Difference between revisions of "2022 AMC 12A Problems"
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{{AMC12 Problems|year=2022|ab=A}} | {{AMC12 Problems|year=2022|ab=A}} | ||
==Problem 1== | ==Problem 1== | ||
− | + | ||
+ | What is the value of <cmath>3+\frac{1}{3+\frac{1}{3+\frac13}}?</cmath> | ||
+ | <math>\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{33}{10}\qquad\textbf{(D)}\ \frac{109}{33}\qquad\textbf{(E)}\ \frac{15}{4}</math> | ||
[[2022 AMC 12A Problems/Problem 1|Solution]] | [[2022 AMC 12A Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
− | + | The sum of three numbers is <math>96.</math> The first number is <math>6</math> times the third number, and the third number is <math>40</math> less than the second number. What is the absolute value of the difference between the first and second numbers? | |
+ | |||
+ | <math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5</math> | ||
[[2022 AMC 12A Problems/Problem 2|Solution]] | [[2022 AMC 12A Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
− | These | + | |
+ | Five rectangles, <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math>, are arranged in a square as shown below. These rectangles have dimensions <math>1\times6</math>, <math>2\times4</math>, <math>5\times6</math>, <math>2\times7</math>, and <math>2\times3</math>, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle? | ||
+ | <asy> | ||
+ | size(150); | ||
+ | currentpen = black+1.25bp; | ||
+ | fill((3,2.5)--(3,4.5)--(5.3,4.5)--(5.3,2.5)--cycle,gray); | ||
+ | draw((0,0)--(7,0)--(7,7)--(0,7)--(0,0)); | ||
+ | draw((3,0)--(3,4.5)); | ||
+ | draw((0,4.5)--(5.3,4.5)); | ||
+ | draw((5.3,7)--(5.3,2.5)); | ||
+ | draw((7,2.5)--(3,2.5)); | ||
+ | </asy> | ||
+ | <math>\textbf{(A) }A\qquad\textbf{(B) }B \qquad\textbf{(C) }C \qquad\textbf{(D) }D\qquad\textbf{(E) }E</math> | ||
[[2022 AMC 12A Problems/Problem 3|Solution]] | [[2022 AMC 12A Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | + | The least common multiple of a positive integer <math>n</math> and <math>18</math> is <math>180</math>, and the greatest common divisor of <math>n</math> and <math>45</math> is <math>15</math>. What is the sum of the digits of <math>n</math>? | |
+ | |||
+ | <math>\textbf{(A) } 3 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 12</math> | ||
[[2022 AMC 12A Problems/Problem 4|Solution]] | [[2022 AMC 12A Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | + | ||
+ | The <math>\textit{taxicab distance}</math> between points <math>(x_1, y_1)</math> and <math>(x_2, y_2)</math> in the coordinate plane is given by <cmath>|x_1 - x_2| + |y_1 - y_2|.</cmath> | ||
+ | For how many points <math>P</math> with integer coordinates is the taxicab distance between <math>P</math> and the origin less than or equal to <math>20</math>? | ||
+ | |||
+ | <math>\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \qquad\textbf{(C)} \, 841 \qquad\textbf{(D)} \, 921 \qquad\textbf{(E)} \, 924 </math> | ||
[[2022 AMC 12A Problems/Problem 5|Solution]] | [[2022 AMC 12A Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | + | A data set consists of <math>6</math> (not distinct) positive integers: <math>1</math>, <math>7</math>, <math>5</math>, <math>2</math>, <math>5</math>, and <math>X</math>. The | |
+ | average (arithmetic mean) of the <math>6</math> numbers equals a value in the data set. What is | ||
+ | the sum of all possible values of <math>X</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40</math> | ||
[[2022 AMC 12A Problems/Problem 6|Solution]] | [[2022 AMC 12A Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | + | A rectangle is partitioned into <math>5</math> regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible? | |
+ | |||
+ | <asy> size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); </asy> | ||
+ | |||
+ | <math>\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720</math> | ||
[[2022 AMC 12A Problems/Problem 7|Solution]] | [[2022 AMC 12A Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | + | ||
+ | The infinite product | ||
+ | <cmath>\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \cdots</cmath> | ||
+ | evaluates to a real number. What is that number? | ||
+ | |||
+ | <math>\textbf{(A) }\sqrt{10}\qquad\textbf{(B) }\sqrt[3]{100}\qquad\textbf{(C) }\sqrt[4]{1000}\qquad\textbf{(D) }10\qquad\textbf{(E) }10\sqrt[3]{10}</math> | ||
[[2022 AMC 12A Problems/Problem 8|Solution]] | [[2022 AMC 12A Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | + | On Halloween <math>31</math> children walked into the principal's office asking for candy. They | |
+ | can be classified into three types: Some always lie; some always tell the truth; and | ||
+ | some alternately lie and tell the truth. The alternaters arbitrarily choose their first | ||
+ | response, either a lie or the truth, but each subsequent statement has the opposite | ||
+ | truth value from its predecessor. The principal asked everyone the same three | ||
+ | questions in this order. | ||
+ | |||
+ | "Are you a truth-teller?" The principal gave a piece of candy to each of the <math>22</math> | ||
+ | children who answered yes. | ||
+ | |||
+ | "Are you an alternater?" The principal gave a piece of candy to each of the <math>15</math> | ||
+ | children who answered yes. | ||
+ | |||
+ | "Are you a liar?" The principal gave a piece of candy to each of the <math>9</math> children who | ||
+ | answered yes. | ||
+ | |||
+ | How many pieces of candy in all did the principal give to the children who always | ||
+ | tell the truth? | ||
+ | |||
+ | <math>\textbf{(A) } 7 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 31</math> | ||
[[2022 AMC 12A Problems/Problem 9|Solution]] | [[2022 AMC 12A Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | + | How many ways are there to split the integers <math>1</math> through <math>14</math> into <math>7</math> pairs such that in each pair, the greater number is at least <math>2</math> times the lesser number? | |
+ | |||
+ | <math>\textbf{(A) } 108 \qquad \textbf{(B) } 120 \qquad \textbf{(C) } 126 \qquad \textbf{(D) } 132 \qquad \textbf{(E) } 144</math> | ||
[[2022 AMC 12A Problems/Problem 10|Solution]] | [[2022 AMC 12A Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
− | + | ||
+ | What is the product of all real numbers <math>x</math> such that the distance on the number line between <math>\log_6x</math> and <math>\log_69</math> is twice the distance on the number line between <math>\log_610</math> and <math>1</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 25 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 81</math> | ||
[[2022 AMC 12A Problems/Problem 11|Solution]] | [[2022 AMC 12A Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | Let <math>M</math> be the midpoint of <math>\overline{AB}</math> in regular tetrahedron <math>ABCD</math>. What is <math>\cos(\angle CMD)</math>? | |
+ | |||
+ | <math>\textbf{(A) } \frac14 \qquad \textbf{(B) } \frac13 \qquad \textbf{(C) } \frac25 \qquad \textbf{(D) } \frac12 \qquad \textbf{(E) } \frac{\sqrt{3}}{2}</math> | ||
[[2022 AMC 12A Problems/Problem 12|Solution]] | [[2022 AMC 12A Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | + | Let <math>\mathcal{R}</math> be the region in the complex plane consisting of all complex numbers <math>z</math> that can be written as the sum of complex numbers <math>z_1</math> and <math>z_2</math>, where <math>z_1</math> lies on the segment with endpoints <math>3</math> and <math>4i</math>, and <math>z_2</math> has magnitude at most <math>1</math>. What integer is closest to the area of <math>\mathcal{R}</math>? | |
+ | |||
+ | <math>\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17</math> | ||
[[2022 AMC 12A Problems/Problem 13|Solution]] | [[2022 AMC 12A Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | + | What is the value of <cmath>(\log 5)^3+(\log 20)^3+(\log 8)(\log 0.25)</cmath> where <math>\log</math> denotes the base-ten logarithm? | |
+ | |||
+ | <math>\textbf{(A) } \frac{3}{2} \qquad \textbf{(B) } \frac{7}{4} \qquad \textbf{(C) } 2 \qquad \textbf{(D) } \frac{9}{4} \qquad \textbf{(E) } 3</math> | ||
[[2022 AMC 12A Problems/Problem 14|Solution]] | [[2022 AMC 12A Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | + | The roots of the polynomial <math>10x^3-39x^2+29x-6</math> are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by <math>2</math> units. What is the volume of the new box? | |
+ | |||
+ | <math>\textbf{(A) } \frac{24}{5} \qquad \textbf{(B) } \frac{42}{5} \qquad \textbf{(C) } \frac{81}{5} \qquad \textbf{(D) } 30 \qquad \textbf{(E) } 48</math> | ||
[[2022 AMC 12A Problems/Problem 15|Solution]] | [[2022 AMC 12A Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
− | + | A triangular number is a positive integer that can be expressed in the form <math>t_n=1+2+3+\cdots+n</math>, for some positive integer <math>n</math>. The three smallest triangular numbers that are also perfect squares are <math>t_1=1=1^2, t_8=36=6^2,</math> and <math>t_{49}=1225=35^2</math>. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square? | |
+ | |||
+ | <math>\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 </math> | ||
[[2022 AMC 12A Problems/Problem 16|Solution]] | [[2022 AMC 12A Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
− | + | ||
+ | Suppose <math>a</math> is a real number such that the equation <cmath>a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}</cmath> | ||
+ | has more than one solution in the interval <math>(0, \pi)</math>. The set of all such <math>a</math> that can be written | ||
+ | in the form <cmath>(p,q) \cup (q,r),</cmath> | ||
+ | where <math>p, q,</math> and <math>r</math> are real numbers with <math>p < q< r</math>. What is <math>p+q+r</math>? | ||
+ | |||
+ | <math>\textbf{(A) } {-}4 \qquad \textbf{(B) } {-}1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 4</math> | ||
[[2022 AMC 12A Problems/Problem 17|Solution]] | [[2022 AMC 12A Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
− | + | Let <math>T_k</math> be the transformation of the coordinate plane that first rotates the plane <math>k</math> degrees counterclockwise around the origin and then reflects the plane across the <math>y</math>-axis. What is the least positive integer <math>n</math> such that performing the sequence of transformations <math>T_1, T_2, T_3, \dots, T_n</math> returns the point <math>(1,0)</math> back to itself? | |
+ | |||
+ | <math>\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721</math> | ||
[[2022 AMC 12A Problems/Problem 18|Solution]] | [[2022 AMC 12A Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
− | + | Suppose that <math>13</math> cards numbered <math>1, 2, 3, \ldots, 13</math> are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards <math>1, 2, 3</math> are picked up on the first pass, <math>4</math> and <math>5</math> on the second pass, <math>6</math> on the third pass, <math>7, 8, 9, 10</math> on the fourth pass, and <math>11, 12, 13</math> on the fifth pass. For how many of the <math>13!</math> possible orderings of the cards will the <math>13</math> cards be picked up in exactly two passes? | |
+ | |||
+ | <asy> | ||
+ | size(11cm); | ||
+ | draw((0,0)--(2,0)--(2,3)--(0,3)--cycle); | ||
+ | label("7", (1,1.5)); | ||
+ | draw((3,0)--(5,0)--(5,3)--(3,3)--cycle); | ||
+ | label("11", (4,1.5)); | ||
+ | draw((6,0)--(8,0)--(8,3)--(6,3)--cycle); | ||
+ | label("8", (7,1.5)); | ||
+ | draw((9,0)--(11,0)--(11,3)--(9,3)--cycle); | ||
+ | label("6", (10,1.5)); | ||
+ | draw((12,0)--(14,0)--(14,3)--(12,3)--cycle); | ||
+ | label("4", (13,1.5)); | ||
+ | draw((15,0)--(17,0)--(17,3)--(15,3)--cycle); | ||
+ | label("5", (16,1.5)); | ||
+ | draw((18,0)--(20,0)--(20,3)--(18,3)--cycle); | ||
+ | label("9", (19,1.5)); | ||
+ | draw((21,0)--(23,0)--(23,3)--(21,3)--cycle); | ||
+ | label("12", (22,1.5)); | ||
+ | draw((24,0)--(26,0)--(26,3)--(24,3)--cycle); | ||
+ | label("1", (25,1.5)); | ||
+ | draw((27,0)--(29,0)--(29,3)--(27,3)--cycle); | ||
+ | label("13", (28,1.5)); | ||
+ | draw((30,0)--(32,0)--(32,3)--(30,3)--cycle); | ||
+ | label("10", (31,1.5)); | ||
+ | draw((33,0)--(35,0)--(35,3)--(33,3)--cycle); | ||
+ | label("2", (34,1.5)); | ||
+ | draw((36,0)--(38,0)--(38,3)--(36,3)--cycle); | ||
+ | label("3", (37,1.5)); | ||
+ | </asy> | ||
+ | <math>\textbf{(A) } 4082 \qquad \textbf{(B) } 4095 \qquad \textbf{(C) } 4096 \qquad \textbf{(D) } 8178 \qquad \textbf{(E) } 8191</math> | ||
[[2022 AMC 12A Problems/Problem 19|Solution]] | [[2022 AMC 12A Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
− | + | ||
+ | Isosceles trapezoid <math>ABCD</math> has parallel sides <math>\overline{AD}</math> and <math>\overline{BC},</math> with <math>BC < AD</math> and <math>AB = CD.</math> There is a point <math>P</math> in the plane such that <math>PA=1, PB=2, PC=3,</math> and <math>PD=4.</math> What is <math>\tfrac{BC}{AD}?</math> | ||
+ | |||
+ | <math>\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}</math> | ||
[[2022 AMC 12A Problems/Problem 20|Solution]] | [[2022 AMC 12A Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
− | + | ||
+ | Let <cmath>P(x) = x^{2022} + x^{1011} + 1.</cmath> Which of the following polynomials is a factor of <math>P(x)</math>? | ||
+ | |||
+ | <math>\textbf{(A)} \, x^2 -x + 1 \qquad\textbf{(B)} \, x^2 + x + 1 \qquad\textbf{(C)} \, x^4 + 1 \qquad\textbf{(D)} \, x^6 - x^3 + 1 \qquad\textbf{(E)} \, x^6 + x^3 + 1 </math> | ||
[[2022 AMC 12A Problems/Problem 21|Solution]] | [[2022 AMC 12A Problems/Problem 21|Solution]] | ||
==Problem 22== | ==Problem 22== | ||
− | + | ||
+ | Let <math>c</math> be a real number, and let <math>z_1</math> and <math>z_2</math> be the two complex numbers satisfying the equation | ||
+ | <math>z^2 - cz + 10 = 0</math>. Points <math>z_1</math>, <math>z_2</math>, <math>\frac{1}{z_1}</math>, and <math>\frac{1}{z_2}</math> are the vertices of (convex) quadrilateral <math>\mathcal{Q}</math> in the complex plane. When the area of <math>\mathcal{Q}</math> obtains its maximum possible value, <math>c</math> is closest to which of the following? | ||
+ | |||
+ | <math>\textbf{(A) }4.5 \qquad\textbf{(B) }5 \qquad\textbf{(C) }5.5 \qquad\textbf{(D) }6\qquad\textbf{(E) }6.5</math> | ||
[[2022 AMC 12A Problems/Problem 22|Solution]] | [[2022 AMC 12A Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
− | + | Let <math>h_n</math> and <math>k_n</math> be the unique relatively prime positive integers such that <cmath>\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{h_n}{k_n}.</cmath> Let <math>L_n</math> denote the least common multiple of the numbers <math>1, 2, 3, \ldots, n</math>. For how many integers with <math>1\le{n}\le{22}</math> is <math>k_n<L_n</math>? | |
+ | |||
+ | <math>\textbf{(A) }0 \qquad\textbf{(B) }3 \qquad\textbf{(C) }7 \qquad\textbf{(D) }8\qquad\textbf{(E) }10</math> | ||
[[2022 AMC 12A Problems/Problem 23|Solution]] | [[2022 AMC 12A Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
− | + | How many strings of length <math>5</math> formed from the digits <math>0</math>, <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math> are there such that for each <math>j \in \{1,2,3,4\}</math>, at least <math>j</math> of the digits are less than <math>j</math>? (For example, <math>02214</math> satisfies this condition | |
+ | because it contains at least <math>1</math> digit less than <math>1</math>, at least <math>2</math> digits less than <math>2</math>, at least <math>3</math> digits less | ||
+ | than <math>3</math>, and at least <math>4</math> digits less than <math>4</math>. The string <math>23404</math> does not satisfy the condition because it | ||
+ | does not contain at least <math>2</math> digits less than <math>2</math>.) | ||
+ | |||
+ | <math>\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296</math> | ||
[[2022 AMC 12A Problems/Problem 24|Solution]] | [[2022 AMC 12A Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
− | + | A circle with integer radius <math>r</math> is centered at <math>(r, r)</math>. Distinct line segments of length <math>c_i</math> connect points <math>(0, a_i)</math> to <math>(b_i, 0)</math> for <math>1 \le i \le 14</math> and are tangent to the circle, where <math>a_i</math>, <math>b_i</math>, and <math>c_i</math> are all positive integers and <math>c_1 \le c_2 \le \cdots \le c_{14}</math>. What is the ratio <math>\frac{c_{14}}{c_1}</math> for the least possible value of <math>r</math>? | |
+ | |||
+ | <math>\textbf{(A)} ~\frac{21}{5} \qquad\textbf{(B)} ~\frac{85}{13} \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~\frac{39}{5} \qquad\textbf{(E)} ~17 </math> | ||
[[2022 AMC 12A Problems/Problem 25|Solution]] | [[2022 AMC 12A Problems/Problem 25|Solution]] | ||
==See also== | ==See also== | ||
− | {{AMC12 box|year=2022|ab=A|before=[[2021 AMC 12B Problems]]|after=[[2022 AMC 12B Problems]]}} | + | {{AMC12 box|year=2022|ab=A|before=[[2021 Fall AMC 12B Problems]]|after=[[2022 AMC 12B Problems]]}} |
[[Category:AMC 12 Problems]] | [[Category:AMC 12 Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 00:32, 9 March 2024
2022 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
What is the value of
Problem 2
The sum of three numbers is The first number is
times the third number, and the third number is
less than the second number. What is the absolute value of the difference between the first and second numbers?
Problem 3
Five rectangles, ,
,
,
, and
, are arranged in a square as shown below. These rectangles have dimensions
,
,
,
, and
, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
Problem 4
The least common multiple of a positive integer and
is
, and the greatest common divisor of
and
is
. What is the sum of the digits of
?
Problem 5
The between points
and
in the coordinate plane is given by
For how many points
with integer coordinates is the taxicab distance between
and the origin less than or equal to
?
Problem 6
A data set consists of (not distinct) positive integers:
,
,
,
,
, and
. The
average (arithmetic mean) of the
numbers equals a value in the data set. What is
the sum of all possible values of
?
Problem 7
A rectangle is partitioned into regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
Problem 8
The infinite product
evaluates to a real number. What is that number?
Problem 9
On Halloween children walked into the principal's office asking for candy. They
can be classified into three types: Some always lie; some always tell the truth; and
some alternately lie and tell the truth. The alternaters arbitrarily choose their first
response, either a lie or the truth, but each subsequent statement has the opposite
truth value from its predecessor. The principal asked everyone the same three
questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the
children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the
children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the children who
answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
Problem 10
How many ways are there to split the integers through
into
pairs such that in each pair, the greater number is at least
times the lesser number?
Problem 11
What is the product of all real numbers such that the distance on the number line between
and
is twice the distance on the number line between
and
?
Problem 12
Let be the midpoint of
in regular tetrahedron
. What is
?
Problem 13
Let be the region in the complex plane consisting of all complex numbers
that can be written as the sum of complex numbers
and
, where
lies on the segment with endpoints
and
, and
has magnitude at most
. What integer is closest to the area of
?
Problem 14
What is the value of where
denotes the base-ten logarithm?
Problem 15
The roots of the polynomial are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by
units. What is the volume of the new box?
Problem 16
A triangular number is a positive integer that can be expressed in the form , for some positive integer
. The three smallest triangular numbers that are also perfect squares are
and
. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
Problem 17
Suppose is a real number such that the equation
has more than one solution in the interval
. The set of all such
that can be written
in the form
where
and
are real numbers with
. What is
?
Problem 18
Let be the transformation of the coordinate plane that first rotates the plane
degrees counterclockwise around the origin and then reflects the plane across the
-axis. What is the least positive integer
such that performing the sequence of transformations
returns the point
back to itself?
Problem 19
Suppose that cards numbered
are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards
are picked up on the first pass,
and
on the second pass,
on the third pass,
on the fourth pass, and
on the fifth pass. For how many of the
possible orderings of the cards will the
cards be picked up in exactly two passes?
Problem 20
Isosceles trapezoid has parallel sides
and
with
and
There is a point
in the plane such that
and
What is
Problem 21
Let Which of the following polynomials is a factor of
?
Problem 22
Let be a real number, and let
and
be the two complex numbers satisfying the equation
. Points
,
,
, and
are the vertices of (convex) quadrilateral
in the complex plane. When the area of
obtains its maximum possible value,
is closest to which of the following?
Problem 23
Let and
be the unique relatively prime positive integers such that
Let
denote the least common multiple of the numbers
. For how many integers with
is
?
Problem 24
How many strings of length formed from the digits
,
,
,
,
are there such that for each
, at least
of the digits are less than
? (For example,
satisfies this condition
because it contains at least
digit less than
, at least
digits less than
, at least
digits less
than
, and at least
digits less than
. The string
does not satisfy the condition because it
does not contain at least
digits less than
.)
Problem 25
A circle with integer radius is centered at
. Distinct line segments of length
connect points
to
for
and are tangent to the circle, where
,
, and
are all positive integers and
. What is the ratio
for the least possible value of
?
See also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2021 Fall AMC 12B Problems |
Followed by 2022 AMC 12B Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.