Difference between revisions of "2022 AMC 10B Problems"
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==Problem 1 == | ==Problem 1 == | ||
− | |||
− | <math>\textbf{(A)}\ | + | Define <math>x\diamond y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y.</math> What is the value of <cmath>(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?</cmath> |
+ | |||
+ | <math> \textbf{(A)}\ {-}2 \qquad | ||
+ | \textbf{(B)}\ {-}1 \qquad | ||
+ | \textbf{(C)}\ 0 \qquad | ||
+ | \textbf{(D)}\ 1 \qquad | ||
+ | \textbf{(E)}\ 2</math> | ||
+ | |||
[[2022 AMC 10B Problems/Problem 1|Solution]] | [[2022 AMC 10B Problems/Problem 1|Solution]] | ||
+ | |||
==Problem 2 == | ==Problem 2 == | ||
− | |||
− | <math>\textbf{(A)}\ | + | In rhombus <math>ABCD</math>, point <math>P</math> lies on segment <math>\overline{AD}</math> so that <math>\overline{BP}</math> <math>\perp</math> <math>\overline{AD}</math>, <math>AP = 3</math>, and <math>PD = 2</math>. What is the area of <math>ABCD</math>? (Note: The figure is not drawn to scale.) |
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | size(180); | ||
+ | real r = 3, s = 5, t = sqrt(r*r+s*s); | ||
+ | defaultpen(linewidth(0.6) + fontsize(10)); | ||
+ | pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); | ||
+ | draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); | ||
+ | label("$A$",A,SW); | ||
+ | label("$B$", B, NW); | ||
+ | label("$C$",C,NE); | ||
+ | label("$D$",D,SE); | ||
+ | label("$P$",P,S); | ||
+ | </asy> | ||
+ | |||
+ | <math>\textbf{(A) }3\sqrt 5 \qquad | ||
+ | \textbf{(B) }10 \qquad | ||
+ | \textbf{(C) }6\sqrt 5 \qquad | ||
+ | \textbf{(D) }20\qquad | ||
+ | \textbf{(E) }25</math> | ||
[[2022 AMC 10B Problems/Problem 2|Solution]] | [[2022 AMC 10B Problems/Problem 2|Solution]] | ||
+ | |||
==Problem 3 == | ==Problem 3 == | ||
− | + | How many three-digit positive integers have an odd number of even digits? | |
− | <math>\textbf{(A)} | + | <math>\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550</math> |
[[2022 AMC 10B Problems/Problem 3|Solution]] | [[2022 AMC 10B Problems/Problem 3|Solution]] | ||
==Problem 4 == | ==Problem 4 == | ||
− | A donkey suffers an attack of hiccups and the first hiccup happens at <math> | + | A donkey suffers an attack of hiccups and the first hiccup happens at <math>4:00</math> one afternoon. Suppose that |
− | the donkey hiccups regularly every <math>5</math> seconds. At what time does the donkey’s <math> | + | the donkey hiccups regularly every <math>5</math> seconds. At what time does the donkey’s <math>700</math>th hiccup occur? |
− | <math>\textbf{(A) } | + | <math>\textbf{(A) }15 \text{ seconds after } 4:58</math> |
− | <math>\textbf{(B) } | + | <math>\textbf{(B) }20 \text{ seconds after } 4:58</math> |
− | <math>\textbf{(C)} | + | <math>\textbf{(C) }25 \text{ seconds after } 4:58</math> |
− | <math>\textbf{(D) } | + | <math>\textbf{(D) }30 \text{ seconds after } 4:58</math> |
− | <math>\textbf{(E) } | + | <math>\textbf{(E) }35 \text{ seconds after } 4:58</math> |
[[2022 AMC 10B Problems/Problem 4|Solution]] | [[2022 AMC 10B Problems/Problem 4|Solution]] | ||
==Problem 5 == | ==Problem 5 == | ||
− | + | What is the value of <cmath>\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?</cmath> | |
− | + | <math>\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt{15} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{105}</math> | |
− | <math>\textbf{(A)}\ | ||
[[2022 AMC 10B Problems/Problem 5|Solution]] | [[2022 AMC 10B Problems/Problem 5|Solution]] | ||
==Problem 6 == | ==Problem 6 == | ||
− | How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ | + | How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ldots</math> are prime numbers? |
<math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4</math> | <math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4</math> | ||
Line 51: | Line 77: | ||
==Problem 7 == | ==Problem 7 == | ||
− | + | For how many values of the constant <math>k</math> will the polynomial <math>x^{2}+kx+36</math> have two distinct integer roots? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16</math> |
[[2022 AMC 10B Problems/Problem 7|Solution]] | [[2022 AMC 10B Problems/Problem 7|Solution]] | ||
==Problem 8 == | ==Problem 8 == | ||
− | + | Consider the following <math>100</math> sets of <math>10</math> elements each: | |
+ | <cmath>\begin{align*} | ||
+ | &\{1,2,3,\ldots,10\}, \\ | ||
+ | &\{11,12,13,\ldots,20\},\\ | ||
+ | &\{21,22,23,\ldots,30\},\\ | ||
+ | &\vdots\\ | ||
+ | &\{991,992,993,\ldots,1000\}. | ||
+ | \end{align*}</cmath> | ||
+ | How many of these sets contain exactly two multiples of <math>7</math>? | ||
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 43\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 50</math> |
[[2022 AMC 10B Problems/Problem 8|Solution]] | [[2022 AMC 10B Problems/Problem 8|Solution]] | ||
==Problem 9 == | ==Problem 9 == | ||
− | + | The sum | |
+ | <cmath>\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}</cmath> can be expressed as <math>a-\frac{1}{b!}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? | ||
− | <math>\textbf{(A)}\ | + | <math> \textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024</math> |
[[2022 AMC 10B Problems/Problem 9|Solution]] | [[2022 AMC 10B Problems/Problem 9|Solution]] | ||
==Problem 10 == | ==Problem 10 == | ||
− | + | Camila writes down five positive integers. The unique mode of these integers is <math>2</math> greater than their median, and the median is <math>2</math> greater than their arithmetic mean. What is the least possible value for the mode? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 13</math> |
[[2022 AMC 10B Problems/Problem 10|Solution]] | [[2022 AMC 10B Problems/Problem 10|Solution]] | ||
− | ==Problem 11 == | + | ==Problem 11== |
− | + | All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"? | |
+ | |||
+ | <math>\textbf{(A) }</math> All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. | ||
+ | |||
+ | <math>\textbf{(B) }</math> No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. | ||
+ | |||
+ | <math>\textbf{(C) }</math> All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. | ||
+ | |||
+ | <math>\textbf{(D) }</math> All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. | ||
− | <math> | + | <math>\textbf{(E) }</math> All schools smaller than Euclid HS sold more T-shirts than Euclid HS. |
[[2022 AMC 10B Problems/Problem 11|Solution]] | [[2022 AMC 10B Problems/Problem 11|Solution]] | ||
Line 93: | Line 136: | ||
==Problem 13 == | ==Problem 13 == | ||
− | + | The positive difference between a pair of primes is equal to <math>2</math>, and the positive difference between the cubes of the two primes is <math>31106</math>. What is the sum of the digits of the least prime that is greater than those two primes? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 16</math> |
[[2022 AMC 10B Problems/Problem 13|Solution]] | [[2022 AMC 10B Problems/Problem 13|Solution]] | ||
==Problem 14 == | ==Problem 14 == | ||
− | + | Suppose that <math>S</math> is a subset of <math>\left\{ 1, 2, 3, \cdots , 25 \right\}</math> such that the sum of any two (not necessarily distinct) elements of <math>S</math> is never an element of <math>S.</math> What is the maximum number of elements <math>S</math> may contain? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16</math> |
[[2022 AMC 10B Problems/Problem 14|Solution]] | [[2022 AMC 10B Problems/Problem 14|Solution]] | ||
==Problem 15 == | ==Problem 15 == | ||
− | + | Let <math>S_n</math> be the sum of the first <math>n</math> terms of an arithmetic sequence that has a common difference of <math>2</math>. The quotient <math>\frac{S_{3n}}{S_n}</math> does not depend on <math>n</math>. What is <math>S_{20}</math>? | |
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420</math> |
[[2022 AMC 10B Problems/Problem 15|Solution]] | [[2022 AMC 10B Problems/Problem 15|Solution]] | ||
==Problem 16 == | ==Problem 16 == | ||
− | + | The diagram below shows a rectangle with side lengths <math>4</math> and <math>8</math> and a square with side length <math>5</math>. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle? | |
+ | |||
+ | <asy> | ||
+ | size(5cm); | ||
+ | filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray); | ||
+ | draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1)); | ||
+ | draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1)); | ||
+ | label("$4$", (8,2), E); | ||
+ | label("$8$", (4,0), S); | ||
+ | label("$5$", (3,11/2), NW); | ||
+ | draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(1.1)); | ||
+ | </asy> | ||
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A) }15\dfrac{1}{8} \qquad |
+ | \textbf{(B) }15\dfrac{3}{8} \qquad | ||
+ | \textbf{(C) }15\dfrac{1}{2} \qquad | ||
+ | \textbf{(D) }15\dfrac{5}{8} \qquad | ||
+ | \textbf{(E) }15\dfrac{7}{8} </math> | ||
[[2022 AMC 10B Problems/Problem 16|Solution]] | [[2022 AMC 10B Problems/Problem 16|Solution]] | ||
==Problem 17 == | ==Problem 17 == | ||
− | + | One of the following numbers is not divisible by any prime number less than <math>10.</math> Which is it? | |
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 2^{606}-1 \qquad\textbf{(B) } 2^{606}+1 \qquad\textbf{(C) } 2^{607}-1 \qquad\textbf{(D) } 2^{607}+1\qquad\textbf{(E) } 2^{607}+3^{607}</math> |
[[2022 AMC 10B Problems/Problem 17|Solution]] | [[2022 AMC 10B Problems/Problem 17|Solution]] | ||
==Problem 18 == | ==Problem 18 == | ||
− | + | Consider systems of three linear equations with unknowns <math>x</math>, <math>y</math>, and <math>z</math>, | |
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | a_1 x + b_1 y + c_1 z & = 0 \\ | ||
+ | a_2 x + b_2 y + c_2 z & = 0 \\ | ||
+ | a_3 x + b_3 y + c_3 z & = 0 | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | where each of the coefficients is either <math>0</math> or <math>1</math> and the system has a solution other than <math>x=y=z=0</math>. | ||
+ | For example, one such system is <cmath>\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}</cmath> | ||
+ | with a nonzero solution of <math>\{x,y,z\} = \{1, -1, 1\}</math>. How many such systems of equations are there? | ||
+ | (The equations in a system need not be distinct, and two systems containing the same equations in a | ||
+ | different order are considered different.) | ||
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 302 \qquad\textbf{(B)}\ 338 \qquad\textbf{(C)}\ 340 \qquad\textbf{(D)}\ 343 \qquad\textbf{(E)}\ 344</math> |
[[2022 AMC 10B Problems/Problem 18|Solution]] | [[2022 AMC 10B Problems/Problem 18|Solution]] | ||
==Problem 19 == | ==Problem 19 == | ||
− | + | Each square in a <math>5 \times 5</math> grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules: | |
+ | |||
+ | * Any filled square with two or three filled neighbors remains filled. | ||
+ | |||
+ | * Any empty square with exactly three filled neighbors becomes a filled square. | ||
+ | |||
+ | * All other squares remain empty or become empty. | ||
− | <math>\textbf{(A)}\ | + | A sample transformation is shown in the figure below. |
+ | <asy> | ||
+ | import geometry; | ||
+ | unitsize(0.6cm); | ||
+ | |||
+ | void ds(pair x) { | ||
+ | filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible); | ||
+ | } | ||
+ | |||
+ | ds((1,1)); | ||
+ | ds((2,1)); | ||
+ | ds((3,1)); | ||
+ | ds((1,3)); | ||
+ | |||
+ | for (int i = 0; i <= 5; ++i) { | ||
+ | draw((0,i)--(5,i)); | ||
+ | draw((i,0)--(i,5)); | ||
+ | } | ||
+ | |||
+ | label("Initial", (2.5,-1)); | ||
+ | draw((6,2.5)--(8,2.5),Arrow); | ||
+ | |||
+ | ds((10,2)); | ||
+ | ds((11,1)); | ||
+ | ds((11,0)); | ||
+ | |||
+ | for (int i = 0; i <= 5; ++i) { | ||
+ | draw((9,i)--(14,i)); | ||
+ | draw((i+9,0)--(i+9,5)); | ||
+ | } | ||
+ | |||
+ | label("Transformed", (11.5,-1)); | ||
+ | </asy> | ||
+ | Suppose the <math>5 \times 5</math> grid has a border of empty squares surrounding a <math>3 \times 3</math> subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.) | ||
+ | <asy> | ||
+ | import geometry; | ||
+ | unitsize(0.6cm); | ||
+ | |||
+ | void ds(pair x) { | ||
+ | filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible); | ||
+ | } | ||
+ | |||
+ | for (int i = 1; i < 4; ++ i) { | ||
+ | for (int j = 1; j < 4; ++j) { | ||
+ | label("?",(i + 0.5, j + 0.5)); | ||
+ | } | ||
+ | } | ||
+ | |||
+ | for (int i = 0; i <= 5; ++i) { | ||
+ | draw((0,i)--(5,i)); | ||
+ | draw((i,0)--(i,5)); | ||
+ | } | ||
+ | |||
+ | label("Initial", (2.5,-1)); | ||
+ | draw((6,2.5)--(8,2.5),Arrow); | ||
+ | |||
+ | ds((11,2)); | ||
+ | |||
+ | for (int i = 0; i <= 5; ++i) { | ||
+ | draw((9,i)--(14,i)); | ||
+ | draw((i+9,0)--(i+9,5)); | ||
+ | } | ||
+ | |||
+ | label("Transformed", (11.5,-1)); | ||
+ | </asy> | ||
+ | <math>\textbf{(A)}\ 14 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 22 \qquad\textbf{(D)}\ 26 \qquad\textbf{(E)}\ 30</math> | ||
[[2022 AMC 10B Problems/Problem 19|Solution]] | [[2022 AMC 10B Problems/Problem 19|Solution]] | ||
==Problem 20 == | ==Problem 20 == | ||
− | + | Let <math>ABCD</math> be a rhombus with <math>\angle{ADC} = 46^{\circ}</math>. Let <math>E</math> be the midpoint of <math>\overline{CD}</math>, and let <math>F</math> be the point on <math>\overline{BE}</math> such that <math>\overline{AF}</math> is perpendicular to <math>\overline{BE}</math>. What is the degree measure of <math>\angle{BFC}</math>? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114</math> |
[[2022 AMC 10B Problems/Problem 20|Solution]] | [[2022 AMC 10B Problems/Problem 20|Solution]] | ||
==Problem 21 == | ==Problem 21 == | ||
− | + | Let <math>P(x)</math> be a polynomial with rational coefficients such that when <math>P(x)</math> is divided by the polynomial <math>x^2 + x + 1</math>, the remainder is <math>x + 2</math>, and when <math>P(x)</math> is divided by the polynomial <math>x^2 + 1</math>, the remainder is <math>2x + 1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial? | |
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23</math> |
[[2022 AMC 10B Problems/Problem 21|Solution]] | [[2022 AMC 10B Problems/Problem 21|Solution]] | ||
Line 163: | Line 304: | ||
==Problem 23 == | ==Problem 23 == | ||
− | + | Ant Amelia starts on the number line at <math>0</math> and crawls in the following manner. For <math>n=1,2,3,</math> Amelia chooses a time duration <math>t_n</math> and an increment <math>x_n</math> independently and uniformly at random from the interval <math>(0,1).</math> During the <math>n</math>th step of the process, Amelia moves <math>x_n</math> units in the positive direction, using up <math>t_n</math> minutes. If the total elapsed time has exceeded <math>1</math> minute during the <math>n</math>th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most <math>3</math> steps in all. What is the probability that Amelia’s position when she stops will be greater than <math>1</math>? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}</math> |
[[2022 AMC 10B Problems/Problem 23|Solution]] | [[2022 AMC 10B Problems/Problem 23|Solution]] | ||
==Problem 24 == | ==Problem 24 == | ||
− | + | Consider functions <math>f</math> that satisfy <cmath>|f(x)-f(y)|\leq \frac{1}{2}|x-y|</cmath> for all real numbers <math>x</math> and <math>y</math>. Of all such functions that also satisfy the equation <math>f(300) = f(900)</math>, what is the greatest possible value of | |
− | + | <cmath>f(f(800))-f(f(400))?</cmath> | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 200</math> |
[[2022 AMC 10B Problems/Problem 24|Solution]] | [[2022 AMC 10B Problems/Problem 24|Solution]] | ||
==Problem 25 == | ==Problem 25 == | ||
− | + | Let <math>x_0,x_1,x_2,\dotsc</math> be a sequence of numbers, where each <math>x_k</math> is either <math>0</math> or <math>1</math>. For each positive integer <math>n</math>, define | |
− | + | <cmath>S_n = \sum_{k=0}^{n-1} x_k 2^k</cmath> | |
− | <math>\textbf{(A)} | + | Suppose <math>7S_n \equiv 1 \pmod{2^n}</math> for all <math>n \geq 1</math>. What is the value of the sum |
+ | <cmath>x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?</cmath> | ||
+ | <math>\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) }12\qquad \textbf{(D) } 14\qquad \textbf{(E) }15</math> | ||
[[2022 AMC 10B Problems/Problem 25|Solution]] | [[2022 AMC 10B Problems/Problem 25|Solution]] | ||
==See also== | ==See also== | ||
− | {{AMC10 box|year=2022|ab=B|before=[[ | + | {{AMC10 box|year=2022|ab=B|before=[[2022 AMC 10A Problems]]|after=[[2023 AMC 10A Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:10, 20 February 2024
2022 AMC 10B (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
Define to be
for all real numbers
and
What is the value of
Problem 2
In rhombus , point
lies on segment
so that
,
, and
. What is the area of
? (Note: The figure is not drawn to scale.)
Problem 3
How many three-digit positive integers have an odd number of even digits?
Problem 4
A donkey suffers an attack of hiccups and the first hiccup happens at one afternoon. Suppose that
the donkey hiccups regularly every
seconds. At what time does the donkey’s
th hiccup occur?
Problem 5
What is the value of
Problem 6
How many of the first ten numbers of the sequence are prime numbers?
Problem 7
For how many values of the constant will the polynomial
have two distinct integer roots?
Problem 8
Consider the following sets of
elements each:
How many of these sets contain exactly two multiples of
?
Problem 9
The sum
can be expressed as
, where
and
are positive integers. What is
?
Problem 10
Camila writes down five positive integers. The unique mode of these integers is greater than their median, and the median is
greater than their arithmetic mean. What is the least possible value for the mode?
Problem 11
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?
All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
Problem 12
A pair of fair -sided dice is rolled
times. What is the least value of
such that the probability that the sum of the numbers face up on a roll equals
at least once is greater than
?
Problem 13
The positive difference between a pair of primes is equal to , and the positive difference between the cubes of the two primes is
. What is the sum of the digits of the least prime that is greater than those two primes?
Problem 14
Suppose that is a subset of
such that the sum of any two (not necessarily distinct) elements of
is never an element of
What is the maximum number of elements
may contain?
Problem 15
Let be the sum of the first
terms of an arithmetic sequence that has a common difference of
. The quotient
does not depend on
. What is
?
Problem 16
The diagram below shows a rectangle with side lengths and
and a square with side length
. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
Problem 17
One of the following numbers is not divisible by any prime number less than Which is it?
Problem 18
Consider systems of three linear equations with unknowns ,
, and
,
where each of the coefficients is either
or
and the system has a solution other than
.
For example, one such system is
with a nonzero solution of
. How many such systems of equations are there?
(The equations in a system need not be distinct, and two systems containing the same equations in a
different order are considered different.)
Problem 19
Each square in a grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
- Any filled square with two or three filled neighbors remains filled.
- Any empty square with exactly three filled neighbors becomes a filled square.
- All other squares remain empty or become empty.
A sample transformation is shown in the figure below.
Suppose the
grid has a border of empty squares surrounding a
subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
Problem 20
Let be a rhombus with
. Let
be the midpoint of
, and let
be the point on
such that
is perpendicular to
. What is the degree measure of
?
Problem 21
Let be a polynomial with rational coefficients such that when
is divided by the polynomial
, the remainder is
, and when
is divided by the polynomial
, the remainder is
. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
Problem 22
Let be the set of circles in the coordinate plane that are tangent to each of the three circles with equations
,
, and
. What is the sum of the areas of all circles in
?
Problem 23
Ant Amelia starts on the number line at and crawls in the following manner. For
Amelia chooses a time duration
and an increment
independently and uniformly at random from the interval
During the
th step of the process, Amelia moves
units in the positive direction, using up
minutes. If the total elapsed time has exceeded
minute during the
th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most
steps in all. What is the probability that Amelia’s position when she stops will be greater than
?
Problem 24
Consider functions that satisfy
for all real numbers
and
. Of all such functions that also satisfy the equation
, what is the greatest possible value of
Problem 25
Let be a sequence of numbers, where each
is either
or
. For each positive integer
, define
Suppose
for all
. What is the value of the sum
See also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2022 AMC 10A Problems |
Followed by 2023 AMC 10A Problems | |
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All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.