry

Source: 2021 Fall AMC10 B P20

11/17/2021
In a particular game, each of 44 players rolls a standard 66{ }-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a 5,5, given that he won the game?
(<spanclass=′latex−bold′>A</span>) 61216(<spanclass=′latex−bold′>B</span>) 3671296(<spanclass=′latex−bold′>C</span>) 41144(<spanclass=′latex−bold′>D</span>) 185648(<spanclass=′latex−bold′>E</span>) 1136(<span class='latex-bold'>A</span>)\: \frac{61}{216}\qquad(<span class='latex-bold'>B</span>) \: \frac{367}{1296}\qquad(<span class='latex-bold'>C</span>) \: \frac{41}{144}\qquad(<span class='latex-bold'>D</span>) \: \frac{185}{648}\qquad(<span class='latex-bold'>E</span>) \: \frac{11}{36}
AMCAMC 10AMC 10 Bsymmetry