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| | ==Problem 21== | | ==Problem 21== |
| − | Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties:
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| − | *<math>P(x)</math> has a leading coefficient <math>1</math>,
| + | XXX |
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| − | *<math>1</math> is a root of <math>P(x)-1</math>,
| + | <math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math> |
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| − | *<math>2</math> is a root of <math>P(x-2)</math>,
| + | [[2024 AMC 10A Problems/Problem 21|Solution]] |
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| − | *<math>3</math> is a root of <math>P(3x)</math>, and
| + | ==Problem 22== |
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| − | *<math>4</math> is a root of <math>4P(x)</math>.
| + | XXX |
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| − | The roots of <math>P(x)</math> are integers, with one exception. The root that is not an integer can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. What is <math>m+n</math>?
| + | <math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math> |
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| − | <math>\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49</math>
| + | [[2024 AMC 10A Problems/Problem 22|Solution]] |
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| − | [[2023 AMC 10A Problems/Problem 21|Solution]]
| + | ==Problem 23== |
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| − | ==Problem 22==
| + | XXX |
| − | Circle <math>C_1</math> and <math>C_2</math> each have radius <math>1</math>, and the distance between their centers is <math>\frac{1}{2}</math>. Circle <math>C_3</math> is the largest circle internally tangent to both <math>C_1</math> and <math>C_2</math>. Circle <math>C_4</math> is internally tangent to both <math>C_1</math> and <math>C_2</math> and externally tangent to <math>C_3</math>. What is the radius of <math>C_4</math>?
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| − | <asy> | + | <math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math> |
| − | import olympiad;
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| − | size(10cm);
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| − | draw(circle((0,0),0.75));
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| − | draw(circle((-0.25,0),1));
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| − | draw(circle((0.25,0),1));
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| − | draw(circle((0,6/7),3/28));
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| − | pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118);
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| − | dot(B^^C);
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| − | draw(B--E, dashed);
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| − | draw(C--F, dashed);
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| − | draw(B--C);
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| − | label("$C_4$", D);
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| − | label("$C_1$", (-1.375, 0));
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| − | label("$C_2$", (1.375,0));
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| − | label("$\frac{1}{2}$", (0, -.125));
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| − | label("$C_3$", (-0.4, -0.4));
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| − | label("$1$", (-.85, 0.70));
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| − | label("$1$", (.85, -.7));
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| − | import olympiad;
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| − | markscalefactor=0.005;
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| − | </asy> | |
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| − | <math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math>
| + | [[2024 AMC 10A Problems/Problem 23|Solution]] |
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| − | [[2023 AMC 10A Problems/Problem 22|Solution]]
| + | ==Problem 24== |
| − | | |
| − | ==Problem 23== | |
| − | If the positive integer <math>c</math> has positive integer divisors <math>a</math> and <math>b</math> with <math>c = ab</math>, then <math>a</math> and <math>b</math> are said to be <math>\textit{complementary}</math> divisors of <math>c</math>. Suppose that <math>N</math> is a positive integer that has one complementary pair of divisors that differ by <math>20</math> and another pair of complementary divisors that differ by <math>23</math>. What is the sum of the digits of <math>N</math>?
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| − | <math>\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19</math>
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| − | [[2023 AMC 10A Problems/Problem 23|Solution]]
| + | XXX |
| | | | |
| − | ==Problem 24==
| + | <math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math> |
| − | Six regular hexagonal blocks of side length <math>1</math> unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is <math>\frac{3}{7}</math> unit. What is the area of the region inside the frame not occupied by the blocks?
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| − | <asy>
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| − | unitsize(1cm);
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| − | draw(scale(3)*polygon(6));
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| − | filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray);
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| − | filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray);
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| − | filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray);
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| − | filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray);
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| − | filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray);
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| − | filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray);
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| − | </asy>
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| − | <math>\textbf{(A)}~\frac{13 \sqrt{3}}{3}\qquad\textbf{(B)}~\frac{216 \sqrt{3}}{49}\qquad\textbf{(C)}~\frac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}</math> | |
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| − | [[2023 AMC 10A Problems/Problem 24|Solution]] | + | [[2024 AMC 10A Problems/Problem 24|Solution]] |
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| | ==Problem 25== | | ==Problem 25== |
| − | If <math>A</math> and <math>B</math> are vertices of a polyhedron, define the distance <math>d(A, B)</math> to be the minimum number of edges of the polyhedron one must traverse in order to connect <math>A</math> and <math>B</math>. For example, <math>\overline{AB}</math> is an edge of the polyhedron, then <math>d(A, B) = 1</math>, but if <math>\overline{AC}</math> and <math>\overline{CB}</math> are edges and <math>\overline{AB}</math> is not an edge, then <math>d(A, B) = 2</math>. Let <math>Q</math>, <math>R</math>, and <math>S</math> be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of <math>20</math> equilateral triangles). What is the probability that <math>d(Q, R) > d(R, S)</math>?
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| − | <math>\textbf{(A) }\frac{7}{22}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{12}\qquad\textbf{(E) }\frac{1}{2}</math>
| + | XXX |
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| − | [[2023 AMC 10A Problems/Problem 25|Solution]] | + | [[2024 AMC 10A Problems/Problem 25|Solution]] |
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| | ==See also== | | ==See also== |
