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2006 AMC 12B Problems/Problem 16
...>C</math> at <math>(0,0)</math> and <math>(7,1)</math>, respectively. What is its area? \mathrm{(A)}\ 20\sqrt {3}

1 KB (203 words) - 16:36, 18 September 2023
2006 AMC 12B Problems/Problem 17
...<math>6</math> on each die are in the ratio <math>1:2:3:4:5:6</math>. What is the probability of rolling a total of <math>7</math> on the two dice? The probability of getting an <math>x</math> on one of these dice is <math>\frac{x}{21}</math>.

1 KB (188 words) - 22:10, 9 June 2016
2006 AMC 12B Problems/Problem 18
...can easily be shown that each location that satisfies these two conditions is indeed reachable. If the object only makes <math>1</math> move, it is obvious that there are only 4 possible points that the object can move to.

2 KB (354 words) - 16:57, 28 December 2020
2006 AMC 12B Problems/Problem 19
..."and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? First, The number of the plate is divisible by <math>9</math> and in the form of

4 KB (696 words) - 09:47, 10 August 2015
2006 AMC 12B Problems/Problem 20
...th>x</math> be chosen at random from the interval <math>(0,1)</math>. What is the probability that Here <math>\lfloor x\rfloor</math> denotes the greatest integer that is less than or equal to <math>x</math>.

3 KB (485 words) - 14:09, 21 May 2021
2006 AMC 12B Problems/Problem 22
...are integers and <math>m</math> is not divisible by <math>10</math>. What is the smallest possible value of <math>n</math>? The power of <math>10</math> for any factorial is given by the well-known algorithm

5 KB (881 words) - 15:52, 23 June 2021
2006 AMC 12B Problems/Problem 23
...ath>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? MP("90^\circ-\alpha",C,3*dir(30),f);

7 KB (1,169 words) - 14:04, 10 June 2022
2006 AMC 12B Problems/Problem 24
...\le \frac{\pi}{2}</math> and <math>0 \le y \le \frac{\pi}{2}</math>. What is the area of the subset of <math>S</math> for which <cmath> \mathrm{(D)}\ \dfrac{3\pi^2}{16}

3 KB (563 words) - 22:45, 24 October 2021
2006 AMC 12B Problems/Problem 25
A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> for <math>n\geq 1</m ...sequence <math>(a_n)</math> completes at <math>i</math> if <math>i</math> is the minimal positive integer such that <math>a_i = a_{i + 1} = 1</math>. Ot

5 KB (924 words) - 12:02, 15 June 2022
2006 AMC 12A Problems/Problem 10
For how many real values of <math>x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer? <math> \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \t

1 KB (167 words) - 23:23, 16 December 2021
2006 AMC 12A Problems/Problem 13
...e centers of three mutually externally tangent [[circle]]s, as shown. What is the sum of the areas of the three circles? <cmath>r_A + r_B = 3</cmath>

1 KB (184 words) - 13:57, 19 January 2021
2006 AMC 12A Problems/Problem 14
...debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? ...mon divisor]]) of <math>a</math> and <math>b</math>. Therefore, the answer is <math>gcd(300,210)=\boxed{\textbf{(C) }30}.</math>

3 KB (442 words) - 03:13, 8 August 2022
2006 AMC 12A Problems/Problem 15
Suppose <math>\cos x=0</math> and <math>\cos (x+z)=1/2</math>. What is the smallest possible positive value of <math>z</math>? <math> \mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6} \

919 bytes (138 words) - 12:45, 4 August 2017
2006 AMC 12A Problems/Problem 16
...d <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>? dotfactor=3;

2 KB (286 words) - 10:16, 19 December 2021
2006 AMC 12A Problems/Problem 17
...th> is tangent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>? ...rac{5}{9}\qquad \mathrm{(C) \ } \frac{3}{5}\qquad \mathrm{(D) \ } \frac{5}{3}\qquad \mathrm{(E) \ } \frac{9}{5}</math>

6 KB (958 words) - 23:29, 28 September 2023
2006 AMC 12A Problems/Problem 20
...s equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vert Therefore, starting at <math>A</math>, the bug has a <math>\frac{3}{3}</math> chance of finding a good path to the next vertex, and call it <math

5 KB (908 words) - 19:23, 22 September 2022
2006 AMC 12A Problems/Problem 22
...sible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>? ...quad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{(E) \ } 6\sqrt{2}-\sqrt{3}</math>

2 KB (343 words) - 15:39, 14 June 2023
2006 AMC 12A Problems/Problem 23
...,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>? <cmath>A^2(S)=\left(\frac{1+2x+x^2}{2^2},\frac{x+2x^2+x^3}{2^2},...,\frac{x^{98}+2x^{99}+x^{100}}{2^2}\right)</cmath>

3 KB (466 words) - 22:40, 29 September 2023
2006 AMC 12A Problems/Problem 24
is simplified by expanding it and combining like terms. How many terms are in if the exponent of <math>y</math> is <math>1</math>, the exponent of <math>z</math> can be all even integers up

8 KB (1,332 words) - 17:37, 17 September 2023
2006 AMC 12A Problems/Problem 25
How many non-[[empty set | empty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> have the following two properties? ...k+1</math>, with no restriction on consecutive numbers. Since this process is easily reversible, we have a [[bijection]].

8 KB (1,405 words) - 11:52, 27 September 2022

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